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Answer Keys Part A

MathWorkbook

© Marshall Cavendish Education

Chapter 1

Whole Numbers

Practice 1.1: Numbers to 10,000,000

Pages 1–4

Practice 1

1 40,000; 50,000; 60,000; 70,000; 80,000 2 900,000;

800,000; 700,000; 600,000; 500,000

3  Standard Form Word Form

hundred thousands 400,000 four hundred thousand

ten thousands

thousands

hundreds

ten

ones

4

2

5

3

1

6

20,000 twenty thousand

5,000 ve thousand

300 three hundred

10 ten

6 six

;

425,316; four hundred twenty-five thousand, three

hundred sixteen

4 239,653 5 835,720 6 816,943 7 605,500 8 103,031

9 870,003 10 300,012

11  Thousands OnesHundred

ThousandsTen

Thousands Hundreds Tens

 one hundred five thousand, three hundred sixty-two

12  Thousands Ones

TenThousands Hundreds TensHundred

Thousands

  five hundred sixty thousand, twenty-one

13 sixty-five thousand, one hundred forty-two 14 three

hundred sixty-eight thousand, four hundred 15 one

16 thousand 17 two 18 nine hundred ninety-nine

thousand, one hundred ninety-eight 19 Three hundred

twelve thousand, eight hundred nineteen 20 Hyde Park;

Nine thousand, five hundred twenty-three or 9,523

Pages 5–6

Practice 2

1  Standard Form Word Form

millions

hundred thousand

ten thousands

thousands

hundreds

tens

ones

9

1

5

6

3

4

2

9,000,000

100,000

50,000

6,000

300

40

2

nine million

one hundred thousand

fty thousand

six thousand

three hundred

forty

two

  9,156,342; nine million, one hundred fifty-six thousand,

three hundred forty-two

2 3,240,000; three million, two hundred forty thousand

3 2,156,004 4 5,238,000 5 7,150,000 6 6,060,050

7 3,000,003 8 five million, fifty thousand 9 eight million,

one hundred forty-seven thousand, six hundred 10 seven

million, two hundred thirty thousand, fourteen 11 five

million, one hundred ninety-two thousand, six hundred

twenty-two 12 nine million, nine thousand, nine

Practice 1.2: Place Value

Pages 7–10

1a 300,000 1b 300,000 2a 40,000 2b 40,000 3a 5,000

3b 5,000

4  2 5 6, 8 6 1

200,000

50,000

6,000

800

60

1

5 300,000 6 6,000 7 20,000

8 20 9 2,000 10 200,000 11 0

12 hundred thousands 13 ten

thousands 14 50,000 15 700,000

16 708,504 17 202,010

18a 1,000,000 18b 1,000,000

19a 8,000 19b 8,000

20 ten thousands

21  7, 5 1 9, 4 5 6

7,000,000

10,000

9,000

400

50

6

500,000

22 millions 23 0

24 9,000,000 25 4,000,000

26 50 27 500,000

28 5,207,070 29 3,029,105

30 9,165,783

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Practice 1.3:

Comparing Numbers to 10,000,000

Pages 11–14

1  Hundred Thousands

TenThousands Thousands Hundreds Tens Ones

1

2

9

2

7

5

2

3

1

0

0

2

2 hundred thousands is greater than 1 hundred

thousand. So, 225,302 is greater than 197,210.

2 > 3 > 4 < 5 <

6  375,061 172,503 127,203 157,203 371,560 371,605

7 315,679; 615,379; 739,615; 795,316 8 97,632; 245,385;

300,596; 805,342 9 6 million is less than 8 million.

6,990,395 is less than 8,079,720. 10 5,096,357 is greater

than 1,083,952. 11 6,438,671 is greater than 6,412,586.

12 > 13 < 14 < 15 > 16 3,190,000; 2,720,000; 2,432,000;

480,000 17 3,150,000; 2,020,000; 913,000; 513,900

18a 200,000 18b 200,000 18c 200,000 more than

1,138,561 is 1,338,561. 18d 1,338,561 19a 100,000

19b 100,000 19c 100,000 less than 4,455,230 is 4,355,230.

