# embeddings_queue

created_at: 2025-05-16 10:05:13
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p25-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 25, "chunk_number": 1, "type": "text", "chroma:document": "Reflexive Property of Set Inclusion\nFor any set A, A \u2286 A.\nKey Concept\nThe symbol \u2286 is nonstrict, while the symbol \u2282 is strict. Note that if A \u2286 B, then \neither A \u2282 B or A = B.\nAnother subset of any set A is the empty set; that is, \u2205 \u2282 A. This means that \u2205 is \na trivial subset of any set. Is \u2205 \u2286 \u2205? \nRemember that any set has at least one of these types of subsets\u2014improper, \nproper, or trivial subset. Now you will study the subsets that a given set could have."}


created_at: 2025-05-16 10:05:13
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p29-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 29, "chunk_number": 0, "type": "text", "chroma:document": "20\nFor items 29 to 37, consider the set V = {0, 10, {100}}. Write True if the statement is \ntrue; otherwise, write False and modify the relation symbol to make it correct.\n 29. 0 \u2208 V\n 30. \u2205 \u2208 V\n 31. 10 \u2282 V\n 32. {10} \u2282 V\n 33. {100} \u2208 V \n 34. {10} \u2282 {100}\n 35. {0} \u2208 V\n 36. {{100}} = {100}\n 37. {0} = \u2205\nFor items 38 to 41, let S be the set of distinct letters in the word \u201csmiles.\u201d Identify all \nthe subsets of S that satisfy each description.\n 38. subsets that have no element\n 39. subsets that contain only one vowel\n 40. subsets that do not contain a vowel\n 41. subsets that contain exactly three elements\nApplications\nAnswer each item completely.\n 42.  Your school is offering the following PE subjects: basketball, volleyball, table \ntennis, badminton, and baseball. Is the set of PE subjects offered in your school \na subset of the set of sports played in the Summer Olympic Games? Explain.\n 43.  Liza can choose three from the five elective subjects offered during a term."}


created_at: 2025-05-16 10:05:13
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p27-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 27, "chunk_number": 0, "type": "text", "chroma:document": "18\nexamPLe 4\nConsider the following sets:\n O = {o|o is an odd number}\n E = {e|e is an even number}\n C = {0, 1, 2, 3, 4, 5}\nTell whether or not each pair of sets are disjoint. If the two sets are not disjoint, \nidentify their common element(s).\na. O and C b. O and E c. E and C\n Answers:\n a. O and C are not disjoint sets. They both contain 1, 3, and 5. \n b. O and E are disjoint sets.\n c.  E and C are not disjoint sets. Their common elements are 0, 2, and 4. \nPractice 1-2\nConcepts and Skills\nWrite True if the statement is true; otherwise, write False.\n 1. An empty set is a subset of itself.\n 2. Any two equivalent sets are not disjoint sets.\n 3. An empty set has no proper subset.\n 4. If A \u2282 B, then A and B are not disjoint.\n 5. An element of a set can be the same as another set.\n 6. The improper subset of any set is equal to the empty set.\n 7. A subset of B is a subset of C only if B is a subset of C.\n 8. If A and B are both equivalent to C, then A and B are equal."}


created_at: 2025-05-16 10:05:13
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p23-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 23, "chunk_number": 1, "type": "text", "chroma:document": "Q is said to be a subset of P if every element of Q is also an element of P. In symbols, it \nis denoted as Q  P. This is read as \u201cQ is a subset of P\u201d or \u201cQ is contained in P.\u201d\nKey Concept\n From Q  P, you also say that \u201cP is a superset of Q.\u201d It can be written as P  Q. If \nat least one element of Q is not contained in P, then Q is not a subset of P. In symbols, \nQ  P. What can you say about the cardinality of a subset and of its superset?\nSet relations can be visualized using a Venn diagram. \nA Venn diagram is a visual representation of logical relationships among sets. It uses \ncircles to represent sets. Each circle is labeled with the name of the set it represents.\nKey Concept\nThe relation that set A is a subset (or not a subset) of another set B can be \nillustrated in the following Venn diagrams:\nA B\nA  B\nAll elements in set A are contained in set B. \nA B\nA  B\nNot all elements in set A are contained in set B."}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p16-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 16, "chunk_number": 1, "type": "text", "chroma:document": "A set that has no elements is called an empty set or a null set.  It is denoted as \u2205 or { }.\nKey Concept\nSince n(A) = 0, you can say that A is an empty set.\nIf the cardinality or number of elements of a set is a counting number, then it is called a finite set. \nA set is defined as a finite set if the cardinality of the set is a fixed or known counting number n. \nOtherwise, the set is not finite and is called an infinite set. \nKey Concept\nThe elements of a finite set are limited; thus, such elements can be completely \nlisted down. For example, V = {a, e, i, o, u} is a finite set since n(V) = 5. Also, although \nan empty set has no element (or zero element), and 0 is not a counting number, an \nempty set is also considered as a finite set. An example of an infinite set is the set of \ncounting numbers, C = {1, 2, 3, ...}. It is an infinite set because there are infinitely \nmany counting numbers.\nIs the set of the stars in the sky finite or infinite? You may think that since there"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p31-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 31, "chunk_number": 0, "type": "text", "chroma:document": "22\nProbe and Learn\nIn a school sports fest, four teams will participate. Each team \nwill be composed of students in each year level. The grade 7 \nstudents will be the Green Team. The grade 8 students will be \nthe Yellow Team. The grade 9 students will be the Red Team. The \ngrade 10 students will be the Blue Team.\nThe four teams will compete with each other in all the events \nexcept for a special event. In this event, only two teams will \ncompete. The grade 7 and grade 10 students will be grouped to \nform the Blue-Green Team. The grade 8 and grade 9 students will \nbe grouped to form the Red-Yellow Team.\nLike in numbers, there are also operations on sets. You can \nperform a specific operation on two or more sets to come up with \nanother set. Study each set operation.\nSet OperationsLESSON\n1-3\nLearning  \nObjectives\nAt the end of the \nlesson, you should be \nable to:\n\u2022\t identify\toperations\t\non sets\n\u2022\t perform\toperations\t\non sets\n\u2022\t illustrate\tset\t\noperations\tusing\t\nVenn\tdiagrams"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p32-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 32, "chunk_number": 0, "type": "text", "chroma:document": "23\nUnion of Sets\nIn the given situation, each team in the sports fest represents a set. Combining \ntwo teams represents a union of two sets. What can you say about the elements of \nthe new set formed?\nFor any sets A and B, the union of A and B, which is denoted as A \u222a B (read as \u201cA union \nB\u201d), is given by:\nA \u222a B = {x|x \u2208 A or x \u2208 B}\nKey Concept\nThe word \u201cor\u201d means that x belongs to A, or to B, or to both A and B. Hence, to \nfind the union of two sets, combine the elements of A and B by writing the elements \nin any order, without repetition.\nexamPLe 1\nConsider the following sets of athletes:\n B =  {Curry, Durant, James, Irving} \n S =  {Messi, Ronaldo, Suarez, Neymar}\n T =  {Yap, Tenorio, Pingris, Romeo} \nFind the elements of each union of sets.\na. S \u222a T\nb. B \u222a S\nc. B \u222a T\nd. T \u222a B\n Solutions:\n a.  S \u222a T =  {Messi, Ronaldo, Suarez, Neymar, Yap, Tenorio, Pingris, Romeo}\n b.  B \u222a S =  {Curry, Durant, James, Irving, Messi, Ronaldo, Suarez, Neymar}"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p27-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 27, "chunk_number": 1, "type": "text", "chroma:document": "6. The improper subset of any set is equal to the empty set.\n 7. A subset of B is a subset of C only if B is a subset of C.\n 8. If A and B are both equivalent to C, then A and B are equal.\n 9.  If B is a proper subset of A, then B does not necessarily contain each element \nof A.\n 10. If A and B are equal to C, then A and B are not necessarily equal."}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p33-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 33, "chunk_number": 0, "type": "text", "chroma:document": "24\nStudy the next example.\nexamPLe 2\nSuppose that the four teams mentioned at the start of the lesson are described \nas follows:\n G  = {x|x is a grade 7 student who is a member of Green Team}\n Y =  {x|x is a grade 8 student who is a member of Yellow Team}\n R =  {x|x is a grade 9 student who is a member of Red Team}\n B =  {x|x is a grade 10 student who is a member of Blue Team}\nIllustrate the Blue-Green Team using a Venn diagram.\n Solution: \n  Note that set B refers to Blue Team and set G refers to Green Team . Thus, \nthe Blue-Green Team refers to the union of B and G, or B \u222a G. Observe that \nB and G are disjoint sets since they do not have any common element. This is \nbecause no grade 7 student is a grade 10 student, and no grade 10 student \nis a grade 7 student. Thus, the Blue-Green Team or B \u222a G is illustrated as \nfollows:\n \nB G\n B \u222a G\nLook at the Venn diagram in the previous example. The shaded region represents"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p30-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 30, "chunk_number": 0, "type": "text", "chroma:document": "21\nEnrichment Exercises\nIn example 3 of this lesson, you have seen that a set with 3 elements has 8 subsets. \nIs there a relationship between the cardinality of a set and the number of subsets of \nthe set? Investigate by answering each question.\n 51.  How many subsets does an empty set have? Give \nthe subsets.\n 52.  What are the subsets of {0}? How many subsets \ndoes this set have?\n 53.  What are the subsets of {0, 1}? How many subsets \ndoes this set have?\n 54.  Suppose that set A is a finite set with 6 elements. \nHow many subsets does A have? Complete the \ntable on the right. Then find a pattern on the \nnumber of subsets of a certain set based on its \ncardinality.\n 55.  Based on the results you obtained in the previous item, make a generalization \nabout the number of subsets of a finite set A, with n(A) = k. \nUse your generalization in the previous item to find the total number of subsets of \neach given set.\n 56. Y = {x|x is a weekday that starts with a vowel}"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p29-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 29, "chunk_number": 1, "type": "text", "chroma:document": "a subset of the set of sports played in the Summer Olympic Games? Explain.\n 43.  Liza can choose three from the five elective subjects offered during a term. \nThe elective subjects offered are Spanish, Leadership, Music Composition, \nPsychology, and Actuarial Science. What are the possible sets of elective \nsubjects she can enroll in?\n 44.  A science teacher divides his class into 8 groups for a project. He wants his \nstudents to research on the planets in the solar system. Can he assign a \ndifferent planet to each group? What set concept is illustrated in this situation? \nExplain.\n 4 5.   Five friends plan a chess tournament. Each person will play against each \nparticipant exactly once. How many possible games can be played?\nConsider sets A, B, and C. Draw a Venn diagram that illustrates each statement.\n 46. A \u2282 B, but B \u2284 C\n 47. A \u2286 B and B \u2286 A\n 48. A \u2282 B, A \u2282 C, and B \u2282 C\n 49. A \u2282 C and B \u2282 C, but A \u2284 B\n 50. A and B are disjoint and B \u2282 C"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p16-c2-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 16, "chunk_number": 2, "type": "text", "chroma:document": "counting numbers, C = {1, 2, 3, ...}. It is an infinite set because there are infinitely \nmany counting numbers.\nIs the set of the stars in the sky finite or infinite? You may think that since there \nare so many stars, then it is impossible to count all of them. However, it has been \nknown that there are approximately 200,000,000 stars in the sky. Although this set \nis very large, it is considered finite since its cardinality is a whole number."}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p26-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 26, "chunk_number": 1, "type": "text", "chroma:document": "In what way can you show the relations among sets M, N, and O?\nNotice that sets M and N have no common elements. Thus, M and N are disjoint \nsets.\nTwo sets are said to be disjoint sets if they have no common element.\nKey Concept\nThe disjoint sets M and N are illustrated in the following diagram:\nM N\nBased on the given sets, can you say that M and O are disjoint?\nM and O have the following common elements: 1, 3, and 5. Therefore, M and O \nare not disjoint."}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p34-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 34, "chunk_number": 0, "type": "text", "chroma:document": "25\nIntersection of Sets\nSuppose that in the sports fest, the presidents of the Science Club and the Math \nClub agree to unite their members. Since some students might be members of both \nclubs, the number of members in the union might be less than the sum of the \noriginal number of members of the two clubs.\nLet S be the set of Science Club members. Let M be the set of Math Club members. \nSince there is a possibility of overlap between S and M, the sets may not be disjoint. \nIn such case, the union of these sets may be illustrated as follows:\nS M\nS \u222a M\nIn this situation, the students who are members of both the Science Club and the \nMath Club form another set. This set is referred to as the intersection of S and M. The \nintersection of S and M is the set of all elements contained in both S and M.\nFor any sets A and B, the intersection of A and B, which is denoted as A \u2229 B (read as \n\u201cA intersection B\u201d), is given by:\nA \u2229 B = {x|x \u2208 A and x \u2208 B}.\nKey Concept"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p32-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 32, "chunk_number": 1, "type": "text", "chroma:document": "b. B \u222a S\nc. B \u222a T\nd. T \u222a B\n Solutions:\n a.  S \u222a T =  {Messi, Ronaldo, Suarez, Neymar, Yap, Tenorio, Pingris, Romeo}\n b.  B \u222a S =  {Curry, Durant, James, Irving, Messi, Ronaldo, Suarez, Neymar}\n c.  B \u222a T = {Curry, Durant, James, Irving, Yap, Tenorio, Pingris, Romeo}\n d.  T \u222a B = {Yap, Tenorio, Pingris, Romeo, Curry, Durant, James, Irving}\nIn the previous example, note that B \u222a T = T \u222a B. This illustrates that the union \nof sets is commutative; that is, the order of the sets does not affect the union. \nCommutative Property of the Union of Sets\nFor any sets A and B,\nA \u222a B = B \u222a A.\nKey Concept"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p33-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 33, "chunk_number": 1, "type": "text", "chroma:document": "is a grade 7 student. Thus, the Blue-Green Team or B \u222a G is illustrated as \nfollows:\n \nB G\n B \u222a G\nLook at the Venn diagram in the previous example. The shaded region represents \nB \u222a G, which is the combined region representing B and G. This means that the \ntotal number of students in the combined set is equal to the sum of the number of \ngrade 7 students and the number of grade 10 students. This illustrates the following \ngeneralization:\nIf A and B are two disjoint finite sets, then A \u222a B is also finite and\nn(A \u222a B) = n(A) + n(B). \nKey Concept\nVerify this fact using example 1."}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p30-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 30, "chunk_number": 1, "type": "text", "chroma:document": "Use your generalization in the previous item to find the total number of subsets of \neach given set.\n 56. Y = {x|x is a weekday that starts with a vowel}\n 57. M = {x|x is a distinct letter in the word \u201cmathematics\u201d}\n 58. C = {x|x is a distinct consonant in the word \u201cconsonant\u201d}\n 59. D = {x|x is a two-digit number that is divisible by 11}\n 60. J = {x|x is a prime number from 1 to 10}\nCardinality Number of \nSubsets\n0\n1\n2\n3 8\n4\n5\n6"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p36-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 36, "chunk_number": 0, "type": "text", "chroma:document": "27\nConsider again the sets V and L. Suppose that the vowels are removed from \nthe set of the letters in the English alphabet. What letters remain? The set of \nthe remaining letters represent V '. Thus, V ' contains the consonants. Since  \nC = {x|x is a consonant in the English alphabet}, then V' = C. Conversely, C' = V. \nexamPLe 3\nConsider the following sets:\n U = {x|x is a digit in the decimal system}\n S = {x|x is an even whole number less than 10}\n T = {x|x is a prime number less than 10}\nDescribe each set using the roster method.\na. S' b. T' c. U'\n Solutions:\n  First, describe sets U, S, and T using the roster method to perform the operation \nmore easily.\n  U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}\n  S = {0, 2, 4, 6, 8}\n  T = {2, 3, 5, 7}\n a. For S', take away the elements of set S from U.\n  S' =  = {1, 3, 5, 7, 9}\n b. T' =  = {0, 1, 4, 6, 8, 9}\n c. U' =  = \u2205\nNotice that for any set A combined with its complement A', the resulting union"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p35-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 35, "chunk_number": 0, "type": "text", "chroma:document": "26\nCommutative Property of the Intersection of Sets\nFor any sets A and B,\nA \u2229 B = B \u2229 A.\nKey Concept\nComplement of a Set\nAnother operation involving sets is the set complement. This operation is performed \nwith respect to a universal set. \nThe universal set is the set that contains all the elements under consideration. This is \nusually denoted as U. \nKey Concept\nFor instance, consider the set V = {x |x is a vowel in the English alphabet}. \nConsider also set L  = {x |x is a letter in the English alphabet}. You know \nthat set V is contained in set L. Thus, the universal set in this situation is  \nL = {x |x is a letter in the English alphabet}. Similarly, when considering the set  \nC = {x|x is a consonant in the English alphabet}, the universal set is L. \nStudy the following definition of the complement of a set:\nLet A \u2282 U. The complement of A is the set of elements of U that are not contained in A.  \nIt is denoted by A' (read as \u201cA complement\u201d). In symbols,\nA' = {x|x \u2208 U and x \u2209 A}."}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p34-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 34, "chunk_number": 1, "type": "text", "chroma:document": "For any sets A and B, the intersection of A and B, which is denoted as A \u2229 B (read as \n\u201cA intersection B\u201d), is given by:\nA \u2229 B = {x|x \u2208 A and x \u2208 B}.\nKey Concept\nThe word \u201cand\u201d means that the elements belong to both A and B.\nUsing a Venn diagram, A \u2229 B can be illustrated as follows:\nA B\nA \u2229 B\nIn the diagram, the shaded region represents A \u2229 B, which is the set of all elements \nthat are common to both A  and B. Notice that the shaded region also represents  \nB \u2229 A. This illustrates that the intersection of sets is also commutative."}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p37-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 37, "chunk_number": 0, "type": "text", "chroma:document": "28\nSet Difference\nAnother operation that is similar to set complement is set difference . Both \noperations involve \u201ctaking away\u201d elements from a set. To find the complement of a \nset, take away the elements of that set from the universal set. To find the difference, \nyou will take away the elements of one set that are in the other set. The set difference \nis defined as follows:\nThe difference of sets A and B, which is denoted as A \u2013 B, is the set of elements of A that \nare not in B. In symbols, \nA \u2013 B = {x|x \u2208 A and x \u2209 B}.\nKey Concept\nSet difference may be illustrated using Venn diagrams as follows: \nA B A B\nA \u2013 B B \u2013 A\nWhat do you observe in the two diagrams?\nConsider again sets S and T in example 3. \n S = {0, 2, 4, 6, 8}\n T = {2, 3, 5, 7}\nNote that the only element of S that is in T is 2. Thus, to find S \u2013 T, take this \nelement (2) away from S. Hence, S \u2013 T = {0, 2, 4, 6, 8} = {0, 4, 6, 8}.\nWhat elements of S will be removed from T to find T \u2013 S ?\nexamPLe 4"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p38-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 38, "chunk_number": 0, "type": "text", "chroma:document": "29\n Solutions:\n a.  The elements a and e of set J are contained in V. Thus, to find J \u2013 V, take \nthese away from set J. Therefore,\n  J \u2013 V = {a , b, c, d, e } = {b, c, d}.\n b. V \u2013 J = {i, o, u}. Can you explain why?\nBased on the previous example, can you say that set difference is commutative?\nSet Product\nConsider the following situation:\nFive players are planning their fencing matches, in such a way that each \nplayer must compete with another player of the opposite gender. There are  \n2 boys and 3 girls. How many matches will there be?\nDo you know?\nFencing,\n\talso\n\tknown\n\tas\n\ts\nword\n\tfighting\n,\t \nis\n\t a\n\t fantastic\n\t w\nay\n\t to\n\t impro\nve\n\t one\u2019\ns\t\nbalance\n,\n\tcoordina\ntion,\n\tand\n\tfle\nxibility.\n\tIn\t\nf\nencing,\n\ttw\no\n\tcompetitor\ns\n\tface\n\tof\nf\tin\n\ta\t\n\u201cbout.\u201d\n\tF\nencers\n\tuse\n\tblade\n\tw\nork,\n\tf\nootwork,\t\nand\n\t\nstrategy\n\t\nto\n\t\nbeat\n\t\ntheir\n\t\nopponents.\nLet the set of boys be given by B = {b1, b2}. Let the set of girls be represented by"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p41-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 41, "chunk_number": 0, "type": "text", "chroma:document": "32\n 40. (A \u2229 B)'\n  \nB\nU\nA\n 41. A \u222a B\n  \nB\nU\nA\n 42. (A \u222a B)'\n  \nB\nU\nA\n 43. A \u2229 B'\n  \nB\nU\nA\n 44. (B \u2013 A)'\n  \nB\nU\nA\n 45. U \u2013 A'\n  \nB\nU\nA\nApplications\nAnswer each item completely.\n 46.  A c anteen offers three kinds of sandwiches and two flavors of fruit shake. How \nmany possible combinations of a sandwich and a fruit shake are there? \nFor items 47 to 50, consider the following situation:\nIn a circle of 12 friends, 10 are varsity players. Ariel, Bart, Carlo, David, Eric, \nFrank, and George are members of the basketball team. Andrei, Brian, Carlo, Daniel, \nand Eric are members of the volleyball team. \n 47. Who are the members of either the volleyball team or the basketball team?\n 48.  Who are the members of both the volleyball team and the basketball team?\n 49.  Who are the members of the basketball team but are not members of the \nvolleyball team?\n 50.  Who are the members of the volleyball team but are not members of the \nbasketball team?"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p40-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 40, "chunk_number": 0, "type": "text", "chroma:document": "31\nFor items 16 to 25, refer to the following diagram: \nU\n4\n20\n10 0\n9\n3\n5\n8 1\n26\nD E\n7\nPerform the indicated operations.\n 16. D \u2229 E\n 17. D'\n 18. D \u2013 E\n 19. (D \u222a E)'\n 20. (E \u2013 D)'\n 21. D' \u222a E' \n 22. D' \u2229 E'\n 23. D' \u222a \u2205' \n 24. E' \u2229 U\n 25. (D \u222a E)' \u00d7 (D \u2229 E)\nFor items 26 to 35 , consider sets A and U, where A \u2282 U. Perform the indicated \noperations.\n 26. A \u222a A \n 27. A \u2229 A \n 28. A \u222a U\n 29. A \u2229 U\n 30. \u2205 \u2229 A\n 31. \u2205 \u2229 U\n 32. \u2205' \u222a U\n 33. \u2205' \u222a \u2205\n 34. \u2205' \u2229 U'\n 35. U \u2229 A'\nShade the region that represents each given set in each Venn diagram.\n 36. A\n  \nB\nU\nA\n 37. B'\n  \nB\nU\nA\n 38. U \u2013 A\n  \nB\nU\nA\n 39. A \u2229 B\n  \nB\nU\nA"}


