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Write the property of real numbers that each equation illustrates.
 19. 105 • 624 = (100 + 5) • 624
 20. (40 + 8) • 25 = 25 • (40 + 8)
 21. 
Compare each pair of values using >, <, or =.
 22. |–24|  |24|
 23.   
 24.   |–7|
 25.   
 26. 0  –102
 27. |–21|  |–32|
 28. –0.3845  –0.3467
 29.   0.27
Perform the indicated operations. Express your answer in simplest form.
 30. 
 31. (–36) – (–28) + 42
 32. 
 33. 
For items 34 to 37, refer to the following table that shows the times recorded by a 
100-m dash sprinter over a one-week training.
Day Time (in seconds)
Monday 10.43
Tuesday 10.39
Wednesday 10.56
Thursday 10.45
Friday 10.46
 34. On which day did the sprinter run fastest?
 35. On which day did the sprinter run slowest?
 36. How many seconds did the sprinter’s time increase from Monday to Wednesday?
 37. What is the average time of the sprinter over the one-week period? 
For items 38 and 39, consider the following situation:
A basketball team won 7 out of 12 games.


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118
For items 40 and 41, refer to the following situation:
A school purchases 1,000 new chairs for its auditorium. The chairs are arranged in 
rows to form a square. 
 40. What is the maximum number of chairs that can be arranged in the auditorium? 
 41. How many chairs are in each row? 
For items 42 and 43, consider the following situation:
Randy receives a monthly salary of P12,000. He allots  of his salary for his house 
rental and saves another . He allots  of the remaining amount for transportation 
and food. He donates the rest to a charity.
 42. What fraction of his salary does he donate to a charity?
 43. How much does he allot for transportation and food?
Answer each item completely.
 44. Graph the points , , –2, , and  on a number line.
  –3 –2 –1 1 2 30
 45.  On a certain day, the temperature dropped 18° in the Fahrenheit scale over a 
three-hour period. Suppose that the temperature dropped at a constant rate. 
How many degrees did the temperature fall each hour?


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no such integer exists. This means that s is neither an integer nor a rational number.
Problems like this involve irrational numbers. The solution to this problem is 
the irrational number 2 . It is equivalent to a nonterminating and nonrepeating 
decimal. The value of 2 , correct to 10 decimal places, is 1.4142135623….
In general, the square root of a non-perfect square is an irrational number. 
Key Concept
For example, 2 , 3 , 5 , 6 , 7 , 8 , and 10 are irrational numbers. You 
can, however, express the square roots of non-perfect square numbers, correct to a 
specific number of decimal places. Study the next example.
2 ft


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 50.  A man is at the top of a mountain and his eyes are approximately 500 ft. above 
the ground. How far can he see to the horizon? Use the formula in example 5 
of this lesson. Express your answer to the nearest whole number.
Enrichment Exercises
Answer each item completely.
 51. Find two integers whose square is 225.
 52. Suppose that a perfect square is divided by 4. What are the possible remainders?
 53.  Write an expression that represents the relationship between a perfect square 
and its square root.
 54. Name all whole numbers whose square roots are between 7 and 8.
 55.  The ones digit of a perfect square is the same as the ones digit of its square 
root. What are the possible ones digits of the number?


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14.  On a particular day, the exchange rate of euro (€) to Philippine peso ( P) is  
€1 = P65.825. The next day, the peso equivalent of €1 is 15 centavos less. How 
much is €10 based on the last exchange rate?
  A. P656.25   B. P656.75   C. P658.10   D. P658.25
 15.  A one-hectare piece of land is square in shape. What is the length of each side 
of the piece of land?
  A. 1 m   B.  10 m   C.  100 m   D.  1,000 m
Locate each number in its correct place 
in the Venn diagram on the right.
 16. 
 17. –20
 18. 2.22… N
W
Z
Q
Q'
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three-hour period. Suppose that the temperature dropped at a constant rate. 
How many degrees did the temperature fall each hour?
 46.  Karin has P23.75 left after buying a notebook and a ball pen. The notebook 
costs P37.50. The ball pen costs half as much as the notebook. How much did 
she have before buying the two items?
 47.  A large handkerchi ef requires 144 square inches of cloth. A small handkerchief 
requires 64 square inches. How much longer is the side of each cloth required 
for a large handkerchief than for a small handkerchief?
 48.  Danny and Jenny, plan to meet, but they have different work schedules. Danny 
is free only every 6 days. Jenny is off duty every 4 days. If they are both free on 
Wednesday this week, when is the next time they could meet?
 49.  Suppose that you are on an airplane flying through a clear sky at an altitude of 
approximately 32,400 ft. How far can you see to the horizon? Use the formula 
that you used in example 5 of Lesson 2-4.


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37. What is the average time of the sprinter over the one-week period? 
For items 38 and 39, consider the following situation:
A basketball team won 7 out of 12 games.
 38. What fraction of the games did the team lose? 
 39. Write this fraction as a decimal.


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approximately 32,400 ft. How far can you see to the horizon? Use the formula 
that you used in example 5 of Lesson 2-4.
 50.  Draw a Venn diagram that shows the set of real numbers and its subsets. Write 
at least two sample elements for each subset.