19d 4,355,230 20 230,180; 231,180; 232,180; 233,180;

234,180. Rule: Count on by 1,000. 21 850,400; 845,400;

840,400; 835,400; 830,400. Rule: Count back by 5,000.

22 2,650,719 3,650,719 4,650,719; 5,650,719; 6,650,719.

Rule: Count on by 1,000,000. 23 6,298,436; 5,198,436;

4,098,436; 2,998,436; 1,898,436. Rule: Count back by

1,100,000. 24 80,000 25 5,602,000 26 562,000

27 1,002,000 28 100,000; S T A M P

Practice 1.4:

Rounding and Estimating

Pages 15–22

1 9,700 9,800

; 9,700

2 31,000 32,000

; 32,000

3 6,000 4 10,000 5 1,000 6 72,000 7 473,000 8 70,000

9 20,000 10 757,000 11a 4,500 11b 90,499

12 7,000 + 4,000 = 11,000 13 7,000 + 7,000 = 14,000

14 5,000 + 6,000 = 11,000 15 3,000 + 10,000 = 13,000

16 7,000 + 3,000 = 10,000 17 5,000 – 4,000 = 1,000

18 7,000 – 4,000 = 3,000 19 5,000 – 1,000 = 4,000

20 4,000 – 3,000 = 1,000 21 6,000 – 1,000 = 5,000

22  2,000 + 6,000 + 1,000 = 9,000

500 + 0 + 600 = 1,100

To the nearest thousand: 1,100 1,000

9,000 + 1,000 = 10,000

23  7,000 + 6,000 + 3,000 = 16,000

800 + 800 + 300 = 1,900

To the nearest thousand: 1,900 2,000

16,000 + 2,000 = 18,000

24  4,000 + 8,000 + 2,000 = 14,000

100 + 900 + 200 = 1,200

To the nearest thousand: 1,200 1,000

14,000 + 1,000 = 15,000

25  6,000 – 3,000 = 3,000

700 – 0 = 700

To the nearest thousand: 700 1,000

3,000 + 1,000 = 4,000

26  8,000 – 3,000 = 5,000

700 – 500 = 200

To the nearest thousand: 200 0

5,000 + 0 = 5,000

27  7,000 – 4,000 = 3,000

800 – 300 = 500

To the nearest thousand: 500 1,000

3,000 + 1,000 = 4,000

28  5,000 – 3,000 = 2,000

900 – 700 = 200

To the nearest thousand: 200 0

2,000 – 0 = 2,000

29  9,000 – 4,000 = 5,000

800 – 100 = 700

To the nearest thousand: 700 1,000

5,000 – 1,000 = 4,000

30  8,000 – 4,000 = 4,000

800 – 200 = 600

To the nearest thousand: 600 1,000

4,000 – 1,000 = 3,000

31 4,000 × 7 = 28,000 32 3,000 × 5 = 15,000

33 5,000 × 6 = 30,000 34 8,000 × 9 = 72,000

35 6,000 × 3 = 18,000 36 6,300 ÷ 7 = 900

37 5,400 ÷ 6 = 900 38 2,700 ÷ 3 = 900

39 6,400 ÷ 8 = 800 40 2,700 ÷ 9 = 300

Math Journal

Pages 23–24

1  Answers vary.

You can estimate by rounding each number to the nearest

thousand.

8,642 rounds to 9,000.

9,328 rounds to 9,000.

9,000 + 9,000 = 18,000

The estimated sum is 18,000.

Kim’s answer is reasonable. Dominic’s answer is too far

from the estimate of 18,000. It is not reasonable.

2a  Look for compatible numbers:

7,986 8,000

8,000 ÷ 8 = 1,000

The quotient is reasonable.

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2b  Look for compatible numbers:

2,659 2,700

2,700 ÷ 3 = 900

The quotient is not reasonable.

