Answers | Investigation 1

Applications

1. 1; Possible explanation: 99 is 1, so 109

is

more than 1. 2. 1

2 ; Possible explanation: Since 8 1 9= ,16 2 16 is

close to 12 .

3. 1; Possible explanation: 56 is only 16 less

than 1 whole. 4. 1

2 ; Possible explanation: 12 of 100 is 50,

so 48100

is closer to .12

5. 12 ; Possible explanation: 12 is 0.50 and 34

is 0.75, so 0.67 is close to .12

6. 0; Possible explanation: 0.0009999 is a very small amount. It does not have any tenths in it, and 12 is equivalent to

5 tenths. 7. 1; Possible explanation: 78 is a little

less than 1 and 49 is a little less than .12

Together, a little less than 1 and a little less than 12 is a little less than 112 or closer to 1

than to 2. 8. 2; Possible explanation: 314 is closest to 2,

and 18 is a small number that will make the

sum closer to, but not greater than, 2. 9. 3; Possible explanation: 113 is a little

more than 1.3. 1.3 and 1.3 is 2.6, which is greater than 12 2

, and closest to 3.

10. 0; Possible explanation: It would take two 14 ’s to equal exactly 12 . Since 18 is less

than 14 , 14 and 18 is less than 12 and closer

to 0. 11. 2; three tenths and eight tenths have a

sum greater than 1, so the total sum here is greater than 2.

12. 2; 4110 is equivalent to 1.4 which, when

added to 0.375, is 1.775, which is close to 2.

13. 0.5; Possible explanation: 35 is a bit more

than half, and 110 is a small fraction, not

big enough to push 35 close to 1.

14. 0.5; Possible explanation: 14 is exactly

in the middle of 0 and 1.2 110 is a small

fraction so the sum is closer to 12 , but not

greater than 1.2

15. 0; Possible explanation: 1 2=4 8. Also, 14 is

exactly halfway between 0 and 1 1 1. <2 9 8, so

1 1 19 8 4+ < .Therefore the sum is closer to 0

than to 1.2

16. a. This is incorrect. 1 1<8 4, so the sum here

is less than 34 .

b. Correct. Each fraction is equal to 12 , so

the sum is 1. c. Correct. 5 3>6 4

, so the sum of 512 and 56

is also greater. d. Correct. 5 1=10 2

and 3 1>8 4, so the sum

is greater than .34

17. Possible answer: 14 and 17

18. Possible answer: 38 and 49

19. Possible answer: 58 and 12

20. Possible answer: 58 and 78

21. a. The greatest possible sum is 1.05 + 9 .10

b. The least possible sum is +1 3 .4 5

22. a. No; In the price list, the whole numbers add up to 9: 2 + 1 + 1 + 1 + 3 + 1 = 9. The cheese is $1.95, which brings the total to $9.95. There’s clearly enough in the cost of other items to put the total over $10. (The exact cost of the groceries is $12.42.)

b. Possible answers: Milk, cheese, avocado ($4.92); eggs, cheese, honey, bread ($4.91)

c. cereal, honey, and avocado ($4.94)

23. 58 is closest to 12 . Possible explanation: .=4 18 2

The other 18 makes 58 just a little more than

12 , but not close to 88 , or 1 whole.

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Let’s Be Rational Investigation 1

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Answers | Investigation 1

24. Soo has enough molding. is less than 1 whole and is more than 1 whole or . Since , when the is added to the it will be greater than 1 yard, and the total will be greater than 2 yards.

25. He had about 8 quarts.

26. No; If you add to , you will get . But is a little less than , so there is not enough.

27. a. Marigolds: ; Lantana: ;

Impatiens: ; Petunias: ; Lilies: ;

Begonias: ; Tulips: ; Daisies: ;

Irises:

b. − = 4 1 320 20 20

c. + + 4 1 1 620 20 20 20= , or 310

d. Incorrect. Possible explanation: The number sentence for the situation is

− = +3 1 1 120 20 10 20

. If you work out the

subtraction problem on the left of the equal sign and the addition problem on the right, the answers are not the same.

e. Possible combinations that total 310 , the

fraction planted with impatiens: Marigolds + Petunias + Lantana:

+ + + + 3 1 1 3 2 1 6 320 10 20 20 20 20 20 10= = =

Lilies + Petunias: + = 4 2 620 20 20

, or 310

Marigolds + Begonias + Tulips + Daisies: 3 1 1 1 6

20 20 20 20 20+ + + = , or 310

28. a. + = 1 1 3 8 16 16 of the page is used for ads.

b. − 3 131 16 16= of the page remains.

29. (three -page ads, or )

plus (four -page ads, or )

plus (ten -page ads, or )

= pages.

30. −3 5 52 1 14 8 8= pages

31. +1 1 3 19 18 18 6= = of the lasagna is eaten,

leaving 56 of the lasagna uneaten.

32. + +3 1 6 1 74 8 8 8 8= = of a small bag of chips

33. 11215

34. 37 8

35. 58 6

36. 2612 or 16 6

37. +3 44 5

is greater: + = + = =2 5 4 5 93 6 6 6 6

=1 101 12 20; + = + =3 4 15 16 1114 5 20 20 20

38. −7 26 3

is greater: − = − =7 2 7 4 36 3 6 6 6 =

=1 52 10

; − = − =3 5 6 5 15 10 10 10 10

39. + =5 1 12 1 46 3 6

40. + =5 5 1115 10 268 6 4

41. + =4 1 294 2 69 5 45

42. − =1 5 56 2 34 6 12

43. − =1 4 73 1 12 5 10

44. − =2 5 208 6 13 7 21

45. 34

46. 36 or 12

47. 38

48. 310

49. 312

50. 314

In all of the problems, you add unit fractions where one fraction is half the size of the other. The fraction in each part with the lesser denominator is twice the value of the unit fraction with the greater denominator. You can think of the unit fraction with the lesser denominator as two unit fractions with the greater denominator. This gives a sum with a 3 in the numerator over the greater denominator.

