CMP3_G6_LBR_ACE1 - Mrs. Southward
Answers | Investigation 1
Applications
1. 1; Possible explanation: [pic] is 1, so [pic] is
more than 1.
2. [pic]; Possible explanation: Since [pic] is close to [pic].
3. 1; Possible explanation: [pic] is only [pic] less
than 1 whole.
4. [pic]; Possible explanation: [pic] of 100 is 50,
so [pic] is closer to [pic]
5. [pic]; Possible explanation: [pic] is 0.50 and [pic]
is 0.75, so 0.67 is close to [pic]
6. 0; Possible explanation: 0.0009999 is
a very small amount. It does not have
any tenths in it, and [pic] is equivalent to
5 tenths.
7. 1; Possible explanation: [pic] is a little
less than 1 and [pic] is a little less than [pic]
Together, a little less than 1 and a little less
than [pic] is a little less than [pic] or closer to 1
than to 2.
8. 2; Possible explanation: [pic] is closest to 2,
and [pic] is a small number that will make the
sum closer to, but not greater than, 2.
9. 3; Possible explanation: [pic] is a little
more than 1.3. 1.3 and 1.3 is 2.6, which is
greater than [pic], and closest to 3.
10. 0; Possible explanation: It would take
two [pic]’s to equal exactly [pic]. Since [pic] is less
than [pic], [pic] and [pic] is less than [pic] 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; [pic] is equivalent to 1.4 which, when
added to 0.375, is 1.775, which is close
to 2.
13. 0.5; Possible explanation: [pic] is a bit more
than half, and [pic] is a small fraction, not
big enough to push [pic] close to 1.
14. 0.5; Possible explanation: [pic] is exactly
in the middle of 0 and [pic] [pic] is a small
fraction so the sum is closer to [pic], but not
greater than [pic]
15. 0; Possible explanation: [pic]. Also, [pic] is
exactly halfway between 0 and [pic], so
[pic].Therefore the sum is closer to 0
than to [pic]
16. a. This is incorrect. [pic], so the sum here
is less than [pic].
b. Correct. Each fraction is equal to [pic], so
the sum is 1.
c. Correct. [pic], so the sum of [pic] and [pic]
is also greater.
d. Correct. [pic] and [pic], so the sum
is greater than [pic]
17. Possible answer: [pic] and [pic]
18. Possible answer: [pic] and [pic]
19. Possible answer: [pic] and [pic]
20. Possible answer: [pic] and [pic]
21. a. The greatest possible sum is 1.05 + [pic]
b. The least possible sum is [pic]
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. [pic] is closest to [pic]. Possible explanation: [pic]
The other [pic] makes [pic] just a little more than
[pic], but not close to [pic], or 1 whole.
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. [pic]
c. [pic], or [pic]
d. Incorrect. Possible explanation: The
number sentence for the situation is
[pic]. 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[pic], the
fraction planted with impatiens:
Marigolds + Petunias + Lantana:
[pic]
Lilies + Petunias: [pic], or [pic]
Marigolds + Begonias + Tulips +
Daisies: [pic], or [pic]
28. a. [pic] of the page is used for ads.
b. [pic] of the page remains.
29. (three -page ads, or )
plus (four -page ads, or )
plus (ten -page ads, or )
= pages.
30. [pic]pages
31. [pic]of the lasagna is eaten,
leaving [pic] of the lasagna uneaten.
32. [pic] of a small bag of chips
33. [pic]
34. [pic]
35. [pic]
36. [pic] or [pic]
37. [pic] is greater: [pic]
[pic]; [pic]
38. [pic] is greater: [pic]
[pic]; [pic]
39. [pic]
40. [pic]
41. [pic]
42. [pic]
43. [pic]
44. [pic]
45. [pic]
46. [pic]or[pic]
47. [pic]
48. [pic]
49. [pic]
50. [pic]
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 [pic] 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 [pic] or [pic] pizzas.
2
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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. [pic] is 0.5 and 50%
[pic] is about 0.33 and 33%
[pic] is 0.25 and 25%
[pic] is about 0.66 or 66% (Some students
will argue for 0.67 and 67%, if the
convention of rounding up is used.)
[pic] is 0.75 and 75%
[pic] is about 0.16 and 16% (Some
students will argue for 0.17 and 17%, if
the convention of rounding up is used.)
[pic] is 0.2 or 20%
[pic] 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 [pic]. 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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Answers | Investigation 1
62. [pic]
63. [pic]
64. [pic]
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.
[pic]
69. a. [pic] of a whole; 3 squares = [pic] of a
whole; 9 squares = 1 whole; so 12
squares = [pic] of a whole.
b. [pic] of a whole; 3 squares = [pic] or [pic] of a
whole; 9 squares = [pic] or [pic] of a whole;
so 7 squares = [pic] of a whole.
70. a. Note: The purple shaded region
represents 1. The green shaded region
represents the [pic] portion of [pic]. The
blue shaded region represents [pic].
Or,
b. Note: The Xs indicate the regions
subtracted. The purple and green
shaded regions each represent 1. The
blue shaded region represents [pic]. The
remaining portion is [pic].
Or,
71. a. Since [pic] and [pic], the sum
of these two numbers is the same no
matter which forms we use.
b. Answers will vary. Possible answers
include: [pic] and [pic].
c. Any problem with common
denominators should be easy to solve.
This includes Maribel’s problem as well
as this one: [pic].
4
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Answers | Investigation 1
Extensions
72. Answers will vary. Possible answers include:
[pic]
73. Answers will vary. Possible answers include:
[pic]
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 [pic] and
[pic] we can rewrite as [pic] and [pic], and then
we know [pic] 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 [pic] and [pic] we can take the average.
[pic]
Now [pic],
so [pic] is between [pic] and [pic].
75. a. [pic] page costs $5, [pic] page costs $10,
[pic] page costs $20, [pic] page costs $40,
[pic] page costs $80, 1 full page costs $160.
b. 40 × 3 + 20 × 4 + 10 × 1 = $210
c. Yes. [pic] page costs $40 and [pic] page
costs $40, for a total of $80.
d. Answers will vary. Possible answers
include: Any combination adding
up to [pic] page. This includes two
[pic]-page ads, or one [pic]-page ad and
two [pic]-page ads.
76. a. 2 acres; 16 people can clear twice as
much in the same amount of time.
b. [pic] 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:[pic].
b. Answers will vary. Possible answers
include: [pic].
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Let’s Be Rational Investigation 1
Let’s Be Rational Investigation 1
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Let’s Be Rational Investigation 1
Let’s Be Rational Investigation 1
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Let’s Be Rational Investigation 1
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