3a  Lisa rounded 763 to the lesser hundred when she ought

to be rounding it to the greater hundred.

The correct answer should have been 800.

3b  Lisa rounded 3,730 to the lesser thousand when she

ought to be rounding it to the greater thousand.

The correct answer should have been 4,000.

Put On Your Thinking Cap!

Page 25

Answers vary. Accept 755,628; 755,682; 755,862; 756,258

Page 26

1  2 3 0 0

3,200 – 2,300 = 900

Subtract 900.

2a 0 2b Answers vary. Samples: 120, 150, 180, 210, 240,

etc. (multiples of 3 with 0 in the ones place)

Chapter 2

Whole Number

Multiplication and Division

Practice 2.1: Using a Calculator

Pages 27–28

1 10,058 2 6,807 3 8,251 4 9,850 5 8,136 6 6,157 7 161

8 16,791 9 4,308 10 12,586 11 9,875 12 56,400 13 56

14 684 15 96 16 978

17  

75 � 16

712 � 32

1,625 � 127 968 � 16

125 � 25

1,708 � 1,372

3,12

5

16,3

72

22,784

336

3,080

1,49

8

1,752

15,4

88

1,044

120

1,20

0

5

Flavio

a soccer ball

Practice 2.2: Multiplying by

Tens, Hundreds, or Thousands

Pages 29–35

1 470 2 380 3 1,090 4 5,210 5 71,400 6 15,030 7 37,020

8 93,420 9 10 10 70 11 10 12 500 13 10 14 402 15 10

16 9,176

17  39 × 30

(39 × 3) × 10

= 117 ×10

= 1,170

18  143 × 90

(143 × 9) × 10

= 1,287 × 10

= 12,870

19  360 × 30

= (360 × 3) × 10

= 1,080 × 10

= 1,800

20  285 × 80

= (285 × 8) × 10

= 2,280 × 10

= 22,280

21 7,000 22 8,600 23 70,000 24 9,500 25 400,000

26 21,700 27 726,000 28 80,300 29 8,032,000

30 381,000 31 3,936,000; P E R S I A N 32 100 33 25

34 478 35 1,000 36 100 37 2,662 38 100 39 5,760

40  12 × 500

= (12 × 5) × 100

= 60 × 100

= 6,000

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41  700 × 900

= (700 × 9) × 100

= 6,300 × 100

= 630,000

42  814 × 700

= (814 × 7) × 100

= 5,698 × 100

= 569,800

43  5,400 × 800

= (5,400 × 8) × 100

= 43,200 × 100

= 4,320,000

44  5 × 7,000

= (5 × 7) × 1000

= 35 × 1000

= 35,000

45  8 × 5,000

= (8 × 5) × 1000

= 40 × 1000

= 40,000

46  12 × 3,000

= (12 × 3) × 1000

= 36 × 1000

= 36,000

47  15 × 2,000

= (15 × 2) × 1000

= 30 × 1000

= 30,000

48  300 × 4,000

= (300 × 4) × 1000

= 1,200 × 1000

= 1,200,000

49  663 × 6,000

= (663 × 6) × 1000

= 3,978 × 1000

= 3,978,000

50 1,190; 11,900; 119,000 51 1,950; 19,500; 195,000

52 3,600; 36,000; 360,000 53 8,120; 81,200; 812,000

54 12,680; 126,800; 1,268,000 55 100 56 3,000 57 30

58 200 59 58 × $219 rounds to 60 × $200 = $12,000

60 652 × $73 rounds to 700 × $70 = $49,000 61 99 × $217

rounds to 100 × $200 = $20,000 62 39 × $4,156 rounds to

40 × $4,000 = $160,000 63 $12,000 + $49,000 + $20,000

+ $160,000 = $241,000

Math Journal

Page 36

184 × 97 = 17,848

To check if my answer is reasonable, I estimate the product

by rounding each factor to its greatest place value.

184 rounds to 200. 97 rounds to 100. 200 × 100 = 20,000

My estimate is 20,000.