created_at: 2025-05-16 10:05:14
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p35-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 35, "chunk_number": 1, "type": "text", "chroma:document": "Let A \u2282 U. The complement of A is the set of elements of U that are not contained in A.  \nIt is denoted by A' (read as \u201cA complement\u201d). In symbols,\nA' = {x|x \u2208 U and x \u2209 A}.\nKey Concept\nTo find the complement of a set, look for the elements in the universal set that \nare not in the given set. For instance, to find A', remove all the elements of A from U. \nThe elements that remain in U are the elements of A'. \nThe shaded region in the following diagram represents A'. In a Venn diagram, the \nuniversal set is represented by a rectangle. Inside the rectangle are the circles that \nrepresent the sets involving the universal set.\nA\nA'\nU"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p39-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 39, "chunk_number": 0, "type": "text", "chroma:document": "30\nIn the given situation,\nB \u00d7 G = {(b1, g1), (b1, g2), (b1, g3), (b2, g1), (b2, g2), (b2, g3)}.\nTherefore, the five players will play a total of 6 matches.\nNote that the order of the elements in an ordered pair is important. The ordered \npairs (a, b) and (c, d) are the same if and only if a = c and b = d. For example, the \nordered pair (1, 2) is different from (2, 1). Does this mean that A \u00d7 B is not the same \nas B \u00d7 A? Study the next example.\nexamPLe 5\nLet C = {a, b, c } and D = {1, 2, 3}. Define each set.\na. C \u00d7 D b. D \u00d7 C\n Solutions:\n a. C \u00d7 D = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}\n b. D \u00d7 C = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), (3, c)}\nBased on the previous example, can you say that set product is commutative? \nExplain.\nPractice 1-3\nConcepts and Skills\nDetermine whether each statement is always, sometimes, or never true.\n 1. \u2205' \u2286 U \n 2. A \u2013 B = B\n 3. A \u222a B = A \n 4. A' \u2229 A = \u2205\n 5. A \u2286 (A \u222a B) \n 6. A \u2286 (A \u2229 B)"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p36-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 36, "chunk_number": 1, "type": "text", "chroma:document": "S' =  = {1, 3, 5, 7, 9}\n b. T' =  = {0, 1, 4, 6, 8, 9}\n c. U' =  = \u2205\nNotice that for any set A combined with its complement A', the resulting union \nis the universal set. The following properties hold true for the complement of a set: \nFor any set A and a universal set U:\nA \u222a A' = U \nA \u2229 A' = \u2205\nKey Concept\nVerify these properties using Venn diagrams."}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p42-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 42, "chunk_number": 0, "type": "text", "chroma:document": "33\nEnrichment Exercises\nShow your answer to each item using a Venn diagram.\n 51. If A \u2282 B, find A \u2229 B.\n 52. If A \u2282 B, find A \u222a B.\n 53. If A \u2282 B and B \u2282 C, find A \u2229 C.\n 54. If A and B are disjoint sets and A \u2282 C, find A \u2229 C.\n 55. If A and B are not disjoint sets and C \u2282 (A \u2229 B), find C \u2229 (A \u2229 B).\nAnswer each item.\n 56. Is (A \u2229 B) \u2229 C = A \u2229 (B \u2229 C)? Give an example to support your answer.\n 57. If A and B are disjoint sets, is A \u2013 B = A? Explain.\n 58. If A and B are disjoint sets, is A \u2229 B = \u2205? Explain.\n 59.  If A and B are finite sets that are not disjoint, is n(A \u222a B) = n(A) + n(B)? Explain.\n 60. How does set difference differ from set complement?"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p43-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 43, "chunk_number": 0, "type": "text", "chroma:document": "34\nProblem Solving Involving Sets\nLearning  \nObjectives\nAt the end of the \nlesson, you should be \nable to:\n\u2022\t illustrate\tproblems\t\ninvolving\tsets\tusing\t\nVenn\tdiagrams\n\u2022\t solve\tproblems\t\ninvolving\tsets\nProbe and Learn\nA teacher conducts a survey among 50 students. She asked \nthem which of the two sports, basketball or volleyball, they \nwould like to play. Each student can choose one or two sports, \nor none of the two sports if they do not like to play either sport. \nThe teacher finds out that 8 students like to play both basketball \nand volleyball, 24 prefer volleyball, and 32 like basketball. How \nmany students do not like basketball or volleyball?\nMany problems involving sets can be solved using Venn \ndiagrams. Recall that a Venn diagram can help you visualize \nthe logical relationships among sets. It is made by drawing a \nrectangle to represent the universal set. Inside the rectangle \nare two or more circles, where each circle represents a specific"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p38-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 38, "chunk_number": 1, "type": "text", "chroma:document": "of\nf\tin\n\ta\t\n\u201cbout.\u201d\n\tF\nencers\n\tuse\n\tblade\n\tw\nork,\n\tf\nootwork,\t\nand\n\t\nstrategy\n\t\nto\n\t\nbeat\n\t\ntheir\n\t\nopponents.\nLet the set of boys be given by B = {b1, b2}. Let the set of girls be represented by  \nG = {g1, g2, g3}. To find the possible matches, pair each boy with each girl. This can \nbe illustrated in a diagram as follows:\n g1\nb1 g2\n g3\n       and       \n g1\nb2 g2\n g3\nBased on the diagram, the possible pairs are (b1, g1), (b1, g2), (b1, g3), (b2, g1), (b2, g2), \nand (b2, g3).\nSuch pairing of the elements between two sets is called set product. \nThe set product or Cartesian product of two sets A and B is the set of all possible \nordered pairs (a, b), where a \u2208 A and b \u2208 B. In symbols, the set product A \u00d7 B (read as \n\u201cA cross B\u201d) is given by:\nA \u00d7 B = {(a, b)|a \u2208 A and b \u2208 B}.\nKey Concept"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p37-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 37, "chunk_number": 1, "type": "text", "chroma:document": "element (2) away from S. Hence, S \u2013 T = {0, 2, 4, 6, 8} = {0, 4, 6, 8}.\nWhat elements of S will be removed from T to find T \u2013 S ?\nexamPLe 4\nLet J = {a, b, c, d, e} and V = {a, e, i, o, u}.\nDefine each set.\na. J \u2013 V\nb. V \u2013 J"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p44-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 44, "chunk_number": 0, "type": "text", "chroma:document": "35\nwherein each region represents a set of elements that share common properties. It \nis important that you know how to illustrate problems using Venn diagrams. Study \nthe following illustrations:\nA Venn diagram involving two sets A and B is shown.\n\u0391\n\u0391 only A \u2229 B B only\nB\nU\n(A \u222a B)'\nThe region where the two circles overlap represents  A \u2229 B, as labeled in the \ndiagram. It contains the elements that belong to both A and B. The region labeled  \n\u201cA only\u201d represents those elements that belong to A but not to B. Similarly, the region \nlabeled \u201cB only\u201d represents those elements that belong to B but not to A. On the \nother hand, the entire region covered by both A and B, which is the colored region, \nrepresents A \u222a B. The uncolored region inside the rectangle represents (A \u222a B)'.\nNow study the logical relations among three sets A, B, and C, as shown here.\n\u0391 only\nC only\nB only\nU\n(A \u2229 B \u2229 C)\n\u0391 B\nC\n(A \u2229 C) \u2212 B\n(A \u222a B \u222a C)'\n(A \u2229 B) \u2212 C\n(B \u2229 C) \u2212 A"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p39-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 39, "chunk_number": 1, "type": "text", "chroma:document": "Practice 1-3\nConcepts and Skills\nDetermine whether each statement is always, sometimes, or never true.\n 1. \u2205' \u2286 U \n 2. A \u2013 B = B\n 3. A \u222a B = A \n 4. A' \u2229 A = \u2205\n 5. A \u2286 (A \u222a B) \n 6. A \u2286 (A \u2229 B)\n 7. (A \u2229 B) \u2286 A \n 8. n(A) + n(A') = n(U)\n 9. n(A \u2013 B) = n(A) \u2013 n(B) \n 10. n(A) + n(B) = n(A \u222a B)\n 11. If A \u2286 U, then A' \u2286 U. \n 12. If A \u2260 \u2205, then \u2205 \u222a A = \u2205.\n 13. If A \u2286 U, then A' = U \u2013 A. \n 14. If A = B, then A \u222a B = A \u2229 B.\n 15. If A ~ B, then A and B are disjoint."}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p46-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 46, "chunk_number": 0, "type": "text", "chroma:document": "37\n \nV\n8\n24 32\n50\nB\nU\n24 \u2212 8\n   = 16\n32 \u2212 8\n  = 24\n  Adding the number of students in the shaded regions gives the number \nof students who like either volleyball or basketball. Hence, there are  \n16 + 8 + 24 = 48 students who like either volleyball or basketball. This number \ncorresponds to n(V \u222a B). So n(V \u222a B)' represents the number of students who \ndo not like volleyball or basketball. Since the universal set has a total of 50 \nstudents, then:\n n(V \u222a B)' = 50 \u2013 48 = 2.\n  Therefore, 2 of the students surveyed do not like any of the two sports. The \ncomplete Venn diagram is shown. \nV\n8\n24 32\n50\nB\nU\n24 \u2212 8\n    = 16\n32 \u2212 8\n  = 24\n50 \u2013 48 = 2\n16 + 8 + 24 = 48\nn(V \u2229 B) =  number of students \nwho like \nboth volleyball \nand basketball\nn(V \u222a B) = \n number of students who like \neither\n basketball or volleyball\nn(V) = \n number of \nstudents who \nlike \nvolleyball\nn(V \u2013 B) = \n number of \nstudents who like \nvolleyball \nonly\nn(U) =  number of students \nsurveyed\nn\n(B) = \n number of \nstudents who"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p45-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 45, "chunk_number": 0, "type": "text", "chroma:document": "36\nexamPLe 1\nOut of the 50 students surveyed by the teacher, how many students do not like \nbasketball or volleyball?\n Solution:\n  If U is the universal set, you can represent the sets in the problem as follows:\n  U = {x|x is a student who was surveyed}\n  V = {v|v is a student who likes volleyball}\n  B = {b|b is a student who likes basketball}\n  To construct the Venn diagram for this problem, draw two circles (representing  \nsets V and B) inside a rectangle (representing set U). Note that some students \nare elements of both V and B. This implies that V and B intersect. Thus,  \nV and B are not disjoint. So you will have the following diagram:\n \nV B\nU\n  In the following diagram, n(U) = 50, n(V) = 24, n(B) = 32, and n(V \u2229 B) = 8\n \nV24 32\n50\nB\nU\n8\n  The next step is to determine the number of students who like volleyball only \nand those who like basketball only. Note that the number of students who like \nvolleyball only refers to n(V \u2212 B). The number of students who like basketball"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p48-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 48, "chunk_number": 0, "type": "text", "chroma:document": "39\n  The universal set U is the same set of students surveyed in example 1. \nThe following Venn diagram illustrates the relationships among sets B, L, \nand T. You can see that the three sets are represented by overlapping circles. \nCan you explain why? \n \nU\nL B\nT\n Based on the given information, you can describe each region as follows:\nstudents who like \nboth badminton \nand lawn tennistotal number of \nstudents surveyed\nstudents who like lawn \ntennis\nstudents who like both lawn \ntennis and table tennis\nstudents who do not like any \nof the three racket sports\nstudents who like all three \nracket sports\nstudents who like \nbadminton \nstudents who like \nboth badminton \nand table tennis\nstudents who like \ntable tennis\nU\nn\nB L\nT\n30 22\n50\n11\n15 10\n25\n6\n  The diagram indicates the number of elements in each labeled set. Consider \nfirst the cardinality of the intersection of any two sets. In each of the boxed \nnumbers in the diagram, the 6 students who like the three racket sports are"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p50-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 50, "chunk_number": 0, "type": "text", "chroma:document": "41\nPractice 1-4\nConcepts and Skills\nFor items 1 to 8, refer to the following situation:\nJay and Ray are both music enthusiasts. The following Venn diagram shows their \ninterests, where J is the set of interests of Jay and R is the set of interests of Ray:\nJ R\nplaying the  \nguitar\nplaying the  \npianosinging\nBased on the diagram, write True if the statement is true; otherwise, write False.\n 1. Jay likes singing.\n 2. Ray does not like singing.\n 3. Both Ray and Jay like singing.\n 4. Jay likes singing while Ray likes playing the piano.\n 5. Ray and Jay have exactly the same interests.\n 6. Ray and Jay like singing, playing the guitar, or playing the piano.\n 7. Both Jay and Ray like singing, while Ray likes playing the guitar.\n 8.  Ray likes both singing and playing the guitar but he does not like playing the \npiano.\nFor items 9 to 25, refer to the following text and diagrams:\nA guidance counselor conducts a survey among her \nstudents about their membership in three clubs\u2014"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p49-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 49, "chunk_number": 0, "type": "text", "chroma:document": "40\n \nU\nB L\nT\n30 22\n50\n11\n15 10\n25\n6\n5\n9 4\n15 \u2212 6 = 9 10 \u2212 6 = 4\n11 \u2212 6 = 5\n  Next solve for the numbers in the unfilled regions in each circle. It is given \nthat the total number of elements in B is 30. So far, there are already  \n5 + 6 + 9 = 20 elements in B that have been accounted for. This implies that \nthere are 10 elements in the unified region of B only. Do the same process for \nsets L and T.\n  Lastly consider the universal set that contains 50 students. To get the total \nnumber of students already indicated in the colored regions, you will have  \n10 + 5 + 6 + 9 + 7 + 4 + 6 = 47 students. Therefore, there are 3 students  \n(that is, 50 \u2013 47 = 3) who do not belong to sets B, L, and T.\n The following is the complete Venn diagram for the problem:\n \nU\nB L\nT\n30 22\n50\n25\n6\n5\n6\n10 7\n9 4\n22 \u2212 (5 + 6 + 4) = 730 \u2212 (5 + 6 + 9) = 10\n25 \u2212 (9 + 6 + 4) = 6\n50 \u2212 (10 + 5 + 6 + 9 + 7 + 4 + 6) = 3\n3\nHence, 3 students do not like to play any of the three racket sports."}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p43-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 43, "chunk_number": 1, "type": "text", "chroma:document": "the logical relationships among sets. It is made by drawing a \nrectangle to represent the universal set. Inside the rectangle \nare two or more circles, where each circle represents a specific \nset. The circles divide the universal set into several regions, \nLESSON\n1-4"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p44-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 44, "chunk_number": 1, "type": "text", "chroma:document": "Now study the logical relations among three sets A, B, and C, as shown here.\n\u0391 only\nC only\nB only\nU\n(A \u2229 B \u2229 C)\n\u0391 B\nC\n(A \u2229 C) \u2212 B\n(A \u222a B \u222a C)'\n(A \u2229 B) \u2212 C\n(B \u2229 C) \u2212 A\nAs you can see in the diagram, each region is specified. The entire colored region \nshows A \u222a B \u222a C. \nA Venn diagram could clearly represent the logical conditions specified in a \ncertain problem. That is why it is a useful tool for solving problems involving sets. \nYou can now solve the problem posed at the start of the lesson. Study the solution \nin the next example."}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p47-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 47, "chunk_number": 0, "type": "text", "chroma:document": "38\nConsider the 8 students who prefer both sports. These 8 students are included \nin both the number of students who like basketball and the number of students \nwho like volleyball; thus, they are counted twice. Hence, you have to subtract 8 from \nthe sum (56) to get a difference of 48. Consequently, there are 50 \u2013 2 = 48 students \nwho like either sport. Note that you obtained the same answer (2) as in the previous \nsolution. In symbols, you may solve for the number of students who like either sport \nas follows:\n24 + 32 \u2013 8 = 48\n\u2193 \u2193 \u2193 \u2193\nn(V) + n(B) \u2013 n(V \u2229 B) = n(V \u222a B)\nThis equation illustrates the counting formula for sets, which can be stated as \nfollows:\nIf A and B are finite sets, then: \nn(A \u222a B) = n(A) + n(B) \u2013 n(A \u2229 B).\nAn equivalent equation to this is:\nn(A \u2229 B) = n(A) + n(B) \u2013 n(A \u222a B).\nKey Concept\nexamPLe 2\nA teacher asks the same set of students in example 1 which racket sport \n(badminton, lawn tennis, table tennis) they would like to play. Each student"}