51. No. If 146 of all the pizza were eaten,

this would be less than one whole pizza. If there are eight sections in each pizza, then people are eating eighths. And all together they ate 148 or 618 pizzas.

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78

18

87117

771 1>7 8

17

78

34

31412 2

58

34

320

120

310

110

15

120

120

120

120

34

14

13× 448

18

1016

116

718

14× 8 110×16

Let’s Be Rational Investigation 1

A C E

Answers | Investigation 1

Connections

52. Divide the rectangle into thirds. Each third of the rectangle represents 50% of the original whole. So, 100% is represented by the blue area.

53. 15 beans. There are 9 beans shown in the

picture representing three fifths. So each fifth must contain 3 beans. Since the whole is five fifths, there must be 15 beans on the counter.

54. a. 12 is 0.5 and 50%

13 is about 0.33 and 33%

14 is 0.25 and 25%

23 is about 0.66 or 66% (Some students

will argue for 0.67 and 67%, if the convention of rounding up is used.) 34 is 0.75 and 75%

16 is about 0.16 and 16% (Some

students will argue for 0.17 and 17%, if the convention of rounding up is used.) 15 is 0.2 or 20%

18 is 0.125 or 12.5%

b. (See Figure 1.)

55. D

56. a and d

a. This set of fractions can be renamed as hundredths because each denominator, 2, 4, and 5, are factors of 100.

b. This set of fractions cannot be renamed as hundredths. The denominator 10 is a factor of 100, but 11 and 12 are not.

c. This set of fractions cannot be renamed as hundredths. The denominators 6 and 8 are not factors of 100.

d. This set of fractions can be renamed as hundredths because all the denominators, 5, 10, and 20, are factors of 100.

57. a. The least possible sum is 141 . This is the

sum of the least numbers in each of these intervals.

b. The greatest possible sum is 2. That is the sum of the greatest numbers in each of these intervals.

58. N = 2

59. N = 3

60. N = 8

61. N = 24

Figure 1

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Let’s Be Rational Investigation 1

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Answers | Investigation 1

62. 2 4=

6 12

63. 8 2 2= =

12 3 6

64. 3 2 6= =

9 6 18

65. 18.156 < 18.17

66. 4.0074 > 4.0008

67. G

68. Answers will vary. Possible answers include: Foley, Burg, Walker, Theule, and Lapp.

5 3 5 3 1 1+ + + + = 1

16 16 16 16 4 4

69. a. 113

of a whole; 3 squares = 113

of a

whole; 9 squares = 1 whole; so 12

squares = 113

of a whole.

b. 7

9 of a whole; 3 squares = 1

3 or 3

9 of a

whole; 9 squares = 33

or 99

of a whole;

so 7 squares = 79

of a whole.

70. a. Note: The purple shaded region represents 1. The green shaded region

represents the 13

portion of 113

. The

blue shaded region represents 16

.

Or,

b. Note: The Xs indicate the regions subtracted. The purple and green shaded regions each represent 1. The

blue shaded region represents 23

. The

remaining portion is 43

.

Or,

71. a. Since 70 7

150 15= and 30 2

150 10= , the sum

of these two numbers is the same no matter which forms we use.

b. Answers will vary. Possible answers

include: 14 4

30 20+ and 14 6

30 30+ .

c. Any problem with common denominators should be easy to solve. This includes Maribel’s problem as well

as this one: 14 6

30 30+ .

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Let’s Be Rational Investigation 1

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Answers | Investigation 1

Extensions

72. Answers will vary. Possible answers include: 2

5

73. Answers will vary. Possible answers include: 2

7

74. It is always possible to find a fraction between any two fractions on the number line. One way to know this is that we can rewrite each given fraction as an equivalent fraction, using common denominators, and then choose a new fraction with a numerator between the two

new numerators. For example, given 13

and

1

2 we can rewrite as 4

12 and 6

12, and then

we know 512

is between these.

Another method would be to take the number that is halfway between the two fractions, which is the average of the two fractions. For example, to find the fraction

between 34

and 45

we can take the average.

( ) ( )1 3 4 1 15 16 1 31 31+ = + = × =

2 4 5 2 20 20 2 20 40

Now 3 30 31 32 4

4 40 40 40 40= < < = ,

so 3140

is between 34

and 45

.

75. a. 1

32 page costs $5, 1

16 page costs $10,

1

8 page costs $20, 1

4 page costs $40,

1

2 page costs $80, 1 full page costs $160.

b. 40 × 3 + 20 × 4 + 10 × 1 = $210

c. Yes. 28

page costs $40 and 416

page

costs $40, for a total of $80.

d. Answers will vary. Possible answers include: Any combination adding

up to 12

page. This includes two

1

4-page ads, or one 1

4-page ad and

two 18

-page ads.

76. a. 2 acres; 16 people can clear twice as much in the same amount of time.

b. 1

4 acre; one fourth of the people can

clear one fourth as much in the same amount of time.

c. 24 people; three times as many people are needed to clear three times the area in the same amount of time.

d. 48 people; twice as many people are needed to complete part (c) in half of the time.

77. a. Answers will vary. Possible answers

include: −1 1 1

3 4 12= .

b. Answers will vary. Possible answers

include: −1 1 1

5 5 30= .

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Let’s Be Rational Investigation 1

A C E