Then I compare my answer with the estimate. My answer is

reasonable. It is close to the estimate of 20,000.

Practice 2.3:

Multiplying by 2-Digit Numbers

Pages 37–40

1 59 × 40 = 2,360; Estimate: 60 × 40 = 2,400

2 91 × 14 = 1,274; Estimate: 90 × 10 = 900

3 96 × 15 = 1,440; Estimate: 100 × 20 = 2,000

4 23 × 17 = 391; Estimate: 20 × 20 = 400

5 750 × 60 = 45,000; Estimate: 800 × 60 = 48,000

6 614 × 31 = 19,034; Estimate: 600 × 30 = 18,000

7 556 × 47 = 26,132; Estimate: 600 × 50 = 30,000

8 843 × 25 = 21,075; Estimate: 800 × 30 = 24,000

9 3,610 × 60 = 216,600; Estimate: 4,000 × 60 = 240,000

10 8,142 × 16 = 130,272; Estimate: 8,000 × 20 = 160,000

11 5,193 × 35 = 181,755; Estimate: 5,000 × 40 = 200,000

12 4,563 × 29 = 132,327; Estimate: 5,000 × 30 = 150,000

13 85 × 45 = 2,975; Estimate: 90 × 40 = 3,600

14 78 × 21 = 1,638; Estimate: 80 × 20 = 1,600

15 738 × 96 = 70,848; Estimate: 700 × 100 = 70,000

16 921 × 57 = 52,497; Estimate: 900 × 60 = 54,000

17 3,072 × 82 = 251,904; Estimate: 3,000 × 80 = 240,000

18 7,846 × 63 = 494,298; Estimate: 8,000 × 60 = 480,000

Math Journal

Pages 41–42

1  a. 2,892 × 21 = 60,732

b. 2,743 × 18 = 49,374

The estimate in (a) is closer to the actual answer. In (a):

• The rounded factors are close to the actual factors. So,

when the rounded factors are multiplied, they give an

estimated product that is close to the actual product.

• One factor is rounded up and the other is rounded

down. These differences from the actual factors are

offset in the estimate.

In (b):

• The rounded factors are not close to the actual factors.

When the rounded factors are multiplied, the difference

between the rounded factors and actual factors is

magnified, giving an estimated product that is not close

to the actual product.

• Not only are the factors farther from their rounded

counterparts, they are also both rounded up. This

magnifies the differences.

2 I would either rework the problem or use another

method to obtain a second estimate. For example, in part

(b), I can also round 2,743 to 2,700. I would then get a

second estimate, 2,700 × 20 = 54,000. Comparing my actual

answer 49,374 against it, I know 49,374 is reasonable.

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Practice 2.4: Dividing by

Tens, Hundreds, or Thousands

Pages 43–48

1 10 2 67 3 10 4 19,740 5 5,226 6 10

7  3,000 ÷ 60

= (3,000 ÷ 10) ÷ 6

= 300 ÷ 6

= 50

8  1,040 ÷ 40

= (1,040 ÷ 10) ÷ 4

= 104 ÷ 4

= 26

9  8,700 ÷ 60

= (8,700 ÷ 10) ÷ 6

= 870 ÷ 6

= 145

10  3,450 ÷ 50

= (3,450 ÷ 10) ÷ 5

= 345 ÷ 5

= 69

11  34,230 ÷ 70

= (34,230 ÷ 10) ÷ 7

= 3,423 ÷ 7

= 489

T R U M A N

12 34 13 560 14 50 15 38 16 77 17 360 18 20 19 415;