created_at: 2025-05-16 10:05:15
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p45-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 45, "chunk_number": 1, "type": "text", "chroma:document": "and those who like basketball only. Note that the number of students who like \nvolleyball only refers to n(V \u2212 B). The number of students who like basketball \nonly refers to n(B \u2212 V).\n Out of 24 elements in V, 8 elements belong to B. Thus,\n n(V \u2013 B) = 24 \u2013 8 = 16.\n  This means that 16 students like volleyball only. Similarly, subtract 8 from 32 \nto find n(B \u2013 V). So the number of students who prefer basketball only is:\n n(B \u2013 V) = 32 \u2013 8 = 24."}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p51-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 51, "chunk_number": 0, "type": "text", "chroma:document": "42\n 9.   How many students are members of all the three clubs? \n 10. How many students are not members of any of the three clubs? \n 11. How many students are members of the Science Club? \n 12. How many students are members of the Math Club? \n 13. How many students are members of the Theater Club? \n 14.  How many students are members of both the Math Club and the Science Club? \n 15.  How many students are members of both the Math Club and the Theater Club? \n 16.  How many students are members of both the Science Club and the Theater Club?\n 17.  How many students are members of the Math Club but not of the Theater Club \nand the Science Club? \n 18.  How many students are members of the Science Club but not of the Theater \nClub and the Math Club?\n 19.  How many students are members of the Theater Club but not of the Math and \nthe Science Club? \n 20.  How many students are members of both the Math Club and the Theater Club \nbut not of the Science Club?"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p49-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 49, "chunk_number": 1, "type": "text", "chroma:document": "30 22\n50\n25\n6\n5\n6\n10 7\n9 4\n22 \u2212 (5 + 6 + 4) = 730 \u2212 (5 + 6 + 9) = 10\n25 \u2212 (9 + 6 + 4) = 6\n50 \u2212 (10 + 5 + 6 + 9 + 7 + 4 + 6) = 3\n3\nHence, 3 students do not like to play any of the three racket sports.\nWhat other conclusions can you make from the diagram in example 2?"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p50-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 50, "chunk_number": 1, "type": "text", "chroma:document": "piano.\nFor items 9 to 25, refer to the following text and diagrams:\nA guidance counselor conducts a survey among her \nstudents about their membership in three clubs\u2014\nScience Club, Math Club, and Theater Club. The \nresults of the survey are shown in the Venn diagram \non the right, where: \n  S = {s|s is a Science Club member};\n  M = {m|m is a Math Club member}; and\n  T = {t|t is a Theater Club member}.\nU\nS M\nT\n3\n8\n23\n8\n27 12\n12 7"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p47-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 47, "chunk_number": 1, "type": "text", "chroma:document": "Key Concept\nexamPLe 2\nA teacher asks the same set of students in example 1 which racket sport \n(badminton, lawn tennis, table tennis) they would like to play. Each student \nchooses at least one of the three racket sports, or none if they do not like to play \nany of the given sports. The results are as follows:\n \u2022  6 students like the three racket sports.  \n \u2022  30 students like badminton.  \n \u2022  22 students like lawn tennis.  \n \u2022  25 students like table tennis.  \n \u2022  10 students like both table tennis and lawn tennis.\n \u2022  15 students like both table tennis and badminton.\n \u2022  11 students like both lawn tennis and badminton.\nHow many students do not like any of the three racket sports? \n Solution:\n  Three different sets of students can be formed. So you will have the following \nsets:\n  B = {b|b is a student who likes badminton}\n  L = {w|w is a student who likes lawn tennis}\n  T = {t|t is a student who likes table tennis}"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p52-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 52, "chunk_number": 0, "type": "text", "chroma:document": "43\n 27. How many fishes are both hardy and resistant to diseases?\n 28. How many fishes are not hardy? \n 29. How many fishes are not resistant to diseases? \n 30. How many fishes are hardy but are not resistant to diseases?\nFor items 31 to 35, refer to the following situation:\nIn a class of 120 students, 45 students are members of dance clubs and 78 \nstudents are members of sports teams. On the other hand, 32 students are not \nmembers of any of the mentioned groups.\n 31. Draw a Venn diagram to illustrate the given information.\n  \n 32. How many students are members of both dance clubs and sports teams? \n 33. How many students are not members of sports teams?\n 34.  How many students are members of dance clubs but are not members of sports \nteams? \n 35.  How many students are either members of dance clubs or members of sports \nteams? \nFor items 36 to 40, refer to the following situation:\nThe science club adviser asks some student members what science courses they"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p48-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 48, "chunk_number": 1, "type": "text", "chroma:document": "first the cardinality of the intersection of any two sets. In each of the boxed \nnumbers in the diagram, the 6 students who like the three racket sports are \nalready counted. Thus, subtract 6 from the boxed number in each of the three \nintersections."}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p46-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 46, "chunk_number": 1, "type": "text", "chroma:document": "n(V) = \n number of \nstudents who \nlike \nvolleyball\nn(V \u2013 B) = \n number of \nstudents who like \nvolleyball \nonly\nn(U) =  number of students \nsurveyed\nn\n(B) = \n number of \nstudents who \nlike \nbasketball\nn(B \u2013 V) = \n number of students \nwho like basketball \nonly\nn\n(V \u222a B)' =  n umber of \nstudents who like \nneither basketball \nnor volleyball\nAs shown in the previous example, a Venn diagram organizes data and shows \nthe relationships between sets. Other information can also be extracted from the \ndiagram, like the number of students who like only one sport.\nIf you will add the 24 students who prefer volleyball and the 32 students who like \nbasketball, you will get 56 students. Note that this sum is greater than the number \nof students surveyed, which does not satisfy the given information in the problem. \nWhat is wrong with this solution?"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p51-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 51, "chunk_number": 1, "type": "text", "chroma:document": "the Science Club? \n 20.  How many students are members of both the Math Club and the Theater Club \nbut not of the Science Club? \n 21 .   H ow many students are members of both the Science Club and the Theater \nClub but not of the Math Club?\n 22.  How many students are members of both the Science Club and the Math Club \nbut not of the Theater Club? \n 23. How many students did the guidance counselor survey? \n 24. How many students are members of exactly one club? \n 2 5.  How many students are members of exactly two clubs?\nApplications\nAnswer each item completely.\nFor items 26 to 30, refer to the following situation:\nIn an aquarium, 20 different freshwater fishes are either hardy or resistant to \ndiseases. Twelve are hardy and 10 are resistant to diseases.\n 26. Draw a Venn diagram to illustrate the given information."}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p54-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 54, "chunk_number": 0, "type": "text", "chroma:document": "45\nFor items 48 and 49, refer to the following situation:\nJose asked his classmates which telecommunication company they subscribed to. \nHe found out that 14 of his classmates subscribed to company A and 18 subscribed \nto company B. He also found out that a total of 25 students subscribed to either \ncompany A or company B. \n 48. Can you tell how many students Jose asked? Explain. \n 49. What other information can you infer from the given data?\n 50.  Let A, B, and C be disjoint finite sets. Is n(A \u222a B  \u222a C) = n(A) + n(B) + n(C)? Give an \nexample to support your answer."}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p55-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 55, "chunk_number": 0, "type": "text", "chroma:document": "46\nWrite True if the statement is true; otherwise, write False.\n 1.  If S = {0, 1}, then {0} \u2208 S.\n 2. An empty set has no cardinality.\n 3.  Any subset of a finite set is finite.\n 4.  Any set is a proper subset of itself.\n 5.  The set of points on a line is an infinite set.\n 6.  The union of two sets is a set that contains their common elements.\n 7.  The complement of a set is a subset of the universal set.\n 8.  Two sets are equal if they have the same number of elements.\n 9.  Two sets A and B are disjoint if no element of A is an element of B.\n 10. The elements of a set product are the products of the elements of the sets.\nChoose the set relation in column B that corresponds to each pair of sets in  \ncolumn A. \n Column A\n 11. A = {g, o, d} and B = {l, o, v, e}\n 12. A = {g, o, d} and B = {f, a, i, t, h}\n 13. A = {m, a, t, h} and B = {l, o, v, e}\n 14.  A = {m, a, t, h} and \nB = {b|b is a distinct letter in \u201cmahatma\u201d}\n 15.  A = {m, a, t, h} and"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p57-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 57, "chunk_number": 0, "type": "text", "chroma:document": "48\nAnswer each question.\n 21. Which is not a set?\n  A.  {m, a, t, h, e, i, c, s}\n  B. {x|x is a very large counting number}\n  C. {x|x is a month with 32 days}\n  D.  {x|x is a bacterium in the human body}\n 22.  What method could be more appropriate to describe the set whose elements \nare Monday, April, 1, fish, apple, and love?\n  A.  rule method\n  B.  roster method\n  C.  either A or B\n  D.  neither A nor B\n 23. Which set is infinite?\n  A. all the points on a line\n  B.  all the people on Earth\n  C.  all the stars in the Milky Way Galaxy\n  D.  all the trees in the Black Forest\n 24.  Let G = {g, o, d} and H = {x|x is a distinct letter in the word \u201cgood\u201d}. Which \nrelation is not true?\n  A. G \u2286 H\n  B. G ~ H\n  C. G = H\n  D. G' = H\n 25. Let W = {0, 1, 2, 3}. Which statement is true?\n  A. { } \u2208 W\n  B.  0 \u2282 W\n  C. {0} \u2282 W\n  D.  {0} \u2208 W\n 26. How many subsets does an empty set have?\n  A. 0\n  B.  1\n  C.  2\n  D.  3\n 27. Let D = {a, b, c, d}. How many subsets of D contain exactly two elements?"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p58-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 58, "chunk_number": 0, "type": "text", "chroma:document": "49\nFor items 31 to 35, refer to the following situation:\nThe first term of a sequence is 2. Its second term is 4. The succeeding terms \ncan be obtained by getting the average of all the preceding terms. Recall that \nthe average is equal to the sum of the terms of a set divided by the number of \nterms. For example, the third term is the average of 2 (the first term) and  \n4 (the second term). It is given by:\nsum of the terms\nnumber of terms  = \n2 + 4\n2  = \n6\n2  = 3\n 31.  Let S be the set of all distinct terms in the sequence. Describe the set using the \nroster method.\n 32. Find the cardinality of S.\n 33. Classify S as a finite set or an infinite set.\n 34. Find the number of subsets of S.\n 35. Let T be the set of the first ten counting numbers. Is S \u2282 T ? Explain.\nFor items 36 to 40, refer to the following diagram:\nX\n2\n5\n3\n7\nY\nU\n1\n9\nLet U = {x|x is a counting number less than 10}.\nPerform the indicated operations.\n 36. X \u2229 Y \n 37. X \u2013 Y\n 38. X'\n 39. Y' \n 40. (X \u222a Y )'"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p52-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 52, "chunk_number": 1, "type": "text", "chroma:document": "teams? \nFor items 36 to 40, refer to the following situation:\nThe science club adviser asks some student members what science courses they \nlike. Twenty-eight members say they like physics, 35 say they like chemistry, and a 34 \nsay they like biology. Also, 10 members say they like physics and chemistry, 12 say \nthey like biology and chemistry, 9 say they like physics and biology, and 4 say they like \nall three. On the other hand, 10 say they do not like any of the three science courses.\n 36. Draw a Venn diagram to illustrate the given information."}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p56-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 56, "chunk_number": 0, "type": "text", "chroma:document": "47\nChoose the diagram in column B that illustrates each set in column A. \n  Column A            Column B\n 16. M' A.\n \nN\nO\nU\nM\n 17. M \u2013 N B.\n \nN\nO\nU\nM\n 18. (N \u222a O)' C.\n \nO\nU\nM\nN\n 19. (O \u2229 M)' D.\n \nO\nU\nM\nN\n 20. M \u222a N \u222a O E.\n \nO\nU\nM\nN"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p53-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 53, "chunk_number": 0, "type": "text", "chroma:document": "44\n 37. How many students did the science club adviser ask? \n 38. How many students like both physics and chemistry but not biology? \n 39. How many students like any of the three courses? \n 40. How many students do not like physics? \nFor items 41 to 45, refer to the following situation:\nA class adviser is arranging the schedules of foreign language classes for  \n70 students. Thirty students prefer to study French, 30 prefer Spanish, and 30 \nprefer Nihongo. Twelve students prefer to study both French and Nihongo, and of \nthese students, 5 prefer to study Spanish as well. Sixteen students prefer to study \nNihongo only, and 20 prefer Spanish only.\n 41. Draw a Venn diagram to illustrate the given information.\n  \n 42. How many students do not prefer to study Spanish, French, or Nihongo?\n 43. How many students prefer to study Spanish and Nihongo but not French? \n 44. How many students prefer to study French and Spanish but not Nihongo?\n 45. How many students prefer to study French only?"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p59-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 59, "chunk_number": 0, "type": "text", "chroma:document": "50\n 41. Draw a Venn diagram to illustrate the given information.\n  \n 42. H ow many students will sing, act, or dance?\n 43. How many students will only act?\n 4 4.  How many students will only sing?\n 4 5.  How many students will only dance?\n 46. How many students w ill act and dance but will not sing?\n 47. H ow many students will sing and act but will not dance?\n 48. H ow many students will sing and dance but will not act?\n 49. How many students will sing or act but will not dance?\n 50. How many students will make the props for the play?\nAnswer each item completely.\n 51.  Let C = {x|x is a counting number divisible by 2 or by 3}. Let U be the set of \ncounting numbers. Does C' contain a multiple of 9? Explain.\n 52.  Let A = {1, 3, 5, 7, 9, 11} and B = {1, 2, 3, 4, 5, 6}. Which elements of A should \nbe removed so that A \u2229 B = \u2205?\n 53.  In a school of 1,400 students, 2\n7  are grade 7 students. Two-fifths of the grade"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p60-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 60, "chunk_number": 0, "type": "text", "chroma:document": "51\nThe Real Number System\nCHAPTER\n2\nLook closely at the head of the sunflower. You would notice two spirals winding in \ntwo directions. The number of spirals in each direction is not the same\u2014some sunflowers \nhave 21 and 34. Others have 34 and 55, 55 and 89, or 89 and 144. These numbers of \nspirals belong to a special sequence of numbers, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ..., \ncalled the Fibonacci sequence. In this sequence, the next number is found by adding the \ntwo numbers preceding it. Fibonacci numbers can also be observed in plants, petals, tree \nbranches, and fruits. On a pineapple, for instance, there are 8 scales in a diagonal in one \ndirection and 13 in another. If you count the number of petals in a flower, you will most \nlikely get a Fibonacci number.  \nMathematics is the language of nature. In this chapter, you will learn about the different \ntypes of numbers and their properties, as you discover the beauty of math and nature.\nLesson 2-1. Real Numbers"}


created_at: 2025-05-16 10:05:16
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p57-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 57, "chunk_number": 1, "type": "text", "chroma:document": "B.  0 \u2282 W\n  C. {0} \u2282 W\n  D.  {0} \u2208 W\n 26. How many subsets does an empty set have?\n  A. 0\n  B.  1\n  C.  2\n  D.  3\n 27. Let D = {a, b, c, d}. How many subsets of D contain exactly two elements?\n  A.  2\n  B.  4\n  C. 6\n  D.  8\n 28.  Which set operation is performed when combining all the elements of two or \nmore sets?\n  A.  intersection of sets\n  B. set difference\n  C. set product\n  D.  union of sets\n 29.  Let U = {x|x is a rainbow color} and S = {blue, green, yellow, red}. Which set is \nthe complement of S?\n  A. {orange, pink, red}\n  B.  {orange, pink, violet}\n  C. {orange, indigo, violet}\n  D. {red, blue, yellow, green}\n 30.  Suppose that S = {s, m, i, l, e} and J = {j, o, v, i, a, l}. What is the cardinality of  \nS \u00d7 J?\n  A.  8\n  B.  10\n  C.  25\n  D.  30"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p58-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 58, "chunk_number": 1, "type": "text", "chroma:document": "X\n2\n5\n3\n7\nY\nU\n1\n9\nLet U = {x|x is a counting number less than 10}.\nPerform the indicated operations.\n 36. X \u2229 Y \n 37. X \u2013 Y\n 38. X'\n 39. Y' \n 40. (X \u222a Y )'\nFor items 41 to 50, refer to the following situation:\nA teacher organizes a musical play. A total of 50 students will participate \nin the play. Suppose that 24 of the students will sing, 25 will act, and 23 \nwill dance. Also, 16 of the students will sing and dance, 12 will act and dance,  \n14 will sing and act, and 6 will sing, act, and dance. Those students who will not  \nperform will create the props for the play."}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p61-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 61, "chunk_number": 0, "type": "text", "chroma:document": "52\nProbe and Learn\nA unique property of Fibonacci numbers is seen when dividing \nany two successive numbers. Look at the quotient of each pair of \nsuccessive numbers ( A and B) in the third column  of the \nfollowing table:\nA B B\nA\n2 3 1.5\n3 5 1.666666666...\n5 8 1.6\n8 13 1.625\nM M M\n144 233 1.618055556...\n233 377 1.618025751...\nM M M\nNote that \nB\nA gets closer and closer to a special number called \nthe golden ratio, which has a value of 1.6180339887.... The \ngolden ratio is denoted by the Greek letter phi \u03c6. Do you know \nthat the golden ratio can also be observed on the characteristics \nof many things around you? Different parts of the human body \nalso show the golden ratio. For instance, the ratio of the length \nof the forearm to the length of the hand is equal to the value of \u03c6.\nReal Numbers\nLearning \nObjectives\nAt the end of the \nlesson, you should be \nable to:\n\u2022\t apply\tthe\t\nproperties\tof\t real \t\nnumbers\n\u2022\t identify\tthe\tsubsets\t\nof\t real \t numbers\n\u2022\t classify\treal\t\nnumbers\nLESSON\n2-1"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p53-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 53, "chunk_number": 1, "type": "text", "chroma:document": "44. How many students prefer to study French and Spanish but not Nihongo?\n 45. How many students prefer to study French only? \nEnrichment Exercises\nAnswer each item completely.\nFor items 46 and 47, refer to the following situation:\nSuppose that 25 basketball players, 30 volleyball players, and 28 soccer players \ncompose a varsity club. \n 46.  If no member plays at least two sports, how many members are there in the \nvarsity club?\n 47.  Suppose that 5 basketball players are also volleyball players. No soccer player \nis also a volleyball or a basketball player. How many members are there in the \nclub?"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p55-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 55, "chunk_number": 1, "type": "text", "chroma:document": "12. A = {g, o, d} and B = {f, a, i, t, h}\n 13. A = {m, a, t, h} and B = {l, o, v, e}\n 14.  A = {m, a, t, h} and \nB = {b|b is a distinct letter in \u201cmahatma\u201d}\n 15.  A = {m, a, t, h} and \nB = {x|x is a distinct letter in \u201cmathematics\u201d}\nColumn B\n A. A ~ B\n B. A = B\n C. A \u2282 B\n D. A \u2229 B = {o}\n E. A \u2229 B = \u2205\nChapter Assessment"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p59-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 59, "chunk_number": 1, "type": "text", "chroma:document": "be removed so that A \u2229 B = \u2205?\n 53.  In a school of 1,400 students, 2\n7  are grade 7 students. Two-fifths of the grade \n7 students are girls. Also, 75 of the girls are into swimming and 87 are into \ngymnastics. Suppose that all the girls are enrolled in at least one of the two \nsports. How many girls are into both swimming and gymnastics?\n 54.  A nurse surveyed 26 children. She learned that 14 of the children have dogs, \n9 have cats, and 6 have rabbits. Also, 4 have dogs and cats, 3 have dogs and \nrabbits, and 1 has cats and rabbits. Suppose that none of them have all three \nkinds of pets. How many children do not have any of the three kinds of pets?\n 55.  In a certain municipality, a market analyst asked 150 mothers about the \ndetergent brand they use. The analyst found out that 102 of the mothers use \nbrand A, 70 use brand B, and 40 use brand C. He also found out that 25 use \nbrand A and brand B, 27 use brand A and brand C, and 30 use brand C and"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p62-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 62, "chunk_number": 0, "type": "text", "chroma:document": "53\n1 1.618\nThe golden ratio is an example of an irrational number. The set of irrational \nnumbers is a subset of the set of real numbers. \nSet of Real Numbers\nThe set of real numbers (R) has many subsets. Refer to the following list of subsets \nof R:\n\u2022 Numbers that are used in counting objects comprise the set of natural numbers \n(N). So, N = {1, 2, 3, ...}.\n\u2022 Natural numbers and 0 (zero) comprise the set of whole numbers (W ).  \nSo, W = {0, 1, 2, 3, ...}.\n\u2022 Natural numbers, their negatives, and 0 comprise the set of integers (Z ).  \nSo, Z = {..., \u20133, \u20132, \u20131, 0, 1, 2, 3, ...}.\n\u2022 Numbers that can be expressed as a ratio of two integers belong to the set \nof rational numbers (Q). Rational numbers can be terminating decimals (like \n0.5 and 0.75), repeating decimals (like 0.333... and 0.2727...), or fractions \nlike \n1\n2 and \n3\n4 .\n\u2022 Nonterminating and nonrepeating decimals (like 1.6180339887... and \n3.1416...) belong to the set of irrational numbers (Q').\nThe Venn diagram on the"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p63-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 63, "chunk_number": 0, "type": "text", "chroma:document": "54\nThe Real Number Line \nAn important characteristic of the set of real numbers is that they can be  \nrepresented by the points on a number line. In fact, the graph of the set of real numbers \nis the entire number line. Every point on the number line corresponds to a unique \nreal number. Look at the following number line:\n\u20134 \u20132\u20133 \u20131 0\n\u20130.5 p\u20132.3\n1 2 3 4\nThe numbers on the right of 0 are positive real numbers. The numbers on the \nleft of 0 are negative real numbers. Some real numbers shown on the number line \nare the positive integers 1, 2, 3, and 4, the negative integers \u20131, \u20132, \u20133, and \u20134, the \nrational numbers \u20132.3, \u20130.5, and 5\n2\n, and the irrational numbers  and \uf006.\nProperties of Real Numbers \nThe real number system consists of the set of real numbers and the numbers that \nresult from binary operations such as the four fundamental operations. Several \nassumptions govern the operations on real numbers. These assumptions are called \naxioms."}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p64-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 64, "chunk_number": 0, "type": "text", "chroma:document": "55\nA set of numbers is said to be closed under an operation if, upon performing the \noperation on any two elements of the set, the result is also an element of that set. \nThe closure property for real numbers guarantees that the sum and the product of \nany two real numbers are also real numbers. Thus, the set of real numbers is said \nto be closed under addition and multiplication.\nexamPLe 1\nEach finger of a human hand is divided \ninto four parts by the knuckles. The \nrelative lengths of these four parts \nare shown on the right. It could be a \ncoincidence, but these four numbers, \n2, 3, 5, and 8 belong to the Fibonacci \nsequence. Suppose that F = {2, 3, 5, 8}. Is \nF closed under addition? \n Solution:\n  To check if set F is closed under addition, add each pair of numbers in the set. \nThen check if each sum also belongs to F. Note that the sum of 2 and 8 is 10, \nwhich is not an element of F. Thus, F is not closed under addition. \nCommutative Property\nLet a and b be real numbers."}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p65-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 65, "chunk_number": 0, "type": "text", "chroma:document": "56\nexamPLe 2\nThe Philippines is recognized as the world\u2019s center of marine biodiversity. Among \nits marine reservoirs are beautiful coral reefs. These coral reefs serve as shelter to \nmore than 900 fish species and 400 coral species in the country. Using the given \nnumbers, show that addition and multiplication are commutative. \n Solutions:\n a. 400 + 900  ? 900 + 400  The two numbers can be added in any \norder without affecting the sum.  1,300  3 1,300\n b. 400 \u2022 900  ? 900 \u2022 400  The two numbers can be multiplied in any \norder without affecting the product.  360,000  3 360,000\nAssociative Property\nLet a, b, and c be real numbers. \n Axiom 5.\n Associative Property of \nAddition\na + (b + c) = (a + b) + c\n Axiom 6.\n Associative Property of Multiplication\na\n \u2022 (b \u2022 c) = (a \u2022 b) \u2022 c\nKey Concept\nThe next example illustrates the Associative Property.\nexamPLe 3\nShow that each equation is correct.\na. 4 + (7 + 1) = (4 + 7) + 1 b. 2 \u2022 (3 \u2022 4) = (2 \u2022 3) \u2022 4\n Solutions:"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p59-c2-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 59, "chunk_number": 2, "type": "text", "chroma:document": "brand A, 70 use brand B, and 40 use brand C. He also found out that 25 use \nbrand A and brand B, 27 use brand A and brand C, and 30 use brand C and \nbrand B. Suppose that all of them use at least one of the three brands. Are \nthere mothers who use all three brands? Show your answer using a Venn \ndiagram."}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p67-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 67, "chunk_number": 0, "type": "text", "chroma:document": "58\nExistence of an Inverse Element\n Axiom 9.\n Additive Inverse Property\n \n    For any real number a\n, there exists a real number \u2013a, called the additive \ninverse of a, such that \n \n   a\n + (\u2013a) = 0  and  (\u2013a) + a = 0. \n Axiom 10.\n Multiplicative Inverse Property\n \n    For \nany nonzero real number a, there exists a multiplicative inverse 1\na ,  \nsuch that \n \n   a\n \u2022 1\na = 1  and  1\na \u2022 a = 1.\nKey Concept\nexamPLe 5\nFind the number that will complete each equation.\na. 7 + ____ = 0 b. 4 \u2022 ____ = 1\n Solutions:\n a. 7 + (\u20137) = 0   The additive inverse of a real number is obtained by changing the  \nsign of the number. Hence, \u20137 is the additive inverse of 7. Equivalently, \n7 is the additive inverse of \u20137.\n b.  4 \u2022 \n1\n4 = 1  The multiplicative inverse of a nonzero real number is its reciprocal. \nTherefore, the multiplicative inverse of 4 is 1\n4 . Conversely, the \nmultiplicative inverse of 1\n4 is 4.\nexamPLe 6\nThe top four eggplant-producing provinces in the country are Pangasinan,"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p63-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 63, "chunk_number": 1, "type": "text", "chroma:document": "result from binary operations such as the four fundamental operations. Several \nassumptions govern the operations on real numbers. These assumptions are called \naxioms. \nAxioms are properties that are assumed to be true without proof.\nKey Concept\nStudy the following axioms for real numbers:\nClosure Property\nLet a and b be real numbers. \n Axiom 1.\n Closure Property of \nAddition\n \n  The sum\n a + b is a real number.\n Axiom 2.\n Closure Property of Multiplication\n \n  The product \na \u2022 b is a real number.\nKey Concept"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p64-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 64, "chunk_number": 1, "type": "text", "chroma:document": "Then check if each sum also belongs to F. Note that the sum of 2 and 8 is 10, \nwhich is not an element of F. Thus, F is not closed under addition. \nCommutative Property\nLet a and b be real numbers. \n Axiom 3.\n Commutative Property of \nAddition\na + b = b + a\n Axiom 4.\n Commutative Property of Multiplication\na\n \u2022 b = b \u2022 a\nKey Concept\nNote that by the Commutative Property, rearranging the addends does not affect \nthe sum. Similarly, rearranging the factors does not affect the product."}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p62-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 62, "chunk_number": 1, "type": "text", "chroma:document": "like \n1\n2 and \n3\n4 .\n\u2022 Nonterminating and nonrepeating decimals (like 1.6180339887... and \n3.1416...) belong to the set of irrational numbers (Q').\nThe Venn diagram on the \nright can help you visualize the \nrelationships among the set of real \nnumbers and its subsets.\nUsing your knowledge of sets, \nwhat generalizations can you \nmake from the given diagram?\n(Q')\nIrrational \nNumbers\nRational Numbers (Q)\nReal Numbers (R)\nWhole \nNumbers (W)\nNatural \nNumbers \n(N)\nIntegers (Z)"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p65-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 65, "chunk_number": 1, "type": "text", "chroma:document": "Key Concept\nThe next example illustrates the Associative Property.\nexamPLe 3\nShow that each equation is correct.\na. 4 + (7 + 1) = (4 + 7) + 1 b. 2 \u2022 (3 \u2022 4) = (2 \u2022 3) \u2022 4\n Solutions:\n a.  4 + (7 + 1)  ? (4 + 7) + 1  When adding numbers, regrouping can be done \nwithout affecting the sum.   4 + 8  ? 11 + 1\n   12  3 12\n b.  2 \u2022 (3 \u2022 4)  ? (2 \u2022 3) \u2022 4  When multiplying numbers, regrouping can be \ndone without affecting the product.  2 \u2022 12  ? 6 \u2022 4\n  24  3 24"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p60-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 60, "chunk_number": 1, "type": "text", "chroma:document": "types of numbers and their properties, as you discover the beauty of math and nature.\nLesson 2-1. Real Numbers\nLesson 2-2. Integers\nLesson 2-3. Rational Numbers\nLesson 2-4. Irrational Numbers"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p67-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 67, "chunk_number": 1, "type": "text", "chroma:document": "Therefore, the multiplicative inverse of 4 is 1\n4 . Conversely, the \nmultiplicative inverse of 1\n4 is 4.\nexamPLe 6\nThe top four eggplant-producing provinces in the country are Pangasinan, \nQuezon, Cebu, and Misamis Oriental. Suppose that you planted 4 rows of eggplant \nseedlings, where each row contains 5 seedlings. Then you added 2 more rows with \nthe same number of eggplant seedlings per row. How many eggplant seedlings \ndid you plant in all?"}


created_at: 2025-05-16 10:05:17
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p68-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 68, "chunk_number": 0, "type": "text", "chroma:document": "59\n Solution:\n  Let \uf0cb represent an eggplant seedling that you planted initially. You already had  \n4 rows, where each row contains 5 seedlings, as shown here.\n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n  Now let \uf0cc represent each additional eggplant seedling you planted. Since you \nadded 2 rows, you will have the following number of eggplant seedlings:\n \n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n\uf0cb \uf0cb \uf0cb \uf0cb \uf0cb\n\uf0cc \uf0cc \uf0cc \uf0cc \uf0cc\n\uf0cc \uf0cc \uf0cc \uf0cc \uf0cc\n There are two ways to solve the problem numerically.\n Method A:\n  Step 1.  Identify the total number of rows. If you added 2 more rows, then  \nthere will be 4 + 2 = 6 rows of seedlings.\n  Step 2.   Find the total number of seedlings by multiplying the number \nof rows by the number of seedlings per row. Hence, there are a \ntotal of 5 \u2022 6 = 30 seedlings.\n  The solution based on this method may be summarized as follows:\n    5 \u2022 (4 + 2)  =  5 \u2022 6\n     =  30\n Method B:\n  Step 1.  Find the original number of seedlings that you planted. There"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p69-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 69, "chunk_number": 0, "type": "text", "chroma:document": "60\n  The solution based on this method is summarized as follows:\n    5 \u2022 4 + 5 \u2022 2  =  20 + 10\n     =  30\n  Note that you obtained the same answer as in the previous method. Hence, \nyou planted a total of 30 seedlings.\nIn the previous example, notice that 5 \u2022 (4 + 2) = 5 \u2022 4 + 5 \u2022 2. This equation \nillustrates the Distributive Property of Multiplication Over Addition.\nDistributive Property of Multiplication Over Addition\nLet a, b, and c be real numbers. \n Axiom 1\n1. \n a\n \u2022 (b + c) = a \u2022 b + a \u2022 c \nKey Concept\nThis property states that when multiplying a number by a sum of two or more \nnumbers, you can multiply first the number by each of the addends. Then add the \nresulting products. You can use this property when multiplying numbers, particularly \nin cases when rewriting one of the factors as a sum could make the computation \neasier. Study the next example. \nexamPLe 7\nPerform the indicated operations. Apply the Distributive Property to make your \ncomputations easier."}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p72-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 72, "chunk_number": 0, "type": "text", "chroma:document": "63\nDetermine the set(s) where each number belongs to. Put a check mark on the correct \ncolumn. You may use a calculator when necessary.\nNumber N W Z Q Q'\n11. 0\n12. 2p\n13. 51\n17\n14. 2\n15. 16\n16. 3.14\n17. \u201313\n1\n7\n18. 0.345\n19. \u20133.8333...\n20. 0.12321213...\nWrite True if the statement is true; otherwise, write False.\n 21. N \u2282 W \n 22. Z \u2282 Q\n 23. Q' \u2282 Q\n 24. Q' = R\n 25. R \u2013 Q' = Q \n 26. Q \u2229 N = Z\n 27. Z \u2229 W = N\n 28. Q \u2229 Q' = \u2205\n 29. N \u2229 W = \u2205 \n 30. W \u2013 N = {0}\nWrite the property of real numbers that each equation shows.\n 31. \n1\n2  2 = 1\n 32. \n1\n2 + 0 = \n1\n2\n 33. 2  x  y = 2  y  x\n 34.  (\u20131) + 1 = 0"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p71-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 71, "chunk_number": 0, "type": "text", "chroma:document": "62\n\u2022\t Subtraction\n If a\n, b, and c are real numbers, the difference a \u2013 b is the number c such that  \n c\n + b = a.\n\u2022\t Division\n \n \nIf a, b, and c are real numbers, the quotient a\nb , where b \u2260 0, is the number c such that  \nc \u2022 b = a.\nKey Concept\nexamPLe 9\nProve each equation using the definition of subtraction or division. \na. 100 \u2013 47 = 53 b. 264 \u00f7 11 = 24\n Solutions:\n a. 100 \u2013 47 = 53, since 47 + 53 = 100.\n b.  264 \u00f7 11 = 24, since 24 \u2022 11 = 264.\nWhich of the presented axioms hold true for subtraction? For division? Verify \nyour answers by giving examples using different pairs of numbers.\nPractice 2-1\nConcepts and Skills\nWrite True if the statement is true; otherwise, write False.\n 1. 0 is a natural number.\n 2.  All numbers have a multiplicative inverse.\n 3. No natural number is an irrational number.\n 4. The additive identity element of the set of integers is 0.\n 5.  A rational number is a quotient of any two integers.\n 6. The product of a number and its additive inverse is 1."}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p73-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 73, "chunk_number": 0, "type": "text", "chroma:document": "64\n 35. z  (x  y) = (x  y)  z\n 36. x + y = y + x\n 37. 2  (x  y) = (2  x)  y\n 38. x  y + 1 = 1 + x  y\n 39. (\u20131) + 1 = 1 + (\u20131)\n 40. (\u20134)  2 = 2  (\u20134)\n 41. (a + b)(c + d) = (c + d)(a + b)\n 42. (a + b)(c + d) = a(c + d) + b(c + d)\nFor items 43 to 45, complete the two-column proof.\nFor any real numbers x, a, b, and c, prove that: \nx  (a + b + c) = x  a + x  b + x  c.\nStatement Reason\n x  (a + b + c) = x  [(a + b) + c] 43.\n  =  x  (a + b) + x  c 44.\n  =  x  a + x  b + x  c 45.\nApplications\nAnswer each item completely.\n 46.  Jaymie took an examination that is divided into three parts. She got 13 correct \nanswers in the first part, 25 in the second part, and 27 in the third part. Apply \nthe properties of real numbers to find her total score in the exam. Solve for the \nanswer mentally.\n 47.  An elevator can carry a maximum of 9 persons with an average weight of 50 kg \neach. Suppose that a box of items weighs 9 kg. How many boxes of items can \nthe elevator carry? Explain your answer."}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p66-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 66, "chunk_number": 0, "type": "text", "chroma:document": "57\nYou can apply the Associative and the Commutative Properties when doing \nmental computations. When adding three or more numbers, you may first combine \nthe addends that add up to 10 or to any multiple of 10 to make computations easier. \nLook at the following illustration:\n7   +  4  +  3   =  14\n10\nWhen multiplying three or more numbers, you may first multiply factors that give \na product of 10 or any multiple of 10. Refer to the following illustration:\n5   \u2022  7  \u2022   2   =  70\n10\nOther properties of real numbers include the existence of the identity element \nand the inverse element. Study the following axioms:\nExistence of an Identity Element\nLet a be any real number. \n Axiom 7.\n Additive Identity Property\n \n    There exists a real number 0, called the \nadditive identity element, such that \n \n   a\n + 0 = a  and  0 + a = a.\n Axiom 8. \n Multiplicative Identity Property\n \n    There exists a real number 1, \ncalled the multiplicative identity element , \nsuch that \n \n   a"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p68-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 68, "chunk_number": 1, "type": "text", "chroma:document": "The solution based on this method may be summarized as follows:\n    5 \u2022 (4 + 2)  =  5 \u2022 6\n     =  30\n Method B:\n  Step 1.  Find the original number of seedlings that you planted. There \nare 4 rows, where each row contains 5 eggplant seedlings. Thus, \nthere are 5 \u2022 4 = 20 seedlings.\n  Step 2.  Find the number of seedlings you added. You added 2 rows, \nwith 5 eggplant seedlings in every row. This means that you \nadded 5 \u2022 2 = 10 seedlings.\n  Step 3.  Find the total number of seedlings. Add the results in steps 1 \nand 2. So you will have 20 + 10 = 30 seedlings."}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p74-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 74, "chunk_number": 0, "type": "text", "chroma:document": "65\nFor items 49 to 51, consider the following situation:\n Janice buys 12 cupcakes that cost P8.50 each. She also buys 8 muffins that cost \nP11 each. Instead of doing the conventional multiplication process, she computes \nfor the total price of the cupcakes and muffins as shown here. Write the properties \nof real numbers that she used.\n8.5(10 + 2) + 11(10 \u2013 2) = 85 + 17 + 110 \u2212 22 49.   \n= 85 + 110 + 22 \u2013 5 \u2013 22 Commutative Property of \nAddition\n= 195 \u2013 5 + 22 + (\u221222) Commutative Property of \nAddition\n= 190 + 0 50.   \n= 190 51.   \nEnrichment Exercises\nAnswer each item completely.\n 52. Is any integer a rational number? Justify your answer.\n 53.  Is the set of odd whole numbers closed under addition? Why or why not?\n 54. Consider the following equation: \n  (3 \u2022 4) + (5 \u2022 6) = 3 \u2022 4 + 5 \u2022 6. \n  Can you say that 3 \u2022 (4 + 5) = 3 \u2022 4 + 5? Justify your answer.\n 55.  Can you find values for a, b, and c that will make the given equation true. \nExplain your answer?\n  ab \u2013 ac = a \u2013 bc"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p71-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 71, "chunk_number": 1, "type": "text", "chroma:document": "4. The additive identity element of the set of integers is 0.\n 5.  A rational number is a quotient of any two integers.\n 6. The product of a number and its additive inverse is 1.\n 7.  The set of counting numbers is not closed under division.\n 8. Subtraction is associative, but it is not commutative.\n 9. Subtracting the additive identity element from any number results in the given \nnumber.\n 10. The union of the sets of integers and rational numbers is the set of real numbers."}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p69-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 69, "chunk_number": 1, "type": "text", "chroma:document": "easier. Study the next example. \nexamPLe 7\nPerform the indicated operations. Apply the Distributive Property to make your \ncomputations easier.\na. 120 \u2022 64 + 120 \u2022 36\nb. 80 \u2022 104\n Solutions:\n a. 120 \u2022 64 + 120 \u2022 36  = 120 \u2022 (64 + 36)\n   = 120 \u2022 100\n   = 12,000\n   Notice that 64 and 36 are added first to get 100, which can be easily \nmultiplied by 120. \n b.  To find the product of 80 \u2022 104, you may think of 104 as 100 + 4 since it \nis easier to multiply a number by multiples of 10.\n   80 \u2022 104  = 80 \u2022 (100 + 4)\n    = 80 \u2022 100 + 80 \u2022 4\n    = 8,000 + 320\n    = 8,320"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p75-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 75, "chunk_number": 0, "type": "text", "chroma:document": "66\nLearning \nObjectives\nAt the end of the \nlesson, you should be \nable to:\n\u2022\t define\tintegers\n\u2022\t locate\tintegers\ton\t\na\tnumber\tline\n\u2022\t compare\tand\torder\t\nintegers\n\u2022\t find\tthe\tabsolute\t\nvalue\tof\t an \t integer\t\nor\texpression\t\ninvolving\tintegers\n\u2022\t perform\toperations\t \non\tintegers\nProbe and Learn\nThe Marianas Trench in the Pacific Ocean is considered as \nthe deepest part of the ocean. It has a depth of about 10,994 m.  \nThe Philippines\u2019 deepest point, which is also the world\u2019s second \ndeepest spot, is the Mindanao Trench. It is about 10,540 m \nbelow sea level. Like ocean depths, heights of mountains are also \nmeasured from sea level. Mount Apo, the highest mountain in \nthe Philippines, has a height of 2,954 m. Mount Everest, the \nhighest mountain in the world, has a height of 8,848 m.\nOcean depths are measured using sound navigation and \nranging (sonar). This is done by sending sound waves from a ship \ntowards the ocean bottom. The time it takes for sound waves to"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p73-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 73, "chunk_number": 1, "type": "text", "chroma:document": "each. Suppose that a box of items weighs 9 kg. How many boxes of items can \nthe elevator carry? Explain your answer.\n 48.  Bob teaches 4 hours a day, 4 days a week. His hourly rate is P150. How many \nweeks should he work to earn at least  7,000?"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p77-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 77, "chunk_number": 0, "type": "text", "chroma:document": "68\nexamPLe 1\nExpress each numerical value as an integer.\na.  The 2018 General Wholesale Price Index (GWPI) in the Philippines dropped by \nabout 2% during a certain month. \nb.  The 2013 functional literacy rate of the country grew by about 8 percentage \npoints from 2008.\nc. Fuelwood consumption in rural areas declined by 6%.\n Answers:\n a.  \u20132  The word dropped suggests a negative integer.\n b. 8 The word grew indicates a positive integer.\n c. \u20136 Why?\nexamPLe 2\nThe following table shows the differences in the grades of a student from one \ngrading period to the next grading period:\n Subject Difference\nEnglish +2\nFilipino 0\nScience \u20131\nMath \u20133\nAraling Panlipunan +4\nPlot each integer on a number line. Label the points properly.\n Solutions:\n  Use the following letters to label the points: E for English, F for Filipino, S for \nScience, M for Math, and A for Araling Panlipunan. So you will have:\n \u20134\u20135 \u20132\u20133 \u20131 0 1 2 3 4 5\nM S F E A"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p66-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 66, "chunk_number": 1, "type": "text", "chroma:document": "a\n + 0 = a  and  0 + a = a.\n Axiom 8. \n Multiplicative Identity Property\n \n    There exists a real number 1, \ncalled the multiplicative identity element , \nsuch that \n \n   a\n \u2022 1 = a  and  1 \u2022 a = a.\nKey Concept\nexamPLe 4\nFind the number that will complete each equation.\na. 4 + ____ = 4 b. 19 \u2022 ____ = 19\n Solutions:\n a. 4 + 0 = 4  Adding 0 to a number gives a sum equal to that number.\n b. 19 \u2022 1 = 19   Multiplying a number by 1 gives a product equal to that number."}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p70-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 70, "chunk_number": 0, "type": "text", "chroma:document": "61\nYou have seen that applying the properties of real numbers could help make your \ncomputations easier. You can also use these basic properties to derive or prove other \nproperties of real numbers.\nProperties that can be proven using axioms are called theorems.\nKey Concept\nThere are no absolute rules in proving theorems. To prove a theorem, you may \nwrite a sequence of statements, each of which follows directly from the preceding \nstatement. Each statement must be supported by a reason, which can be an axiom, \na definition, or a previously proven theorem. A clear understanding of the properties \nof real numbers would be a big help when proving theorems. Study the next example.\nexamPLe 8\nProve that c \u2022 (a + b) = c \u2022 b + c \u2022 a, for any real numbers a, b, and c.\n Solution:\n  To prove the given property, manipulate the expression on one side of \nthe equation to yield the expression on the other side. In this case, you"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p76-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 76, "chunk_number": 0, "type": "text", "chroma:document": "67\nof a mountain when it is brought at the top of the mountain. Such measures of \nheights and depths are good examples of the existence of positive and negative values. \nLook at the following number line. Describe the labeled numbers on the left of 0. \nHow do these numbers compare with the set of natural numbers?\n\u20135 \u20134 \u20133 \u20132 \u20131 1 2 3 4 5 0\nSet of Integers\nThe set of integers, which is denoted as Z, is composed of the natural numbers, their \nadditive inverses, and 0.\nKey Concept\nThe set of integers may be described as follows: \nZ = {\u2026, \u20133, \u20132, \u20131, 0, 1, 2, 3, ...}.\nNote that the set of integers is the union of three disjoint sets, namely, the set of \npositive integers Z+ = {1, 2, 3, 4, 5, ...}, the set containing 0, and the set of negative \nintegers Z\u2212 = {\u2026, \u20135, \u20134, \u20133, \u20132, \u20131}. It may be represented as follows:\nZ = Z\u2013 \u222a {0} \u222a Z+.\nNote that the set of positive integers is equal to the set of natural numbers."}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p78-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 78, "chunk_number": 0, "type": "text", "chroma:document": "69\nComparing and Ordering Integers\nLook at the following number line: \n\u20135 \u20134 \u20133 \u20132 \u20131 1 2 3 4 5 0\nA number line can help you in comparing and ordering integers. Note that the \nvalues of integers are increasing from left to right on the number line. This means \nthat the integer on the right of a given integer is greater, and any other integer on its \nleft is lesser in value. Also, remember that any negative integer is less than 0, while \nany positive integer is greater than 0. Can you say that any positive integer is greater \nthan any negative integer?\nexamPLe 3\nRefer to the given table in the previous example. In which subject does the student \nhave the greatest increase in grades? How about the greatest decrease?\n Solution:\n  You can use the number line you made in the previous example to arrange the \nintegers from least to greatest. Consider that the integers are written in order \nfrom left to right on the number line.\nM S F E A\n\u20133 \u20131 0 2 4"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p79-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 79, "chunk_number": 0, "type": "text", "chroma:document": "70\nAbsolute Value\nThe distance of a number from 0 on a number line is called the absolute value of the \nnumber. The absolute value of an integer n is written as |n|.\nKey Concept\nNote that since the absolute value refers to a distance, it is always nonnegative. \n|n|  =  distance of n from 0\n0 n\nLet n be any integer.\n\u2022 |n|  =   n if n is a positive integer.\n\u2022 |n|   =  0 if n equals 0.\n\u2022 |n|  =   \u2013n if n is a negative integer.\nexamPLe 5\nFind |4|, |\u20134|, and |0|.\n Solutions:\n  First, locate each integer on the number line. Then count how many units \naway each integer is from 0.\n \u20134\u20135 \u20132\u20133 \u20131 0 1 2 3 4 5\n4 units 4 units\n Since 4 is 4 units away from 0, then |4|   =  4.\n Since \u20134 is 4 units away from 0, then |\u20134|   =  4.\n Notice that |0| = 0. Why?\nOperations on Integers\nLike with whole numbers, performing operations on integers follows certain rules. \nYour knowledge of the concept of absolute value will be of great help in understanding"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p75-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 75, "chunk_number": 1, "type": "text", "chroma:document": "Ocean depths are measured using sound navigation and \nranging (sonar). This is done by sending sound waves from a ship \ntowards the ocean bottom. The time it takes for sound waves to \ntravel to the seafloor and back to the ship is accurately measured \nand recorded. On the other hand, a surveying altimeter is used to \nmeasure the height of mountains. It directly records the altitude \nIntegersLESSON\n2-2"}