A M P H I B I A N S

20  1,600 ÷ 400

= (1,600 ÷ 100) ÷ 4

= 16 ÷ 4

= 4

21  81,000 ÷ 900

= (81,000 ÷ 100) ÷ 9

= 810 ÷ 9

= 90

22  31,500 ÷ 500

= (31,500 ÷ 100) ÷ 5

= 315 ÷ 5

= 63

23  56,000 ÷ 7,000

= (56,000 ÷ 1,000) ÷ 7

= 56 ÷ 7

= 8

24  133,000 ÷ 7,000

= (133,000 ÷ 1,000) ÷ 7

= 133 ÷ 7

= 19

25  120,000 ÷ 8,000

= (120,000 ÷ 1,000) ÷ 8

= 120 ÷ 8

= 15

26 9; 9; 9 27 17; 17; 17 28 634; 634; 634 29 290; 290; 290

30 10 31 300 32 7,000 33 20 34 400 35 5,000

36 7,865 ÷ 41 rounds to 8,000 ÷ 40 = 200

37 9,125 ÷ 345 rounds to 9,000 ÷ 300 = 30

38 9,825 ÷ 206 rounds to 10,000 ÷ 200 = 50

39 7,226 ÷ 871 rounds to 7,200 ÷ 900 = 8

40 5,299 ÷ 49 rounds to 5,000 ÷ 50 = 100

41 3,654 ÷ 27 rounds to 3,600 ÷ 30 = 120

Practice 2.5:

Dividing by 2-Digit Numbers

Pages 49–54

1 7 2 10 R 30 3 19 R 20 4 2 R 10 5 7 R 10 6 6 R 2

7 2 R 17 8 2 R 20 9 6 R 2 10 3 R 43 11 9 R 58 12 4 R 37

13 50 14 21 R 21 15 25 16 21 R 33 17 125 18 640 R 7

19 278 R 3 20 146 R 3 21 174 R 8 22 2 R 3 23 3 R 1

24 16 R 9 25 16 R 6 26 12 R 2 27 11 R 4 28 92

29 61 R 5 30 36 R 7

Practice 2.6: Order of Operations

Pages 55–62

1  26 + 8 − 19 = 15

Step 1 26 + 8 = 34

Step 2 34 – 19 = 15

2  12 + 16 − 9 + 3 = 22

Step 1 12 + 16 = 28

Step 2 28 – 9 = 19

Step 3 19 + 3 = 22

3  58 − 23 + 11 − 6 = 40

Step 1 58 – 23 = 35

Step 2 35 + 11 = 46

Step 3 46 – 6 = 40

4 

Numeric ExpressionOrder of Operations Performed

First Second Third

12 � 14 � 9 = 35 + +

60 � 18 � 7

70 � 15 � 49

23 � 16 � 7 � 12

15 � 12 � 17 � 6

= 71

= 6

= 44

= 14

+

–

+

–

–

–

– +

+ –

5

6

7

8  25 × 3 ÷ 5 = 15

Step 1 25 × 3 = 75

Step 2 75 ÷ 5 = 15

9  200 ÷ 10 × 3 ÷ 5 = 12

Step 1 200 ÷ 10 = 20

Step 2 20 × 3 = 60

Step 3 60 ÷ 5 = 12

10  250 ÷ 5 ÷ 10 × 2 = 10

Step 1 250 ÷ 5 = 50

Step 2 50 ÷ 10 = 5

Step 3 5 × 2 = 10

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11 

Numeric ExpressionOrder of Operations Performed

First Second Third

30 � 2 � 5 = 300 × ×

6 � 10 � 5

28 � 7 � 4

40 � 8 � 5

20 � 10 � 8 � 2

120 � 12 � 2 � 16

= 12

= 16

= 1

= 8

= 80

÷×

÷ ×

÷ ÷

÷ × ÷

÷ ÷ ×

12

13

14

15

16   14 + 9 × 7 = 77

Step 1 9 × 7 = 63

Step 2 14 + 63 = 77

17  200 ÷ 20 + 5 = 15

Step 1 200 ÷ 20 = 10

Step 2 10 + 5 = 15

18  80 − 16 ÷ 4 = 76

Step 1 16 ÷ 4 = 4

Step 2 80 – 4 = 76

19 

Numeric ExpressionOrder of Operations Performed

First Second

25 � 5 � 3 = 10 × −

90 � 16 � 8

83 � 72 � 6

5 � 90 � 7

240 � 20 � 15

7 � 80 � 160

= 92

= 71

= 635

= 27

= 400

÷ +

÷ –

× +

÷ +

× –

20

21

22

23

24  40 − 6 + 10 × 3 = 64

Step 1 10 × 3 = 30

Step 2 40 – 6 = 34

Step 3 34 + 30 = 64

25  36 ÷ 6 − 25 ÷ 5 = 1

Step 1 36 ÷ 6 = 6

Step 2 25 ÷ 5 = 5

Step 3 6 – 5 = 1

26  25 × 4 − 36 ÷ 9 = 96

Step 1 25 × 4 = 100

Step 2 36 ÷ 9 = 4

Step 3 100 – 4 = 96

27 

Numeric ExpressionOrder of Operations Performed

First Second Third Fourth

60 � 3 � 14 � 2 = 48 ÷ × +

20 � 5 � 2 � 6

13 � 6 � 2 � 12 � 4

27 � 3 � 40 � 6

64 � 60 �12 � 3

42 � 7 � 2 � 7 = 11

= 16

= 4

= 249

= 40

÷

+× –

× – +

÷ × +

+–×

÷ – +

28

29

30

31

32  (11 + 5) ÷ 16 = 1

Step 1 11 + 5 = 16

Step 2 16 ÷ 16 = 1

33  63 − (9 × 7) = 0

Step 1 9 × 7 = 63

Step 2 63 – 63 = 0

34  32 ÷ (14 + 2) = 2

Step 1 14 + 2 = 16

Step 2 32 ÷ 16 = 2

35 

Numeric ExpressionOrder of Operations Performed

First Second

3 � (72 � 8) = 27 (÷) ×

(40 � 5) � 11

(36 � 15) � 2

36 � (15 � 2)

(62 � 10) � 6

70 � (16 �9)

(÷)

(–)

(×)

(+)

(–)

= 88

= 42

= 6

= 12

= 10

×

×

–

÷

÷

36

37

38

39

40  7 + (8 − 4) × 10 = 47

Step 1 8 – 4 = 4

Step 2 4 × 10 = 40

Step 3 7 + 40 = 47

41  32 ÷ (7 + 1) × 9 − 5 = 31

Step 1 7 + 1 = 8

Step 2 32 ÷ 8 = 4

Step 3 4 × 9 = 36

Step 4 36 – 5 = 31

42  (47 + 12) − 10 ÷ 5 × 3 = 53

Step 1 47 + 12 = 59

Step 2 10 ÷ 5 = 2

Step 3 2 × 3 = 6

Step 4 59 – 6 = 53

43 

Numeric ExpressionOrder of Operations Performed

First Second Third Fourth

100 � (720 � 200) � 2 = 560 (+) ÷ +

24 � 5 � (125 � 80)

360 � (98 � 22) � 19 � 30

11 � (34 � 16) � 5

7 � 6 � (18 � 6)

21 � (2 � 5) � 12 � 8= 28

= 75

= 27

= 21

= 30

(–)

(+)

(–)

×

÷

÷

×

÷

–

+(+)

(+)

× –

–

–×

44

45

46

47

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Practice 2.7: Real-World Problems:

Multiplication and Division

Pages 63–68

Practice 7

1 118 ÷ 9 = 13 R 1; 13 pages of the album are full and the

last page has 1 card.

2  146 × $30 = $4,380

The club collected $4,380 a month.

12 × $4,380 = $52,560

The club collected $52,560 in fees for the year.

3 1,250 ÷ 30 = 41 R 20; She needs 42 egg trays to hold all

the eggs.

4  Total amount of pineapple juice = 18 × 40

= 720 oz

Cost of pineapple juice = 720 ÷ 16 × $1

= $45

He has to pay $45 altogether.

5a 4,500 ÷ 25 = 180; There are 180 families.

5b  Method 1

180 × $32 = $5,760

$5,760 – $4,500 = $1,260

Method 2

$32 – $25 = $7

180 × $7 = $1,260

The charity will need $1,260 more.