created_at: 2025-05-16 10:05:18
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p82-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 82, "chunk_number": 0, "type": "text", "chroma:document": "73\n Solution:\n  Represent each integer using counters. Then remove any zero pair, if there are \nany. The number that represents the remaining counters is the sum.\n \u201334 + (\u20137)\n Hence, 4 + (\u20137)  =  \u20133.\nIn examples 6 and 7, you found out that (\u20134) + 7 = 3 and 4 + (\u20137) = \u20133. Notice that \nthe sum in each of the given equations is the difference of the absolute values of the \naddends, and the sign of the sum is the sign of the addend with the greater absolute \nvalue. This shows the next rule.\nWhen adding integers with unlike signs, subtract their absolute values. Then attach the \nsign of the integer with the greater absolute value to the sum.\nKey Concept\nThe rule means that the sum of a positive integer and a negative integer may be \neither negative or positive, depending on the sign of the addend that has the greater \nabsolute value. Can their sum be equal to 0?\nexamPLe 8\nFind the sum of (\u22129) + 4.\n Solution:\n  |\u20139| \u2013 |4|  = 5  Find the difference of the absolute values of the integers."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p81-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 81, "chunk_number": 0, "type": "text", "chroma:document": "72\n Solutions:\n a. (\u20134) + 7\n  \nu Start at \u20134.\nv\n Move 7 units to the right.\n\u2013\n4\u20135 \u20132\u20133 \u20131 0 1 2 3 4 5\n  Thus, (\u20134) + 7 = 3.\n b. (\u22124) + (\u22127)\n  \nv Move 7 units to the left.\nu\tStart at \u20134.\n\u2013\n12\u201313 \u201310\u201311 \u20139 \u20138 \u20137 \u20136 \u20135 \u20134 \u20133\n  Thus, (\u20134) + (\u20137) = \u201311.\nIn item b of the previous example, you found out that (\u20134) + (\u20137) = \u201311. This \nmeans that the sum of two negative integers is also negative. Use the following rule \nwhen adding integers with like signs:\nWhen adding integers with like signs, add their absolute values. Then attach the common \nsign to the sum.\nKey Concept\nAnother way to illustrate addition of integers is by using counters. Counters are \nuseful in showing the rule in adding integers with different signs. \nLet  represent +1. Let  represent \u22121. Note that a  and a  make a zero pair. \nSince a zero pair has a value of 0, you can remove zero pairs without affecting the \nresulting answer. Study next example.\nexamPLe 7\nFind the sum of 4 + (\u20137) using counters."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p70-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 70, "chunk_number": 1, "type": "text", "chroma:document": "Solution:\n  To prove the given property, manipulate the expression on one side of \nthe equation to yield the expression on the other side. In this case, you \nmay start with the left side of the equation. Then apply the properties \nof real numbers to end up with the expression on the right side.  \nYou may use a two-column proof. The first column gives the statements, while \nthe second column gives the corresponding reasons. So you will have the \nfollowing:\nStatement Reason\nc \u2022 (a + b)  =  c \u2022 (b + a) Commutative Property of Addition\n=  c \u2022 b + c \u2022 a Distributive Property\n  Since you already obtained the expression on the right side of the equation, \nyou have shown that the property is true. \nHave you noticed that the previously presented properties of real numbers involve \nonly addition and multiplication? Can you explain why?\nIn earlier grade levels, you have learned that subtraction is the inverse of"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p76-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 76, "chunk_number": 1, "type": "text", "chroma:document": "integers Z\u2212 = {\u2026, \u20135, \u20134, \u20133, \u20132, \u20131}. It may be represented as follows:\nZ = Z\u2013 \u222a {0} \u222a Z+.\nNote that the set of positive integers is equal to the set of natural numbers. \nMountain heights and ocean depths are measured with respect to sea level, which \nmay be referred to as the reference point. Thus, the sea level may be represented  \nby 0. The phrases above sea level and below sea level  indicate values that have \nopposite signs. Distances below sea level may be represented by negative numbers. \nDistances above sea level may be represented by positive numbers. Hence, the given \nocean depths and mountain heights may be written using signed numbers as follows:\nPlace Distance from  \nSea Level (in meters)\nAbove Sea Level\nMount Everest  8,848\nMount Apo  2,954\nBelow Sea Level\nMindanao Trench  \u201310,540\nMarianas Trench  \u201310,994\nPositive numbers may be written without a plus sign (+). Thus, +4 is the same \nas 4. Negative numbers are written with a minus sign (\u2013) before them. For example,"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p78-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 78, "chunk_number": 1, "type": "text", "chroma:document": "integers from least to greatest. Consider that the integers are written in order \nfrom left to right on the number line.\nM S F E A\n\u20133 \u20131 0 2 4\n  Therefore, the greatest increase is 4, which is in Araling Panlipunan. The \ngreatest decrease is \u20133, which is in Math.\nTo compare two integers, the following symbols may be used:\nSymbol Meaning\n= is equal to\n> is greater than\n< is less than\nexamPLe 4\nCompare each pair of integers using >, <, or =.\na. 1 _____ 5 b. \u20131 _____ \u20132 c. \u20134 _____ 0\n Solutions:\n a. 1 < 5  1 is located on the left side of 5 on the number line.\n b.  \u20131 > \u20132  \u20131 is located on the right side of \u20132 on the number line.\n c. \u20134 < 0  Why?"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p79-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 79, "chunk_number": 1, "type": "text", "chroma:document": "Operations on Integers\nLike with whole numbers, performing operations on integers follows certain rules. \nYour knowledge of the concept of absolute value will be of great help in understanding \nthese rules. Study how addition, subtraction, multiplication, and division on integers \nare performed. \nDo you know?\nYou\tcan \tthink \tof\tthe \ta bsolute \tvalue \tof\ta \t\nnumber\tas \tthe \tnumber \titself , \tb ut \twithout \t\nthe\tsign.\tFor\texample,\t|\u201347|\t\t=\t\t47."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p83-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 83, "chunk_number": 0, "type": "text", "chroma:document": "74\nNote that the addends in the equation include positive and negative integers. \nRecall that addition of real numbers is commutative and associative. Thus, you can \ngroup together and add integers of the same sign. Then add the resulting sums. So \nyou will have:\n N = (1,426 + 320) + [(\u2013292) + (\u2013585)]\n  =  1,746 + (\u2212877)\n  =  869\nThe sum is positive. This means that Amy has a profit of P869 in those four \nweeks.\nSubtraction of Integers\nDifferent cooperatives help \npeople through their loan programs. \nSuppose that Susan borrows P10,000 \nfrom a credit cooperative to start a \nsmall business. After a few months, \nshe pays P5,000 back. How much \ndoes she still owe the cooperative, \nassuming no interest will be applied?\nAn amount of borrowed money \nimplies a negative number. In this \nproblem, the amount Susan borrows can be represented by \u201310,000. The act of \npaying back may be expressed by subtracting a negative number. Since Susan pays"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p82-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 82, "chunk_number": 1, "type": "text", "chroma:document": "absolute value. Can their sum be equal to 0?\nexamPLe 8\nFind the sum of (\u22129) + 4.\n Solution:\n  |\u20139| \u2013 |4|  = 5  Find the difference of the absolute values of the integers.\n  (\u20139) + 4  = \u20135   Attach to the sum the sign of the integer with the greater absolute \nvalue. Since \u20139 has the greater absolute value, the sign of the sum is \nnegative.\nYou can now solve the problem posed at the start of the section on adding integers. \nThe problem asked for Amy\u2019s gain or loss from her sari-sari store in four weeks. To \nsolve it, you have to find the value of N in the following equation:\nN = 1,426 + (\u2013292) + (\u2013585) + 320."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p84-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 84, "chunk_number": 0, "type": "text", "chroma:document": "75\nexamPLe 9\nFind each difference.\na. (\u20135) \u2013 (\u20132) b. (\u20135) \u2013 2\n Solutions:\n a. (\u20135) \u2013 (\u20132)\n  \nu Start at \u20135.\nv Move 2 units to the right.\n\u20135\u20136 \u20133\u20134 \u20132 \u20131 0 1\n  Thus, (\u20135) \u2013 (\u20132) = \u20133.\n b. (\u20135) \u2013 2\n  \nu Start at \u20135.\nv Move 2 units to the left.\n\u20137\u20138 \u20135\u20136 \u20134 \u20133 \u20132 \u20131\n  Thus, (\u20135) \u2013 2 = \u20137.\nIn item a of the previous example, notice that the number line also shows  \n(\u20135) + 2. Thus, \n(\u20135) \u2013 (\u20132) = (\u20135) + 2.\nIn item b, the number line also shows (\u20135) + (\u20132). Thus,\n(\u20135) \u2013 2 = (\u20135) + (\u20132).\nThese illustrate that you can rewrite a subtraction problem into an addition \nproblem. This is done by changing the subtrahend to its additive inverse, and then \nchanging the minus sign to a plus sign. Therefore, the rule for subtracting integers \nmay be stated as follows: \nWhen subtracting integers, change the subtrahend to its additive inverse, and change the \noperation to addition. Then follow the rules in adding integers. \nKey Concept"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p85-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 85, "chunk_number": 0, "type": "text", "chroma:document": "76\nStudy the following illustration:\n(\u20135)  +    2  =  \u20133\n(\u20135) \u2013 (\u20132)\nw Add the integers.\nu\n Get the additive inverse.v\n Change to addition.\nexamPLe 10\nSubtract.\na. 7 \u2013 (\u20139)\nb. 6 \u2013 18\nc. \u20133 \u2013 12\nd. \u20139 \u2013 (\u20134)\n Solutions:\n a. 7 \u2013 (\u20139) = 7 + 9 = 16 \n b. 6 \u2013 18 = 6 + (\u201318) = \u201312\n c. \u20133 \u2013 12 = \u20133 + (\u201312) = \u201315 \n d. \u20139 \u2013 (\u20134) = \u20139 + 4 = \u20135\nIn general, if a and b are real numbers, the following equivalence relations exist:\n\t \u2022\t a + (\u2013b) = a \u2212 b\n\t \u2022\t a \u2013 (+b) = a \u2212 b\n\t \u2022\t a \u2212 (\u2212b) = a + b\nKey Concept\nMultiplication of Integers \nSuppose that a bus travels at an average \nrate of 80 kph. It decreases its speed by  \n5 kph after every hour. What is its speed \nafter 4 hours? \nTo find the speed of the bus after  \n4 hours, determine by how much the bus \nhas decreased its speed after 4 hours. \nA decrease of 5 kph in speed may be \nrepresented by \u20135. Now determine how \nmany times the bus decreases its speed in \n4 hours. Since its speed decreases every \nhour, the bus decreases its speed 4 times."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p70-c2-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 70, "chunk_number": 2, "type": "text", "chroma:document": "only addition and multiplication? Can you explain why?\nIn earlier grade levels, you have learned that subtraction is the inverse of \naddition. You have also learned that division is the inverse of multiplication. Study \nhow subtraction and division are defined based on their inverses."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p86-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 86, "chunk_number": 0, "type": "text", "chroma:document": "77\n4 \u2022 (\u20135) \u201320\nHence, the speed of the bus after 4 hours is given by:\n80 + (\u201320) = 60 kph.\nYou may also use counters to verify the following rules in multiplying integers:\nWhen multiplying integers, get the product of their absolute values. Then apply the \nfollowing rules for the sign of the product:\n \u2022\n The product of two integers with like signs is positive. \n \u2022\n The product of two integers with unlike signs is negative. \nKey Concept\nexamPLe 11\nMultiply.\na. (\u201315)(\u20138) b. 9 \u2022 (\u201312)\n Solutions:\n a. (\u201315)(\u20138) = 120   The two factors have like signs. Hence, their product is positive.\n b. 9 \u2022 (\u201312) = \u2013108   The two factors have unlike signs. Hence, their product is negative.\nWhen multiplying three or more integers, get the product of their absolute values. \nThen determine the sign of the product by counting the negative factors. An odd \nnumber of negative factors gives a negative product. An even number of negative \nfactors gives a positive product. Can you explain why?\nexamPLe 12"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p76-c2-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 76, "chunk_number": 2, "type": "text", "chroma:document": "Marianas Trench  \u201310,994\nPositive numbers may be written without a plus sign (+). Thus, +4 is the same \nas 4. Negative numbers are written with a minus sign (\u2013) before them. For example, \nnegative 4 is written as \u20134. Zero is neither positive nor negative, so it is written \nwithout a + or \u2013 sign."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p80-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 80, "chunk_number": 0, "type": "text", "chroma:document": "71\nAddition of Integers\nYou can find sari-sari stores anywhere \nin the Philippines. Amy owns a sari-sari \nstore. Every week, she lists down her profits \nand losses. The following is a summary of \nher record for four weeks:\nWeek 1 a profit of  1,426\nWeek 2 a loss of  292\nWeek 3 a loss of  585\nWeek 4 a profit of  320\nDid Amy earn or lose money in those \nfour weeks? How much did she earn or \nlose?\nProfits imply positive numbers, while losses imply negative numbers. Hence, each \namount in the table may be represented by an integer. You can solve the problem \nby finding the sum of the integers. Let N represent the sum. So you will have the \nfollowing equation:\nN = 1,426 + (\u2013292) + (\u2013585) + 320 \nHow do you solve for N? First, identify the rules in adding integers. Remember \nthat positive integers are counting numbers. This means that the sum of two positive \nintegers is also positive.\nHow about the sum of two negative integers? Or the sum of a positive integer and"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p83-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 83, "chunk_number": 1, "type": "text", "chroma:document": "implies a negative number. In this \nproblem, the amount Susan borrows can be represented by \u201310,000. The act of \npaying back may be expressed by subtracting a negative number. Since Susan pays \nP5,000 back, you have to subtract \u20135,000 from \u201310,000.\nLet N be the amount that Susan still has to pay. Hence,\nN = (\u201310,000) \u2013 (\u20135,000).\nTo find N, you need to subtract integers. Like in adding integers, you may use \ncounters in subtracting integers. For this problem, let  represent P1,000.\n\u201310,000 \u2013 (\u20135,000) = \u20135,000\u201310,000\nTherefore, Susan still owes the cooperative P5,000.\nYou may also use a number line to subtract integers. First, locate the minuend \non the number line. Suppose that n is the subtrahend.\n\u2022 If n is negative, move |n| units to the right of the minuend.\n\u2022 If n is positive, move |n| units to the left of the minuend.\nConsider the next examples."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p87-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 87, "chunk_number": 0, "type": "text", "chroma:document": "78\nDivision of Integers\nBaguio City is considered as the \nsummer capital of the Philippines due \nto its relatively low temperature. Its \ntemperatures range from 15\u00b0C to 26\u00b0C. \nOn a particular day in December 2018, the \ntemperature reading in Baguio at 3 p.m. \nis 21\u00b0C. Suppose that the temperature \ndecreases by 2\u00b0C every hour. At what  \ntime will the temperature reading be \n15\u00b0C? \nA decrease in temperature implies \na negative number. If the temperature \ndrops from 21\u00b0C to 15\u00b0C, then the \ntotal decrease in temperature may be \nrepresented by 15 \u2013 21 = \u20136, or \u22126\u00b0C. After how many hours will the temperature be \n15\u00b0C? Recall that the temperature decreases by 2\u00b0C every hour. A decrease of 2\u00b0C \nmay be represented by \u22122.\nTo solve this problem, you have to divide integers; that is, divide \u20136 by \u20132. Since \ndivision is the inverse of multiplication, the same rules of signs apply. You can verify \nthese rules using counters."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p88-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 88, "chunk_number": 0, "type": "text", "chroma:document": "79\nPractice 2-2\nConcepts and Skills\nExpress each numerical value using integers.\n 1. Today is warmer than yesterday by 5\u00b0C.\n 2.  Joy lost 2 kg after months of jogging.\n 3. John deposited P20,000 in his savings account.\n 4. Mary needs to run 5 more meters to reach the finish line.\n 5. The mystical Mount Banahaw is 2,169 m above sea level.\nWrite True if the statement is true; otherwise, write False.\n 6.  The additive inverse of 4 is \u20134.\n 7. The quotient of any two integers is an integer.\n 8. The absolute value of any integer is positive.\n 9.  The product of three negative integers is negative.\n 10. The sum of two integers is greater than each of the addends.\n 11. The quotient of a positive number and its additive inverse is \u20131.\n 12.  If a is a negative number and b is a positive number, then a \u2022 (\u2013b)  \nis positive.\n 13.  If the product of two integers is positive, then one of the integers could be \nnegative.\nPerform the indicated operations.\n 14. 36 + 9\n 15. 36 + (\u20139)\n 16. (\u201336) + 9"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p86-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 86, "chunk_number": 1, "type": "text", "chroma:document": "number of negative factors gives a negative product. An even number of negative \nfactors gives a positive product. Can you explain why?\nexamPLe 12\nFind the product of (\u20135)(\u20132)(\u20133).\n Solution:\n 5 \u2022 2 \u2022 3 = 30  Multiply the absolute values of the given integers.\n (\u20135)(\u20132)(\u20133) = \u201330  The product is negative since there are three negative factors."}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p90-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 90, "chunk_number": 0, "type": "text", "chroma:document": "81\n 47.  What is the change in the number of enrollees in each grade level from  \nSY 2009\u22122010 to SY 2010\u22122011? Express each change as an integer.\nGrade Level Change in the Number \nof Enrollees\nGrade 7\nGrade 8\nGrade 9\nGrade 10\n 48.  Plot the data you obtained in the previous item on a number line. Use the grade \nlevels to label the points.\n 49. In which grade level is the increase in the number of enrollees greatest? \n 50. In which grade level is the decrease in the number of enrollees greatest?\n 51.  Which level has a greater change in the number of enrollees from SY 2009\u22122010 \nto SY 2010\u22122011: grade 8 or grade 9? By how many students? \n 52.  What is the change in the total number of enrollees from SY 2009\u22122010 to  \nSY 2010\u22122011?\n 53.  What is the average change in the number of enrollees from SY 2009\u22122010 to \nSY 2010\u22122011?\nAnswer each item completely.\n 54.  What is the distance between the lowest point and the highest point in the"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p89-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 89, "chunk_number": 0, "type": "text", "chroma:document": "80\nWrite the letter of the statement that best describes each pair of quantities in \nthe two columns.\n  A.  The quantity in column 1 is greater than the quantity in column 2.\n  B.  The quantity in column 2 is greater than the quantity in column 1.\n  C.  The two quantities are equal. \n  D.  The two quantities cannot be compared.\nColumn 1 Column 2\n 37. |0| 0 \u2022 (\u20135)\n 38. \u2013(\u20131) |\u20133| \u2013 2\n 39. 5 \u2022 (\u20134) (\u201380) \u00f7 (\u20134)\n 40. 15 \u2013 (\u20135) \u201315 + 5\n 41. n |n|\nFind the next two terms in each sequence.\n 42. \u201316, \u201311, \u20136, \u20131, 4, , \n 43. 729, \u2013243, 81, \u201327, 9, , \n 44. 1, \u20131, 2, \u20132, 4, \u20134, 8, \u20138, , \n 45. \u20132, 8, 2, 5, 6, 2, 10, \u20131, , \n 46. 0, 0, 1, \u20131, 2, \u20134, 3, \u20139, , \nApplications\nFor items 47 to 53, refer to the following table that shows the number of enrollees  \nin a certain high school for the school year (SY) 2009\u22122010 and SY 2010\u22122011.\nGrade Level SY 2009\u20132010 SY 2010\u20132011\nGrade 7 258 342\nGrade 8 310 295\nGrade 9 365 370\nGrade 10 375 360"}


created_at: 2025-05-16 10:05:19
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p85-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 85, "chunk_number": 1, "type": "text", "chroma:document": "represented by \u20135. Now determine how \nmany times the bus decreases its speed in \n4 hours. Since its speed decreases every \nhour, the bus decreases its speed 4 times. \nTo find the total decrease in speed after  \n4 hours, get the product of 4 \u2022 (\u20135). You may \nsolve this using counters. Let  represent \n\u20131."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p91-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 91, "chunk_number": 0, "type": "text", "chroma:document": "82\nEnrichment Exercises\nAnswer each item completely.\n 58.  If |x| < |y|, is it true that x < y ? Justify your answer.\n 59.  Consider the equation | x| + | y| = | x + y|. Could x and y  have different \nsigns?\n 60.  Suppose that a negative number is subtracted from a positive number. Is the \ndifference positive or negative? Explain your answer."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p87-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 87, "chunk_number": 1, "type": "text", "chroma:document": "division is the inverse of multiplication, the same rules of signs apply. You can verify \nthese rules using counters.\nWhen dividing integers, get the quotient of their absolute values. Then apply the following \nrules for the sign of the quotient:\n \u2022\n The quotient of two integers with like signs is positive.\n \u2022\n The quotient of two integers with unlike signs is negative.\nKey Concept\nTherefore, the solution to the previous problem is:\n(\u20136) \u00f7 (\u20132) = 3.\nThis means that it will take 3 hours for the temperature to decrease from 21\u00b0C to \n15\u00b0C. Hence, the temperature reading will be 15\u00b0C at 6 p.m.\nexamPLe 13\nDivide.\na. (\u201315) \u00f7 (\u20133) b. 132 \u00f7 (\u201312)\n Solutions:\n a. (\u201315) \u00f7 (\u20133) = 5   The two integers have like signs. Hence, their quotient is positive.\n b. 132 \u00f7 (\u201312) = \u201311   The two integers have unlike signs. Hence, their quotient is \nnegative."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p88-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 88, "chunk_number": 1, "type": "text", "chroma:document": "is positive.\n 13.  If the product of two integers is positive, then one of the integers could be \nnegative.\nPerform the indicated operations.\n 14. 36 + 9\n 15. 36 + (\u20139)\n 16. (\u201336) + 9\n 17. (\u201336) + (\u20139)\n 18. 36 \u2013 9\n 19. 36 \u2013 (\u20139) \n 20. (\u201336) \u2013 9 \n 21. (\u201336) \u2013 (\u20139)\n 22. 36 \u2022 9 \n 23. 36 \u2022 (\u20139) \n 24. (\u201336) \u2022 9 \n 25. (\u201336) \u2022 (\u20139)\n 26. 36 \u00f7 9 \n 27. 36 \u00f7 (\u20139) \n 28. (\u201336) \u00f7 9 \n 29. (\u201336) \u00f7 (\u20139)\n 30. 5 \u2022 [10 \u2013 (\u20132)]\n 31. (\u20139) + 10 + (\u201311) \n 32. 100 \u2013 (\u201325) \u2013 50 \n 33. 20 \u2022 (\u201325) \u00f7 (\u20135) \n 34. 15 \u2022 30 + (\u201310) \n 35. 30 \u00f7 (\u20132) \u2013 (\u201315)\n 36. 3 \u2022 [17 \u2013 (\u201313)] \u00f7 (\u20132)"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p92-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 92, "chunk_number": 0, "type": "text", "chroma:document": "83\nLearning \nObjectives\nAt the end of the \nlesson, you should be \nable to:\n\u2022\t define\trational\t\nnumbers\n\u2022\t graph\trational\t\nnumbers\ton\ta\t\nnumber\tline\t\n\u2022\t compare\tand\torder\t\nrational\tnumbers\n\u2022\t state\tthe\tdensity\t\nproperty\tof\t rational \t\nnumbers\n\u2022\t express\tfractions\tas\t\ndecimals,\tand\tvice\t\nversa\n\u2022\t perform\toperations\t\ninvolving\trational\t\nnumbers\n\u2022\t simplify\tcomplex\t\nfractions\n\u2022\t solve\tword\tproblems\t\ninvolving\trational\t\nnumbers\nRational Numbers\nProbe and Learn\nPlants grow new cells in spirals so that the new leaves do \nnot block the older leaves from getting sunlight. Each new cell \nis formed about 137.5\u00ba away from the previous cell. This angle \nis called the golden angle. In geometry, the golden angle  is the \nsmaller of the two angles formed by sectioning the circumference \nof a circle based on the golden ratio.\nThe number 137.5 is the same as the fraction \n275\n2 , which is \na quotient of two integers. Recall that the set of integers is closed"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p90-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 90, "chunk_number": 1, "type": "text", "chroma:document": "SY 2010\u22122011?\nAnswer each item completely.\n 54.  What is the distance between the lowest point and the highest point in the \nPhilippines? Refer to the data given at the start of this lesson.\n 55.  In a 20-item test, a student receives 5 points for each question answered \ncorrectly. Two points are subtracted from the total score for each question \nanswered incorrectly. One point is subtracted for each unanswered question. \nIf a student has 10 correct answers and 6 incorrect answers, what is his total \nscore in the test?\n 56.  The Philippine national debt rose to about 4 trillion pesos in 2008. Suppose \nthat there are approximately 75 million Filipinos. What is the average debt of \neach Filipino?\n 57.  Nora records the temperatures in her place at different times. At midnight, \nthe temperature is 15\u00b0C. By 4 a.m., it drops by 6\u00b0C. It then begins to rise at a \nrate of 2\u00b0C every hour until noon, when the rate of increase becomes 3\u00b0C every"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p93-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 93, "chunk_number": 0, "type": "text", "chroma:document": "84\nSet of Rational Numbers\nThe set of rational numbers, denoted by Q, is composed of numbers that can be \nexpressed as a ratio of two integers. In symbols,\nQ = \n  \np\nq  p and q are integers, and q \u2260 0 .\nKey Concept\nFrom the given definition, you can say that fractions are rational numbers. Consider \nthe fraction \n1\n2, which is the ratio of the integers 1 and 2. Dividing 1 by 2 gives a decimal \nquotient of 0.5. Note that any fraction can be expressed either as a terminating decimal \nor a nonterminating but repeating decimal. \nFractions and Decimals\nA terminating decimal consists of a finite number of decimal digits. Numbers, \nsuch as 4, 1.2345, and 0.987654321, that have finite nonzero digits on the right of \nthe decimal point are terminating decimals. \nA nonterminating but repeating decimal has an infinitely repeating digit or group \nof digits. An ellipsis is used to denote that the decimal has an infinite number of"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p94-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 94, "chunk_number": 0, "type": "text", "chroma:document": "85\n Solutions:\na. Divide 1 by 4.\n   \n  Thus, 1\n4 = 0.25, which  \nis a terminating decimal.\nb. Divide 1 by 3.\n   \n  Therefore, \n1\n3 = 0.3, which is a \nnonterminating but repeating \ndecimal.\nexamPLe 2\nSome scientific studies say that smiling requires fewer muscles than frowning \ndoes\u201417 muscles to smile and 43 to frown. Write a fraction representing the \nnumber of needed muscles to smile as a part of the number of needed muscles to \nfrown. Then express the fraction as a decimal.\n Solution:\n  The number of needed muscles to smile as a part of the number of needed \nmuscles to frown can be represented as . To write this fraction as a decimal, \ndivide 17 by 43. The quotient is the nonterminating but repeating decimal \n0.3953488\u2026. Observe that the decimal does not seem to show a pattern in \nits first 7 digits. For a decimal like this, you may round it off to the nearest \nhundredths as 0.4.\nConversely, you can write any terminating or nonterminating but repeating"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p96-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 96, "chunk_number": 0, "type": "text", "chroma:document": "87\nexamPLe 4\nSimplify \n30\n45 .\n Solution:\n  First, find the prime factorization of 30 and 45. Then cancel out all of their \ncommon factors. \n \nSince 2 and 3 are relatively prime, \n2\n3 is the simplest form of \n30\n45 .\nNow, how do you write a nonterminating but repeating decimal as a fraction? \nNote that in this lesson, you will only consider decimals that are made up \nentirely of repeating digits. You will not deal with decimals that have a combination \nof nonrepeating digits and repeating digits. In the next example, you will learn a \nmethod for converting nonterminating but repeating decimals into fractions.\nexamPLe 5\nExpress each decimal as a fraction in simplest form.\na. 0.333\u2026 b. 0.545454\u2026\n Solutions:\n a. \nf 3 is the repeating digit.\nf\n W\nrite a 9 for every repeating digit.\nf\n Simplify 3\n9.\n0.333...\n b.  0.545454\u2026 f 54 is the repeating group of digits.\nf\n W\nrite a 9 for every repeating digit.\nf\n Simplify 54\n99."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p95-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 95, "chunk_number": 0, "type": "text", "chroma:document": "86\nexamPLe 3\nExpress each decimal as a fraction. Write your answer in simplest form.\na. \u20131.2 b. 0.625\n Solutions:\n a. \u20131.2 is read as \u201cnegative 1 and 2-tenths.\u201d\n   \n\u20131.2 = \u20131\n2\n10\n = \u20131\n1\n5\n\u2190 The single decimal place in \u20131.2 corresponds to 1 zero.\n  Hence, \u20131.2 = \u20131\n1\n5.\n b. 0.625 is read as \u201c625-thousandths.\u201d\n   \n\u2190 The three decimal places in 0.625 correspond to 3 zeros.0.625 = \n625\n1,000\n \n= \n5\n8\n  Therefore, 0.625 = \n5\n8.\nYou can express a rational number as a fraction in infinitely many ways. How do \nyou know if a fraction is already in its simplest form?\nA fraction is in simplest form if its numerator and denominator are relatively \nprime; that is, their greatest common factor (GCF) is 1. For example, the fraction \n5\n7 \nis in simplest form since 5 and 7 are relatively prime. \nTo simplify fractions, you can apply the Fundamental Law of Fractions.\nThe Fundamental Law of Fractions\nIf a, b, and c are real numbers, b \u2260 0 and c \u2260 0, then ac\nbc = a\nb.\nKey Concept"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p97-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 97, "chunk_number": 0, "type": "text", "chroma:document": "88\nexamPLe 6\nPlot each rational number on a number line.\na. b. \u20132.3\n Solutions:\n a. You know that \n   To plot  on a number line, divide the units of the number line into \nhalves.\n  \n\u20131 0 1 2 3 4 5\n7\n2\n b. \u20132.3\n   To plot \u20132.3 on a number line, divide the units of the number line into \ntenths.\n  \n\u20134 \u20133\n\u20132.3\n\u20132 \u20131 0\nComparing Rational Numbers\nCan you tell how many rational numbers there are between \n1\n4  and \n1\n3? The answer \ncan be explained using the next property of rational numbers.\nDensity Property of Rational Numbers\nBetween any two rational numbers, there is always another rational number.\nKey Concept\nThis means that between any two rational numbers, there are infinitely many \nrational numbers. \nKnowing the positions of rational numbers or their decimal representations on a \nnumber line may help you compare fractions. However, you may not need to plot the \nnumbers on a number line to be able to compare them. The following are the rules \nwhen comparing rational numbers:"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p90-c2-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 90, "chunk_number": 2, "type": "text", "chroma:document": "the temperature is 15\u00b0C. By 4 a.m., it drops by 6\u00b0C. It then begins to rise at a \nrate of 2\u00b0C every hour until noon, when the rate of increase becomes 3\u00b0C every \nhour. At 4 p.m., the temperature begins to decrease by 2\u00b0C every hour. What \nis the temperature at 6 p.m.?"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p94-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 94, "chunk_number": 1, "type": "text", "chroma:document": "its first 7 digits. For a decimal like this, you may round it off to the nearest \nhundredths as 0.4.\nConversely, you can write any terminating or nonterminating but repeating \ndecimal as a fraction. Study the next examples."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p92-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 92, "chunk_number": 1, "type": "text", "chroma:document": "of a circle based on the golden ratio.\nThe number 137.5 is the same as the fraction \n275\n2 , which is \na quotient of two integers. Recall that the set of integers is closed \nunder addition, subtraction, and multiplication, but not under \ndivision. For example, the quotient of the integers 275 and 2 is \nnot an integer. Such quotients are part of another subset of the \nset of real numbers\u2014the set of rational numbers. \nLESSON\n2-3"}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p97-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 97, "chunk_number": 1, "type": "text", "chroma:document": "number line may help you compare fractions. However, you may not need to plot the \nnumbers on a number line to be able to compare them. The following are the rules \nwhen comparing rational numbers:\n\u2022  When comparing two positive rational numbers, compare as you would with \nwhole numbers."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p96-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 96, "chunk_number": 1, "type": "text", "chroma:document": "f\n W\nrite a 9 for every repeating digit.\nf\n Simplify 3\n9.\n0.333...\n b.  0.545454\u2026 f 54 is the repeating group of digits.\nf\n W\nrite a 9 for every repeating digit.\nf\n Simplify 54\n99.\nKnowing how to express rational numbers as fractions or as decimals can help \nyou plot them on a number line. Study the next example."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p95-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 95, "chunk_number": 1, "type": "text", "chroma:document": "To simplify fractions, you can apply the Fundamental Law of Fractions.\nThe Fundamental Law of Fractions\nIf a, b, and c are real numbers, b \u2260 0 and c \u2260 0, then ac\nbc = a\nb.\nKey Concept\nThis property means that if the numerator and the denominator of a fraction  \nhave a common factor (c), you can cancel out the common factor without changing \nthe value of the fraction. Also, when you cancel the GCF of the terms of the fraction, \nthe result is the simplest form of the fraction."}