6  Total cost for adults = 10 × $13

= $130

Total cost for children = 18 × $7

= $126

Total cost for adults and children = $130 + $126

= $256

They pay $256 altogether.

7a  Length = 26 + 10 = 36 cm

Area of board = 36 × 26 = 936 cm2

Area of each piece = 936 ÷ 9 = 104 cm2

The area of each piece is 104 square centimeters.

7b  To give 9 smaller equal-sized pieces, the board

would have to be divided evenly along its length.

Possible dimensions of each piece: 4 cm × 26 cm

Check: 4 × 26 = 104

8  Number of blue chairs = 36 × 12 = 432

Number of yellow chairs = 912 – 432 = 480

Number of rows of yellow chairs = 480 ÷ 20 = 24

There are 24 rows of yellow chairs.

9  Number of weekdays worked each week = 4 days

Number of Saturdays and Sundays worked each

week = 2 days

1 weekday $186

4 weekdays 4 × $186 = $744

1 day of the weekend $248

1 weekend 2 × $248 = $496

$744 + $496 = $1,240

He earns $1,240 in 1 week.

10a  Parking fee from 9.30 a.m. to 10.30 a.m. = $8

Parking fee from 10:30 a.m. to 11 a.m. = $3

Total parking fee = $8 + $3 = $11

Sharona had to pay $11.

10b  Parking fee from 9 a.m. to 10 a.m. = $8

Parking fee from 10 a.m. to 12:30 p.m. = 5 × $3

= $15

Total parking fee = $8 + $15 = $23

Daryll had to pay $23.

Pages 69–72

Practice 8

1 

5 units $230 – $120 = $110

1 unit $22

$120 – $22 = $98

Hannah has $98.

Hannah

Francine

Peter

$120

$230

Hannah

5 units $230 – $120 = $110

1 unit $22

$120 – $22 = $98

Hannah has $98.

2  Larry’s Age Sister’s Age Total Age

10 7 17

10 + 1 = 11 7 + 1 = 8 19 (too little)

10 + 2 = 12 7 + 2 = 9 21 (too little)

10 + 3 = 13 7 + 3 = 10 23 (too little)

10 + 4 = 14 7 + 4 = 11 25

Their total age will be 25 years in 4 years’ time.

3 

2 staplers

1 box of chalk

$10

$18

3 boxes of chalk

$18 – $10 = $8

$8 ÷ 2 = $4

A box of chalk costs $4.

$10 – $4 = $6

$6 ÷ 2 = $3

A stapler costs $3.

$4 + $3 = $7

The total cost of 1 box of chalk and 1 stapler is $7.

ANSWER KEY MATH WORKBOOK

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4 

After18 postcards sold

Sally

Marta

Sally

Marta

Before

3 units 18

1 unit 18 ÷ 3 = 6

4 units 4 × 6 = 24

Each girl had 24 postcards.

5 

basket

basket

12 apples

7 apples

3,105 g

1,980 g

Difference in number of apples = 12 – 7 = 5

Mass of each apple = (3,105 – 1,980) ÷ 5

= 225 grams

Mass of basket = 3,105 – (12 × 225)

= 405 grams

The mass of the basket is 405 grams.

Math Journal

Pages 73–74

1 The quotient tells the number of pages that Kelly can

possibly allocate to each month. The number of pages that

Kelly can possibly allocate to each month is 30. The

remainder tells the number of pages left over. There are 10

pages left over.

2 No. He should multiply before he adds in a numeric

expression that contains both operations.

The correct solution: 6 + 4 × 2 = 6 + (4 × 2)

= 6 + 8 = 14

3  The models were drawn correctly but the worked

solution was incorrect. The correct answer should be:

$408 – $7 – $2 = $399

$399 ÷ 3 = $133

$133 + $2 = $135

Abel has $135.

Put On Your Thinking Cap!

Pages 75–76

1  Number of packets of stickers = 37 ÷ 8 = 4 R 5

Cost of 4 packets of stickers = 4 × $1 = $4

Cost of 5 stickers = 5 × 15¢ = 75¢

Cost of 4 packets of stickers and 5 stickers

= $4 + 75¢ = $4.75

The least amount that Clement spends is $ 4.75.