created_at: 2025-05-16 10:05:20
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p80-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 80, "chunk_number": 1, "type": "text", "chroma:document": "that positive integers are counting numbers. This means that the sum of two positive \nintegers is also positive.\nHow about the sum of two negative integers? Or the sum of a positive integer and \na negative integer? You could better understand how to add integers if you use a \nnumber line. Study how it could be done. \nFirst, locate one of the addends on the number line. Suppose that n is the other \naddend.\nIf n is positive, move |n| units to the right of the first addend.\nIf n is negative, move |n| units to the left of the first addend.\nStudy the next example.\nexamPLe 6\nFind each sum.\na. (\u20134) + 7 b. (\u22124) + (\u22127)\nA sari-sari store is a small convenience store where \nyou can buy items by piece or by small packages."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p93-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 93, "chunk_number": 1, "type": "text", "chroma:document": "A nonterminating but repeating decimal has an infinitely repeating digit or group \nof digits. An ellipsis is used to denote that the decimal has an infinite number of \ndigits. For example, 0.33333... has an infinite number of 3s on the right of the \ndecimal point, while the decimal 0.456456456456\u2026 has an infinitely repeating \ngroup of digits, 456. The repeating digit or group of digits may also be written once \nwith a bar on top, called a vinculum. Thus, you can rewrite 0.3333... as 0. 3 and \n0.456456... as 0.456.\nexamPLe 1\nWrite each fraction as a decimal.\na. 1\n4\nb. 1\n3"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p98-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 98, "chunk_number": 0, "type": "text", "chroma:document": "89\n\u2022  Any negative rational number is less than any positive rational  number.\n\u2022  When comparing two negative rational numbers, the greater the absolute \nvalue of a number, the lesser the number.\nexamPLe 7\nCompare each pair of numbers using <, >, or =.\na. 5.05 ____ 5.005 b. \u20132.5 ____ 1.5 c. \u20136.4 ____ \u201310.2\n Solutions:\n a. 5.05 > 5.005   The hundredths digit of 5.05 is greater than the hundredths digit of 5.005\n b. \u20132.5 < 1.5  Any negative rational number is less than any positive rational number.\n c. \u20136.4 > \u201310.2  Since |\u20136.4| < |\u201310.2|, then \u20136.4 > \u201310.2.\nYou have learned that to compare fractions, you may need to rename the \nfractions as similar fractions using the Least Common Denominator (LCD) as their \ndenominator. Recall that the LCD of the fractions is the least common multiple of the \ndenominators. You may then compare the numerators of the similar fractions. The \nfraction with a greater numerator is the greater fraction.\nexamPLe 8\nCompare \n4\n9  and \n5\n12 .\n Solution:"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p100-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 100, "chunk_number": 0, "type": "text", "chroma:document": "91\nOrdering Rational Numbers\nYou already know how to compare rational numbers. Now you are ready to order \nrational numbers from least to greatest, or vice versa. Study the next example.\nexamPLe 10\nOrder the following rational numbers from least to greatest:\n1.5, \n10\n7 , 1\n7\n8 , \n4\n9, 0.3, \u20130.3, \u2013\n1\n3\n Solution:\n  Observe that two of the given numbers are negative. Recall that negative \nnumbers are less than positive numbers. So, which is less, \u20130.3 or \u2013\n1\n3? To \nanswer it, write the numbers in the same form, say decimal form. You may \nexpress the fraction \u2013\n1\n3  as \u20130.3. Thus, \u2013\n1\n3 < \u20130.3.\n  Now consider the five remaining numbers: 1.5, \n10\n7 , 1\n7\n8, \n4\n9, and 0.3. You can \nsee that 0.3 < 1.5, since both are in decimal form.\n  Of the three remaining fractions, \n4\n9 is the least since it is a proper fraction, \nwhich means it is less than 1.\n  On the other hand, \n10\n7  = 1\n3\n7, which is less than 1\n7\n8. Thus, the three fractions \nmay be written from least to greatest as \n4\n9, 1\n3\n7, 1"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p99-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 99, "chunk_number": 0, "type": "text", "chroma:document": "90\nAnother way to compare fractions is by performing cross-multiplication. The \nfollowing diagrams illustrate how cross-products are used to compare the fractions \na\nb and \nc\nd:\na \u2022 d  >  c \u2022 b\na\nb   \n> \n \nc\nd\na \u2022 d  <  c \u2022 b\na\nb  \n< \n \nc\nd\na \u2022 d  =  c \u2022 b\na\nb  \n= \n \nc\nd\nWhy can you compare fractions by comparing the resulting cross-products?\nexamPLe 9\nCompare each pair of fractions by getting their cross-products.\na. 5\n8  \u2013\u2013\u2013\u2013\u2013 6\n7 b. \u2013  5\n11  \u2013\u2013\u2013\u2013\u2013 \u2013 4\n9 \n Solutions:\n a. Compare the cross-products.\n  \n5 \u2022 7  =  35  <  6 \u2022 8  =  48\n5\n8   \n< \n \n6\n7\n  Since 35 < 48, then .\n b.  Notice that both fractions are negative. You can write a negative rational \nnumber in any of the following forms:\n  \n\u2013a\nb , a\n\u2013b , or \u2013 a\nb .\n  Hence, you may rewrite \u2013 5\n11 as \u20135\n11 and \u2013 4\n9  as \u20134\n9 . \n  (\u20135) \u2022 9  =  \u201345  <  (\u20134) \u2022 11  =  \u201344\n  Since \u201345 < \u201344, then ."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p101-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 101, "chunk_number": 0, "type": "text", "chroma:document": "92\nAddition and Subtraction of Rational Numbers\nRecall that when adding or subtracting decimals, you have to align the decimal \npoints and the digits having the same place value.\nexamPLe 11\nAdd or subtract.\na. 10.23 + 5.3 + 27.456 b. 11.346 + (\u20133.09) c. \u201320.5 \u2013 7.12\n Solutions:\n a.  10.23\n \n5.3\n+\n 27.456\n 42.986\nb.  11.346\n+\n (\u20133.09) \n \n 8.256\nc.  \u2013 20.5\n\u2013\n 7.12\n \uf067  \u201320.50\n+\n \n(\u20137.12)\n \u201327.62\nRecall that when adding or subtracting similar fractions, add or subtract the \nnumerators. Then copy the denominator. Express the sum or difference in simplest \nform, if necessary.\nexamPLe 12\nAdd or subtract.\na. b. \n Solutions:\n a.  = \n   =  Rename 9\n7  as 1 2\n7.\n b.   \uf067  Rename 4 4\n15  as 319\n15. Why?\n   \n  Thus, , which may be written in simplest form as ."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p102-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 102, "chunk_number": 0, "type": "text", "chroma:document": "93\nTo find the sum or the difference of dissimilar fractions, rename the fractions using \nthe LCD. Then follow the rules in adding and subtracting similar fractions. \nexamPLe 13\nAdd or subtract.\na. b. c. \n Solutions:\n a.  =   The LCD of 10 and 15 is 30.\n   =  \n b.  =   Rename 10 1\n2  as 9 3\n2 \n   =  \n   =  \n c.  =   Rewrite it into an addition problem.\n   =  \n   =  \nNow study the next example that shows how to solve a word problem involving \naddition and subtraction of fractions.\nexamPLe 14\nAmong the Philippines\u2019s major agricultural exports are canned pineapples. Jose \nowns a pineapple farm. He wants to fence his rectangular farm, which has a \nlength of  ft. and a width of 21 1\n3  ft. Suppose that fencing wires are sold in \nrolls of 20 yards. How many rolls of wire should he buy? How many feet of wire \nwill be left over?"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p104-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 104, "chunk_number": 0, "type": "text", "chroma:document": "95\nexamPLe 15\nMultiply.\na. 24.68 \u00d7 0.5 b. (\u2013213.6) \u00d7 (\u20132.5)\n Solutions:\n a.  24.68\n\u00d7\n \n0.5\n \n12.340\n\u2190  2 decimal places\n\u2190 1 decimal place\n\u2190 3 decimal places\n  You may write the product as 12.34. Why?\n b.   \u2013213.6\n\u00d7\n \n  (\u20132.5)\n \n534.00\n\u2190  1 decimal place\n\u2190 1 decimal place\n\u2190 2 decimal places\n  The product is positive. Why?\nRecall that when multiplying fractions, multiply the numerators to get the \nnumerator of the product. Then multiply the denominators to get the denominator \nof the product.\nIf the numerator and the denominator of the fractions that you need to multiply \nhave common factors, you may cancel out the common factors first before multiplying \nthe fractions.\nWhen multiplying mixed numbers, write each mixed number as an improper \nfraction before multiplying.\nexamPLe 16\nMultiply. Write your answer in simplest form.\na. b. \n Solutions:\n a.  Write \u20136 6\n7 as \u2013 48\n7 .\n    Since the factors have unlike signs, attach a negative sign \nto the product."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p103-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 103, "chunk_number": 0, "type": "text", "chroma:document": "94\n Solution:\n  To find the length of the required wire to fence his rectangular farm, find \nthe perimeter of the farm. Recall that you can solve for the perimeter P of a \nrectangle by adding the measures of all its sides.\n P  =  l + l + w + w, where l  =  length and w  =  width\n It is given that l = 32\n1\n4  ft. and w = 21\n1\n3 ft. So you will have the following:\n \n  Thus, Jose needs 107\n1\n6 ft. of fencing wire. Recall that a roll of fencing wire is \n20 yards long. Since 1 yd. = 3 ft., then 20 yd. = 60 ft. Thus, he needs 2 rolls of \nfencing wire, with a total length of 120 ft.\n  To find how much of the wire will be left over after fencing  his farm, subtract \n107\n1\n6  ft. from 120 ft.\n \n Therefore, 12\n5\n6 ft. of wire will be left over.\nMultiplication of Rational Numbers\nRecall that when multiplying decimals, multiply the numbers as you would \nwith whole numbers. To determine the position of the decimal point in the product,"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p98-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 98, "chunk_number": 1, "type": "text", "chroma:document": "denominators. You may then compare the numerators of the similar fractions. The \nfraction with a greater numerator is the greater fraction.\nexamPLe 8\nCompare \n4\n9  and \n5\n12 .\n Solution:\n  The LCD of \n4\n9 and \n5\n12  is 36. Find fractions equivalent to the given fractions \nusing the LCD as denominator.\n \n\u00d7 4\n\u00d7 4   \nand\n  \n\u00d7 3\n\u00d7 3  \n Compare the numerators of the similar fractions. Since 16 > 15, then ."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p100-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 100, "chunk_number": 1, "type": "text", "chroma:document": "which means it is less than 1.\n  On the other hand, \n10\n7  = 1\n3\n7, which is less than 1\n7\n8. Thus, the three fractions \nmay be written from least to greatest as \n4\n9, 1\n3\n7, 1\n7\n8. \n  Now consider 0.3 and 1.5. Is 0.3 less than \n4\n9? You can express 0.3 as \n3\n10 , \nwhich is less than \n4\n9. Since 1.5 = 1\n1\n2, then 1\n3\n7 < 1\n1\n2 < 1\n7\n8. Thus, the  \nfollowing is the correct order of the numbers from least to greatest:\n \u2013\n1\n3, \u20130.3, 0.3, \n4\n9, \n10\n7 , 1.5, 1\n7\n8"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p106-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 106, "chunk_number": 0, "type": "text", "chroma:document": "97\nStudy the next example that shows how to solve a word problem involving division \nof decimals.\nexamPLe 18\nLightning is a massive flow of electric current in the atmosphere. A bolt of lightning \ncan reach temperatures approaching 30,000\u00b0C. It can contain one billion volts of \nelectricity. \nIn electric devices, the amount of electric current can be computed using the \nformula I = \nV\nR , where I is the electric current (in amperes), V is the voltage  \n(in volts), and R is the resistance (in ohms). Compute the electric current flowing \nthrough a light bulb with a resistance of 10.5 ohms and that is plugged on a  \n110-volt outlet.\n Solution:\n  To solve for the amount of electricity flowing through the light bulb, substitute \nthe given values of V and R in the formula for I, then solve.\n It is given that R = 10.5 ohms and V = 110 volts. So you will have: \n I = \n110\n10.5\n To find I, divide 110 by 10.5.\n \n  Hence, the amount of electricity flowing through the light bulb is about"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p108-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 108, "chunk_number": 0, "type": "text", "chroma:document": "99\nexamPLe 20\nIn 2003, Dagupan City in Pangasinan is the record holder of the world\u2019s longest \nbarbecue. The barbecue measures about 3,300 ft. It was composed of milkfish \nplaced together on grills arranged side by side. Suppose that a milkfish has an \naverage width of 3\n3\n4  in. About how many pieces of milkfish were used for the \nbarbecue?\n Solution:\n  To find the number of milkfish, divide the total length of the barbecue by the \nwidth of each milkfish. Since the units are not the same, you may convert \n3,300 ft. to inches. Since 1 ft. = 12 in., then 3,300 ft. = 39,600 in.\n Now divide 39,600 by 3 3\n4.\n \n Therefore, there were about 10,560 pieces of milkfish in the longest b arbecue.\nexamPLe 21\nSimplify the following complex fraction: .\n Solution:\n  One way to simplify the given expression is to express the numerator as a \nsingle fraction, and then to treat the fraction bar as a division symbol.\n  \nAdd the fractions in the numerator.\n  \nThe fraction bar means \u00f7."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p105-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 105, "chunk_number": 0, "type": "text", "chroma:document": "96\n b.  Write \u20132 5\n8   as  \u2013 21\n8   and  \u20133 3\n7  as  \u2013 24\n7 .\n    \n     Cancel out the common factors.\nexamPLe 17\nJam is a call center agent. She earns P85 per hour for the first 8 hours of \nwork. She also earns an overtime pay of 1.5 times the hourly rate for each hour in \nexcess of 8 hours. Suppose that Jam works for 10 hours each day from Monday \nto Wednesday. She also works for 11 hours each day on Thursday and Friday. \nHow much is her salary for the given 5-day period?\n Solution:\n  To find Jam\u2019s salary for the given period, compute for her salary for each \nspecified day. It is given that for the first 8 hours, she is paid P85 per hour. \nHer overtime pay per hour is P85 \u2022 1.5 = P127.50.\nDay Hours  \nof Work \nPay for the First 8 Hours \n(in pesos)\nOvertime Pay \n(in pesos)\nMonday 10 85 \u2022 8 = 680 127.50 \u2022 2 = 255\nTuesday 10 85 \u2022 8 = 680 127.50 \u2022 2 = 255\nWednesday 10 85 \u2022 8 = 680 127.50 \u2022 2 = 255\nThursday 11 85 \u2022 8 = 680 127.50 \u2022 3 = 382.50\nFriday 11 85 \u2022 8 = 680 127.50 \u2022 3 = 382.50"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p103-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 103, "chunk_number": 1, "type": "text", "chroma:document": "Multiplication of Rational Numbers\nRecall that when multiplying decimals, multiply the numbers as you would \nwith whole numbers. To determine the position of the decimal point in the product, \ncount the decimal places in the factors. The total number of decimal places in the \nfactors is equal to the number of decimal places in the product. Study the given \nexample."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p109-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 109, "chunk_number": 0, "type": "text", "chroma:document": "100\n  \nMultiply the numerator by the reciprocal \nof the divisor, and then simplify.\nPractice 2-3\nConcepts and Skills\nGive one rational number between each pair of rational numbers.\n 1.  and\n  \n2.  and \u20130.428\nPlot each set of rational numbers on a number line. Then order the numbers from \nleast to greatest.\n \n3.\n \n\u20131.9, 2.5,\n \n4\n3,\n \n17\n9  \n \n4.  \n1\n4, \n3\n7, \n4\n19,\n \n7\n20 \n 5.  0.16, 0.1625, 0.163, 0.16\nWrite each decimal as a fraction. Express your answer in simplest form.\n 6.  0.8\n 7. \u20136.75\n 8.  10.35\n 9. 5.66\u2026\n 10. \u20130.3636\u2026\nWrite each fraction as a decimal.\n \n11. \n5\n12\n 12. \u2013\n4\n9\n 13. \n18\n7\n 14. 3\n1\n2\n 15. \u201310\n3\n4"}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p110-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 110, "chunk_number": 0, "type": "text", "chroma:document": "101\nWrite each fraction in simplest form.\n \n16. \n24\n54\n 17. \n3\n51\n 18. \n\u201372\n98\n 19. \u2013\n56\n108\n 20. \n26\n143\nPerform the indicated operations. Express your answer in simplest form. \n 21.  30. \n 22.  31. \n 23.  32. \n 24.  33. \n 25.  34. \n 26.  35. (\u20135.5)(1.09 + 9.91)\n 27.  36. \n 28.  37. \n 29."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p107-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 107, "chunk_number": 0, "type": "text", "chroma:document": "98\nRecall that when dividing fractions, multiply the dividend by the reciprocal or \nmultiplicative inverse of the divisor.\n.\nStudy the next example about solving a word problem involving division of \nfractions.\nexamPLe 19\nLJ, an interior designer, uses patterns to create quilts. \nOne of her designs is shown on the right. What part of \nthe square quilt is the white triangle?\n Solution:\n  Note that the largest part in her quilt is \n1\n2 . The next \nlargest part is obtained by dividing the remaining \npart of the triangle into 2; that is:\n  \n1\n2 \u00f7 2 =  \n1\n2 \u2022 \n1\n2\n   =  \n1\n4\n The next fraction is also obtained in the same way; that is:\n \n1\n4 \u00f7 2 = \n1\n4 \u2022 \n1\n2 = \n1\n8\n So, to solve for the required fraction, divide \n1\n64 by 2.\n \n1\n64 \u00f7 2 = \n1\n64 \u2022 \n1\n2 = \n1\n128\n Hence, the white triangle is \n1\n128  of the square quilt.\nSimplifying Complex Fractions \nIn some cases, a fraction may contain one or more fractions in its numerator \nand/or denominator. Such a fraction is called a complex fraction."}


created_at: 2025-05-16 10:05:21
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p111-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 111, "chunk_number": 0, "type": "text", "chroma:document": "102\nApplications\nAnswer each item completely.\n 38. Mitch scores 45 out of 62 in a Math exam. She scores 36 out of 50 items in an \nEnglish exam. In which subject is her score better?\n 39. Engineers set standard measurements for hand tools such as wrenches of \ndifferent sizes. Engineer Joy has four wrenches with sizes \n9\n16, \n11\n16, \n1\n2, and \n5\n8 \nof an inch. Suppose that she needs the wrench with the second largest size. \nWhich of the wrenches should she use?\n 40. A school year usually has 200 days. There are 365 days in 1 ordinary year. What \nfraction could represent a school year as part of an ordinary year? Express the \nfraction in simplest form. \n 41. The brain of a newborn baby weighs about 400 g. Suppose that a newborn \nbaby weighs 3.5 kg. What fraction of the baby\u2019s weight is his or her brain? \n 42. A hair strand is approximately \n1\n450 in. thick. What is the equivalent value of \nthis fraction in decimal form?"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p112-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 112, "chunk_number": 0, "type": "text", "chroma:document": "103\n 47.  Mr. Angeles shared \n1\n2  of his estate with his wife. Also, he shared \n1\n8 with each \nof his two children, and \n1\n16 with each of his three grandchildren. He donated \nthe remaining P30,000 worth of property to a charity. What is the value of his \nentire estate?\n 48.  A carpenter will cut a 42\n3\n4-in. long board into 5 pieces of equal lengths.  \nIf \n1\n16 in. of board wears down in every cut, how long will each piece be?\n 49. Suppose that 0.408 kg of grapes costs P65.28. How much is 1 kg of grapes?\n 50.  The following table shows the monthly average exchange rate of US dollar to \npeso in the first six months of the year 2016:\nMonth Peso Equivalent \nof 1 US dollar\nJanuary 47.73\nFebruary 47.45\nMarch 46.64\nApril 46.20\nMay 46.67\nJune 43.32\n  What is the average exchange rate for the first six months of that year?\nEnrichment Exercises\nAnswer each item completely.\n 51. Can you find unique integers a, b, and c such that ? If yes, give \na possible set of integers. If no, explain why."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p114-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 114, "chunk_number": 0, "type": "text", "chroma:document": "105\nLearning \nObjectives\nAt the end of the \nlesson, you should be \nable to:\n\u2022\t define\tirrational\t\nnumbers\n\u2022\t locate\tan\tirrational\t\nnumber\ton\ta\t\nnumber\tline\n\u2022\t find\tthe\tsquare\t\nroots\tof\t positive \t\nrational\tnumbers\nProbe and Learn\nRainbows are among the most beautiful natural phenomena. \nA rainbow is formed when light shines through water droplets, \nusually in the form of rain, and diffracts into different colors. A \nrainbow is not a 360\u00ba full circle; it is just a 42\u00ba arc around a spot \nopposite the Sun. \nTo explain natural phenomena that exhibit circles or spheres, \nthe number p (pi) is usually used. This number is an example of \nan irrational number.\nUsing a scientific calculator, you can \nget the value of p  by pressing the p  button. \nA 12-digit scientific calculator will display \n3.14159265359, while a 15-digit calculator will \nshow 3.14159265358979. Notice that when you \nuse a calculator that can display more digits, the \nnumber of displayed digits of p also increases."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p106-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 106, "chunk_number": 1, "type": "text", "chroma:document": "It is given that R = 10.5 ohms and V = 110 volts. So you will have: \n I = \n110\n10.5\n To find I, divide 110 by 10.5.\n \n  Hence, the amount of electricity flowing through the light bulb is about  \n10.48 amperes."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p105-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 105, "chunk_number": 1, "type": "text", "chroma:document": "Tuesday 10 85 \u2022 8 = 680 127.50 \u2022 2 = 255\nWednesday 10 85 \u2022 8 = 680 127.50 \u2022 2 = 255\nThursday 11 85 \u2022 8 = 680 127.50 \u2022 3 = 382.50\nFriday 11 85 \u2022 8 = 680 127.50 \u2022 3 = 382.50\nTotal 3,400 1,530\n  Therefore, Jam\u2019s total salary for the given period is P3,400 + P1,530 = P4,930. \nCould you think of another way of solving the previous problem?\nDivision of Rational Numbers \nRecall the following steps when dividing a number by a decimal: \nStep 1. Make the divisor a whole number. You may do this by moving the decimal \npoint of the divisor to the right until you have a whole number divisor. \nStep 2. Move the decimal point of the dividend the same number of places to the \nright. \nStep 3. Divide as you would with whole numbers. Note that the decimal point of \nthe quotient is placed directly above the decimal point of the dividend."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p115-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 115, "chunk_number": 0, "type": "text", "chroma:document": "106\nSet of Irrational Numbers\nThe set of irrational numbers is the set of numbers that cannot be expressed as p\nq, \nwhere p and q are integers, and q \u2260 0.\nKey Concept\nTo better understand the set of irrational numbers, you need to learn how to find \nthe square roots of positive rational numbers.\nSquare Roots of Positive Rational Numbers\nInteresting patterns may be discovered by representing numbers with figures. \nLook at the following figure: \nObserve how dots were drawn inside each square. Do you notice a pattern? Draw \ndots in the next two squares based on the pattern. Then complete the following table:\nNumber of Dots along \nEach Side of the Square\nTotal Number  \nof Dots \n1 1\n2 4\n3 9\n4 16\n5\n6\n7\n8\n9\n10\n \nThe total number of dots in each square represents a perfect square. The \nnumber of dots in each row or column in the square is called a square root of the \nperfect square. For example, 16 is a perfect square and 4 is its square root.\nDo you know?"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p113-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 113, "chunk_number": 0, "type": "text", "chroma:document": "104\n 53. Find the sum of the following expression:\n  .\n 54. Consider the following equation:\n  \n  Is the equation correct? Explain.\n 55. The following figure is called a magic square:\n6\n1\n4 11\n1\n4\n7\n1\n2\n8\n3\n4\n  Complete the magic square. Each row, column, and diagonal must add up to \nthe same number."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p111-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 111, "chunk_number": 1, "type": "text", "chroma:document": "baby weighs 3.5 kg. What fraction of the baby\u2019s weight is his or her brain? \n 42. A hair strand is approximately \n1\n450 in. thick. What is the equivalent value of \nthis fraction in decimal form?\n 43. Jose has 4 pieces of wire. Two of the pieces of wire measure 9 in. long each. The \nother 2 pieces measure 1\n1\n4  ft. each. If he puts the pieces of wire together, how \nlong is the wire in inches?\n 44.  Dennis has finished painting \n1\n3 of his room using 2\n1\n2 cans of paint. How many \nmore cans of paint will he need to finish painting his room?\n 45.  The fuel tank of Trina\u2019s car is \n2\n3 full. To travel to her destination, she needs \na \n4\n5-full tank. What part of the fuel tank does she need to fill to reach her \ndestination?\n 46. There are 18 baskets to be filled with different kinds of fruits. Suppose that \nthere are 1\n1\n2 dozen pineapples, 6 dozen apples, 3 dozen oranges, 7\n1\n2 dozen \nbananas, and 9 dozen mangoes. If each basket should have the same contents,"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p107-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 107, "chunk_number": 1, "type": "text", "chroma:document": "Simplifying Complex Fractions \nIn some cases, a fraction may contain one or more fractions in its numerator \nand/or denominator. Such a fraction is called a complex fraction.\nComplex fractions may involve a series or combination of operations on rational \nnumbers. In such cases, the GEMDAS rule may be applied. Recall that GEMDAS  \nstands for Grouping Symbols, Exponents, Multiplication, Division, Addition, and \nSubtraction. Study the next examples.\n1\n2\n1\n4\n1\n81\n16\n1\n32\n?1\n64"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p112-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 112, "chunk_number": 1, "type": "text", "chroma:document": "Enrichment Exercises\nAnswer each item completely.\n 51. Can you find unique integers a, b, and c such that ? If yes, give \na possible set of integers. If no, explain why.\n 52.  Look at the following method of renaming fractions:\n  Can you find other fractions for which this method works? Also, give an example \nfor which this method does not work and explain why."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p114-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 114, "chunk_number": 1, "type": "text", "chroma:document": "3.14159265359, while a 15-digit calculator will \nshow 3.14159265358979. Notice that when you \nuse a calculator that can display more digits, the \nnumber of displayed digits of p also increases. \nThis indicates that p is nonterminating. Recall \nthat nonterminating but repeating decimals \nare classified as rational numbers. Take note, \nhowever, that nonterminating decimals, such \nas p, that do not show a repeating pattern of \ndigits are irrational numbers.\nIrrational NumbersLESSON\n2-4"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p116-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 116, "chunk_number": 0, "type": "text", "chroma:document": "107\nIf a and b are nonnegative numbers and b2 = a, then b is a square root of a. In symbols, \nit can be written as:\n a = b.\nKey Concept\nConsider  = 4, which is true since 4 2 = 4 \u2022 4 = 16. The number 16 is called \nthe radicand. The number 4 is the square root. However, note that (\u20134)(\u20134) is also \nequal to 16. Does this mean that \u20134 is also a square root of 16? \nThe expression  is used to refer specifically to the principal square root of x, \nor the nonnegative square root of x. While it is true that (\u20134)(\u20134) = 16, \u20134 is not the \nprincipal square root of 16, but the negative square root of 16.\n  = 4 \u2192  4 is the principal square root of 16.\n \u2013  = \u22124 \u2192   \u20134 is the negative square root of 16.\nPerfect squares also include decimals and fractions. An example is 0.04 since \n0.04 = (0.2)2. Another example is \n9\n16 since .\nexamPLe 1\nFind each indicated square root.\na. b. \n Solutions:\n a.  152 = 225\n b.  \u2013  indicates the negative square root. \nexamPLe 2"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p117-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 117, "chunk_number": 0, "type": "text", "chroma:document": "108\n Replacing A with 400 in the given formula gives . \n  Since 400 = 202 and 400 = (\u201320)2, then s has two possible values: 20 or \u201320. \nHowever, since s represents the measure of each side of the room, it cannot be \nnegative. Therefore, each side of the room measures 20 ft.\nYou have seen that extracting the square root of a perfect square yields a rational \nnumber. Consider another situation.\nA dressmaker designs a cover for square tables that \nmeasure 2 ft. on each side. She drafts the design shown \non the right.\nSince the square table measures 2 ft. on each side, \nthe amount of cloth that she needs for each cover is  \n4 sq. ft. Connecting the midpoints of the sides of the \nsquare forms a small square, which is the shaded part. \nSuch small square has an area of 2 sq. ft. What could be \nthe measure of each side of the smaller square?\nSince the area of the small square is 2 sq. ft., the measure of its side must be a"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p119-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 119, "chunk_number": 0, "type": "text", "chroma:document": "110\n Step 2. Divide the radicand by any of the square roots you obtained in step 1. \nSuppose that you divide 8 by 3.\n   8 \u00f7 3 \u2248 2.67\n   (The quotient may be rounded to the nearest hundredths.)\n Step 3.  Find the average of the quotient in step 2 and the square root you \nused as a divisor. \n   So get the average of 2.67 (which is the quotient) and 3 (which is the \ndivisor).\n   \n Step 4.  Check if the value you obtained in step 3 is a good estimation of the \nrequired square root. To do this, get the square of the obtained value. \nCompare it with the given radicand by getting the absolute value of \nthe difference of the result and the radicand. \n   (2.835)2 \u2248 8.0372\n   The difference between this result and the radicand is: \n   .\n   You can check if you can still find a closer estimation. Do the next \nstep.\n Step 5.  Repeat steps 2 and 3, but this time, use the value you obtained in \nstep 3 as the divisor.\n   So divide 8 by 2.835.\n   8 \u00f7 2.835 \u2248 2.822"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p115-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 115, "chunk_number": 1, "type": "text", "chroma:document": "number of dots in each row or column in the square is called a square root of the \nperfect square. For example, 16 is a perfect square and 4 is its square root.\nDo you know?\nSome\tnumber s \tare \treg arded \tas \thappy \nnumbers.\t Consider \t the \t number \t 13. \t\nAdding\tthe \tsquares \tof\tthe \tdigits \tof\t13 \t\ngives\t12\t+\t 32\t\t= \t\t10. \tR epeating \tthe \tprocess \t\nwith\tt he \tdigits \tof\t10 \tyields \t12\t+\t 02\t\t= \t\t1. \tSince\t\nthe\tresulting \tnumber \tis \t1, \tthen \t13 \tis \ta \t\nhappy\tnumber . \tF or \tan y \tha ppy \tnumber , \tthe \t\nresulting\tnumber \tshould \tbe \t1 \tafter \ta \tfinite \t\nnumber\tof\t steps.\nHow\tman y \tha ppy \tnumber s \tless \tthan \t50 \t\ncan\tyou\tfind?"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p111-c2-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 111, "chunk_number": 2, "type": "text", "chroma:document": "there are 1\n1\n2 dozen pineapples, 6 dozen apples, 3 dozen oranges, 7\n1\n2 dozen \nbananas, and 9 dozen mangoes. If each basket should have the same contents, \nhow many pieces of each fruit must be placed in each basket?"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p120-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 120, "chunk_number": 0, "type": "text", "chroma:document": "111\nKnowing how to approximate square roots of non-perfect squares can help you \nplot irrational numbers on a number line. From the previous example, since  is \napproximately equal to 2.829, you can plot it on the number line as follows:\n0 1 2 3\n8\nexamPLe 5\nOne of Manila\u2019s major attractions is \nthe breathtaking sunset view along the \nBaywalk at Manila Bay. The line that \nseems to separate the bay and the sky is \ncalled the horizon. How far you can see \nto the horizon can be determined using \nthe formula , where d is the \ndistance you can see (in miles), and h is \nthe height (in feet) your eyes are from the \nground. Suppose that your eyes are about \n5 ft. from the ground. Estimate how far \nyou can see to the horizon during a sunset \nif you are standing along the Baywalk. \n Solution:\n Use the formula , where h = 5.\n \n Thus, you can see to the horizon up to a distance of about 2.7 miles."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p118-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 118, "chunk_number": 0, "type": "text", "chroma:document": "109\nexamPLe 3\nFind a whole number approximation for 70.\n Solution:\n  Since 82 = 64 and 9 2 = 81, you can say that 70 is between the two perfect \nsquares, 64 and 81. Hence, 70 is between 8 and 9.\n \n 64 <  70  <  81\n 82 < 70  <  92\n 8 <   <  9\n  Between 8 and 9, which is a closer whole number approximation of 70? \nIn the following 9 \u00d7 9 grid, you can see that the 70 shaded squares represent \na number that is closer to 64 squares than to 81 squares.\nThus, the whole number approximation for 70 is 8. \nFor a close approximation of a square root up to a specific number of decimal \nplaces, you may follow the steps shown in the next example. \nexamPLe 4\nApproximate 8  up to two decimal places.\n Solution: \n Step 1.  Determine between which two perfect squares the radicand lies. \nThen get the square roots of such perfect squares.\n    The radicand is 8, and 8 is between the perfect squares 4 and 9.\n    4 < 8 < 9\n    22 < 8 < 32\n    2 < 8  < 3\n   Therefore, 8  is between 2 and 3."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p121-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 121, "chunk_number": 0, "type": "text", "chroma:document": "112\nPractice 2-4\nConcepts and Skills\nWithout using a calculator, determine whether each rational number is a perfect \nsquare or not.\n 1. 16\n 2. 20\n 3.  121\n 4. 225\n 5.  1,000\n 6. \n 7. \n 8. 0.0004\n 9.  0.0404\n 10. 4.0004\nFind each square root without using a calculator.\n 11. \n 12. \n 13. \n 14. \n 15. \n 16. \n 17. \n 18. \n 19. \n 20. \nWithout using a calculator, find the nearest whole number approximation for each \nirrational number. \n 21. \n 22. \n 23. \n 24. \n 25. \nWithout using a calculator, find an approximation for each square root up to the  \nindicated place value. \n 26.  to the nearest tenths\n 27.  to the nearest hundredths\n 28.  to the nearest tenths"}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p116-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 116, "chunk_number": 1, "type": "text", "chroma:document": "0.04 = (0.2)2. Another example is \n9\n16 since .\nexamPLe 1\nFind each indicated square root.\na. b. \n Solutions:\n a.  152 = 225\n b.  \u2013  indicates the negative square root. \nexamPLe 2\nFour hundred pieces of 1-square-foot tiles are used for the floor tiling of a square \nroom. What is the measure of each side of the room?\n Solution:\n  It is given that 400 pieces of 1-square-foot tiles are \nused for floor tiling. Thus, the area of the room is  \n400 square feet (sq. ft.). To find the measure of each \nside of the square room, use the formula A = s2, where \nA is the area and s is the measure of each side of the \nsquare. But since you need to find s, you must get the \nsquare root of A; that is, s = .\n400 sq. ft.\ns ft.\ns ft."}


created_at: 2025-05-16 10:05:22
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p117-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 117, "chunk_number": 1, "type": "text", "chroma:document": "Such small square has an area of 2 sq. ft. What could be \nthe measure of each side of the smaller square?\nSince the area of the small square is 2 sq. ft., the measure of its side must be a \nnumber s which, when squared, gives 2; that is, s2 = 2. What is the value of s? \nYou may try to guess and check possible values of s. For example, you know that \n12 = 1 and 22 = 4. Since s2 = 2, and 2 is between 1 (or 1 2) and 4 (or 22), you can say \nthat s is between 1 and 2. Note, however, that no integer lies between 1 and 2. This \nimplies that s is not an integer.\nRecall that a rational number can be written as a ratio of two integers, where \nthe denominator is not 0. The equation s2 = 2 is equivalent to s = 2\ns . If 2\ns  is a rational \nnumber, then the denominator s has to be an integer. However, you already know that \nno such integer exists. This means that s is neither an integer nor a rational number.\nProblems like this involve irrational numbers. The solution to this problem is"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p119-c1-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 119, "chunk_number": 1, "type": "text", "chroma:document": "step.\n Step 5.  Repeat steps 2 and 3, but this time, use the value you obtained in \nstep 3 as the divisor.\n   So divide 8 by 2.835.\n   8 \u00f7 2.835 \u2248 2.822\n   Get the average of 2.835 and 2.822.\n   \n Step 6.  Repeat step 4. Then compare the two estimates you obtained. Choose \nthe estimate whose square is closer to the radicand. To do this, get \nthe absolute value of the difference of each square and the radicand.\n   (2.829)2 = 8.0032\n   Between 8.0372 and 8.0032, the number closer to 8 is 8.0032 since:\n   |8.0032 \u2013 8| = 0.0032   and   |8.0372 \u2013 8| = 0.0372.\n   Thus, you can say that 8 is approximately equal to 2.829. When you \ncompute  using a calculator, you will get 2.828427125. As you can \nsee, you obtained a close approximation for ."}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p123-c0-t
encoding: FLOAT32
metadata:
{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 123, "chunk_number": 0, "type": "text", "chroma:document": "114\nFind the next two numbers in each sequence.\n 41.  , \n 42.  , \n 43. \u20131, 1, \u20139, 9, \u201325, 25, \u201349, 49, , \n 44. , , \n 45. \u20131, 0, 0, 1, 1, 2, 4, 5, 25, 26, , \nApplications\nAnswer each item completely.\n 46.  The Great Pyramid of Khufu is known as \nthe largest pyramid ever built in Egypt. \nIts base forms a nearly perfect square \nwith an area of 52,900 m2. What is the \nmeasure of each side of its base?\n 47.  Jon is reading a certain page of a book. The square of the page is 961. On \nwhich pages is Jon\u2019s book open? \n 48.  A farmer grows cabbages in his square-shaped farm. Each cabbage takes  \n1 sq. ft. of land area in his farm. He planted 256 cabbages. How many cabbages \nare planted along each side of his farm?\n 49.  A square pigpen has an area of 576 sq. ft. How many feet of fencing material is \nrequired to enclose it?\n 50.  A man is at the top of a mountain and his eyes are approximately 500 ft. above \nthe ground. How far can he see to the horizon? Use the formula in example 5"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p122-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 122, "chunk_number": 0, "type": "text", "chroma:document": "113\n 29.  to the nearest tenths\n 30.  to the nearest hundredths\nAnswer each question. Choose the letter of your answer. \n 31. Which statement is false?\n  A. \n  B. \n  C. \n  D. \n 32. What is ?\n  A. \u201314   B.  \u201313   C.  13   D.  14\n 33. Between which two integers does  lie?\n  A.  13 and 14\n  B.  14 and 15\n  C.  16 and 17\n  D.  17 and 18\n 34. Which number must be a perfect square?\n  A. sum of two perfect squares\n  B.  product of two odd numbers\n  C.  product of two perfect squares\n  D.  product of an even number and a perfect square\n 35. Which number has a square root that is not a real number?\n  A. a counting number\n  B.  a negative number\n  C.  a positive irrational number\n  D.  zero\nFor items 36 to 40, find the value of each expression.\n36. \n  A.  24   B.  34   C.  36   D.  44\n 37. \n  A. 2.02   B.  2.2   C.  20.02   D.  20.2\n 38. \n  A.  0.08   B.  0.088   C.  0.8   D. 0.88\n 39. \n  A.  3   B.  4   C.  5   D.  7\n 40. \n  A.  5   B.  15   C.  25   D.  125"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p125-c0-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 125, "chunk_number": 0, "type": "text", "chroma:document": "116\n 9. What is the value of ?\n  A. 22   B.  28   C.  32   D.  38\n 10. Which decimal is greater than 0.25 but less than ?\n  A.  0.249   B.   C.   D.  0.29\n 11.  A fruit dealer delivers 4 crates of mangoes. He unloads the heaviest crate from \nhis van last. Which crate does he unload last?\n  A. crate A weighing 27.875 kg\n  B.  crate B weighing 28.105 kg \n  C.  crate C weighing 27.984 kg\n  D.  crate D weighing 28.052 kg\n 12.  Larry withdraws P1,000 from his bank account. Which expression represents \nthe amount in his transaction?\n  A.  \u20131,000   B.  1,000   C.  |\u20131,000|   D.  |1,000|\n 13.  A recipe that makes 12 servings requires  cup of melted butter. Suppose that \nyou will prepare a recipe that makes only 9 servings. How many cups of melted \nbutter do you need?\n  A.  cup   B.  cup   C.  cup   D.  cup\n 14.  On a particular day, the exchange rate of euro (\u20ac) to Philippine peso ( P) is  \n\u20ac1 = P65.825. The next day, the peso equivalent of \u20ac1 is 15 centavos less. How"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p124-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 124, "chunk_number": 0, "type": "text", "chroma:document": "115\nAnswer each question without using a calculator. Choose your answer.\n 1.  What is the simplest form of ?\n  A.   B.   C.   D. \n 2.  Which is equivalent to ?\n  A.   B.   C.   D. \n 3.  What is the LCD of \n9\n28 and \n13\n91?\n  A.  4   B.  7   C.  28   D.  364\n 4.  Which is a rational number?\n  A. \n  B.  4.56565653...\n  C.  0.123456...\n  D.  5.3613613613...\n 5.  Which is an irrational number?\n  A.  3.123123\u2026\n  B.  3.14\n  C. \n  D. \n 6.  Which set of fractions is arranged from greatest to least?\n  A. \n  B. \n  C. \n  D. \n 7.  Which set is not a subset of the set of rational numbers?\n  A. set of terminating decimals\n  B.  set of integers\n  C.  set of non-terminating decimals\n  D.  set of whole numbers\n 8.  Which set is closed under division?\n  A.  set of counting numbers\n  B.  set of integers\n  C.  set of non-zero rational numbers\n  D.  set of whole numbers\nChapter Assessment"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p126-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 126, "chunk_number": 0, "type": "text", "chroma:document": "117\nWrite the property of real numbers that each equation illustrates.\n 19. 105 \u2022 624 = (100 + 5) \u2022 624\n 20. (40 + 8) \u2022 25 = 25 \u2022 (40 + 8)\n 21. \nCompare each pair of values using >, <, or =.\n 22. |\u201324|  |24|\n 23.   \n 24.   |\u20137|\n 25.   \n 26. 0  \u2013102\n 27. |\u201321|  |\u201332|\n 28. \u20130.3845  \u20130.3467\n 29.   0.27\nPerform the indicated operations. Express your answer in simplest form.\n 30. \n 31. (\u201336) \u2013 (\u201328) + 42\n 32. \n 33. \nFor items 34 to 37, refer to the following table that shows the times recorded by a \n100-m dash sprinter over a one-week training.\nDay Time (in seconds)\nMonday 10.43\nTuesday 10.39\nWednesday 10.56\nThursday 10.45\nFriday 10.46\n 34. On which day did the sprinter run fastest?\n 35. On which day did the sprinter run slowest?\n 36. How many seconds did the sprinter\u2019s time increase from Monday to Wednesday?\n 37. What is the average time of the sprinter over the one-week period? \nFor items 38 and 39, consider the following situation:\nA basketball team won 7 out of 12 games."}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p127-c0-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 127, "chunk_number": 0, "type": "text", "chroma:document": "118\nFor items 40 and 41, refer to the following situation:\nA school purchases 1,000 new chairs for its auditorium. The chairs are arranged in \nrows to form a square. \n 40. What is the maximum number of chairs that can be arranged in the auditorium? \n 41. How many chairs are in each row? \nFor items 42 and 43, consider the following situation:\nRandy receives a monthly salary of P12,000. He allots  of his salary for his house \nrental and saves another . He allots  of the remaining amount for transportation \nand food. He donates the rest to a charity.\n 42. What fraction of his salary does he donate to a charity?\n 43. How much does he allot for transportation and food?\nAnswer each item completely.\n 44. Graph the points , \uf006, \u20132, , and  on a number line.\n  \u20133 \u20132 \u20131 1 2 30\n 45.  On a certain day, the temperature dropped 18\u00b0 in the Fahrenheit scale over a \nthree-hour period. Suppose that the temperature dropped at a constant rate. \nHow many degrees did the temperature fall each hour?"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p117-c2-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 117, "chunk_number": 2, "type": "text", "chroma:document": "no such integer exists. This means that s is neither an integer nor a rational number.\nProblems like this involve irrational numbers. The solution to this problem is \nthe irrational number 2 . It is equivalent to a nonterminating and nonrepeating \ndecimal. The value of 2 , correct to 10 decimal places, is 1.4142135623\u2026.\nIn general, the square root of a non-perfect square is an irrational number. \nKey Concept\nFor example, 2 , 3 , 5 , 6 , 7 , 8 , and 10 are irrational numbers. You \ncan, however, express the square roots of non-perfect square numbers, correct to a \nspecific number of decimal places. Study the next example.\n2 ft"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p123-c1-t
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 123, "chunk_number": 1, "type": "text", "chroma:document": "required to enclose it?\n 50.  A man is at the top of a mountain and his eyes are approximately 500 ft. above \nthe ground. How far can he see to the horizon? Use the formula in example 5 \nof this lesson. Express your answer to the nearest whole number.\nEnrichment Exercises\nAnswer each item completely.\n 51. Find two integers whose square is 225.\n 52. Suppose that a perfect square is divided by 4. What are the possible remainders?\n 53.  Write an expression that represents the relationship between a perfect square \nand its square root.\n 54. Name all whole numbers whose square roots are between 7 and 8.\n 55.  The ones digit of a perfect square is the same as the ones digit of its square \nroot. What are the possible ones digits of the number?"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p125-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 125, "chunk_number": 1, "type": "text", "chroma:document": "14.  On a particular day, the exchange rate of euro (\u20ac) to Philippine peso ( P) is  \n\u20ac1 = P65.825. The next day, the peso equivalent of \u20ac1 is 15 centavos less. How \nmuch is \u20ac10 based on the last exchange rate?\n  A. P656.25   B. P656.75   C. P658.10   D. P658.25\n 15.  A one-hectare piece of land is square in shape. What is the length of each side \nof the piece of land?\n  A. 1 m   B.  10 m   C.  100 m   D.  1,000 m\nLocate each number in its correct place \nin the Venn diagram on the right.\n 16. \n 17. \u201320\n 18. 2.22\u2026 N\nW\nZ\nQ\nQ'\nR"}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p127-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 127, "chunk_number": 1, "type": "text", "chroma:document": "three-hour period. Suppose that the temperature dropped at a constant rate. \nHow many degrees did the temperature fall each hour?\n 46.  Karin has P23.75 left after buying a notebook and a ball pen. The notebook \ncosts P37.50. The ball pen costs half as much as the notebook. How much did \nshe have before buying the two items?\n 47.  A large handkerchi ef requires 144 square inches of cloth. A small handkerchief \nrequires 64 square inches. How much longer is the side of each cloth required \nfor a large handkerchief than for a small handkerchief?\n 48.  Danny and Jenny, plan to meet, but they have different work schedules. Danny \nis free only every 6 days. Jenny is off duty every 4 days. If they are both free on \nWednesday this week, when is the next time they could meet?\n 49.  Suppose that you are on an airplane flying through a clear sky at an altitude of \napproximately 32,400 ft. How far can you see to the horizon? Use the formula \nthat you used in example 5 of Lesson 2-4."}


created_at: 2025-05-16 10:05:23
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p126-c1-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 126, "chunk_number": 1, "type": "text", "chroma:document": "37. What is the average time of the sprinter over the one-week period? \nFor items 38 and 39, consider the following situation:\nA basketball team won 7 out of 12 games.\n 38. What fraction of the games did the team lose? \n 39. Write this fraction as a decimal."}


created_at: 2025-05-16 10:05:24
topic: persistent://default/default/4073130c-0e29-4c0f-92a7-6a9545956bc1
id: 528ac757-c0d2-4424-bb89-9db7630f6776-p127-c2-t
encoding: FLOAT32
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{"document_uuid": "528ac757-c0d2-4424-bb89-9db7630f6776", "page_number": 127, "chunk_number": 2, "type": "text", "chroma:document": "approximately 32,400 ft. How far can you see to the horizon? Use the formula \nthat you used in example 5 of Lesson 2-4.\n 50.  Draw a Venn diagram that shows the set of real numbers and its subsets. Write \nat least two sample elements for each subset."}