2  Number of candles made to make up for one member

dropping out = 39 × 3 = 117

So each member makes 117 candles.

Total number of candles made = 40 × 117

= 4,680

They make 4,680 candles altogether.

3  (26 × 3,000 cm) + (27 × 10 cm) = 78,270 cm

= 782.7 m

The length of the fence is 782.7 meters.

4 

Number of

quarters ValueNumber of

dimes ValueTotal Value

Is the totalvalue

$9.25?

2019

500¢475¢

4445

440¢450¢

940¢925¢

NoYes

There are 19 quarters and 45 dimes.

Pages 77–78

1 y g

$20

$268

Darcy

Jason

Maria

$268 + $20 = $288

4 units $288

1 unit $72

(2 × $72) – $20 = $124

Darcy and Jason have $124 altogether.

2  10

Juan

22

Rachel

22 – 10 = 12

2 units 12 marbles

1 unit 6 marbles

3 × 6 +10 = 28 marbles or

22 + 6 = 28 marbles

Each of them had 28 marbles at first.

3 

Number of pens

Number of pencils Total

If one pen is traded for 2 pencils, is the

total number of pencils 48?

101518

201512

303030

10 × 2 + 20 = 40 15 × 2 + 15 = 45 18 × 2 + 12 = 48

Gerry had 18 pens and 12 pencils before the trade.

Cumulative Review for Chapters 1–2

Pages 79–92

1 100,070 2 560,000 3 5,080,005 4 2,400,720 5 One

hundred twenty thousand, four hundred fifty 6 Five

hundred thousand, three hundred twelve 7 One million,

fifty thousand, four hundred 8 Five million, seven

hundred thirty-two thousand, eight hundred 9 8,000

10 900 11 1,000,000 12 30,000 13 300,000 14 7 15 5

16 ten thousands 17 hundred thousands 18 900,000

19 30,000 20 6,030,090 21 > 22 > 23 < 24 >

25 1,280,500; 528,100; 528,010; 258,100 26 276,300;

286,300; 296,300; 306,300; 316,300. Rule: Count on by

10,000. 27 8,000 + 3,000 = 11,000 28 7,000 – 6,000 =

1,000 29 2,000 × 5 = 10,000 30 2,700 ÷ 3 = 900

ANSWER KEY MATH WORKBOOK

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31  4,000 + 3,000 + 9,000 = 16,000

0 + 900 + 100 = 1,000

16,000 + 1,000 = 17,000

32  8,000 + 6,000 + 7,000 = 21,000

600 + 300 + 700 = 1,600 2,000

21,000 + 2,000 = 23,000

33  5,000 – 1,000 = 4,000

800 – 100 = 700 1,000

4,000 + 1,000 = 5,000

34  7,000 – 2,000 = 5,000

900 – 600 = 300 0

5,000 + 0 = 5,000

35 96 × 96 = 9,216; 9,216 in.2 36 $5,651 + $853 = $6,504;

$6,504 37 176 – 19 = 157; 157 gallons 38 2,000 ÷ 25 = 80;

80 pounds 39 3,150 40 2,500 41 238,000

42  7,350;

147 × 50 = 147 × 5 × 10

= 735 × 10

= 7,350

43  12,600;

63 × 200 = 63 × 2 × 100

= 126 × 100

= 12,600

44  6,342,000;

906 × 7,000 = 906 × 7 × 1,000

= 6,342 × 1,000

= 6,342,000

45 Answers vary. Sample: 40 × 60 = 2,400 46 Answers

vary. Sample: 300 × 30 = 9,000 47 Answers vary. Sample:

1,000 × 20 = 20,000 48 Answers vary. Sample: 5,000 × 80

= 400,000 49 3,690; Estimate: 80 × 50 = 4,000 50 1,638;

Estimate: 80 × 20 = 1,600 51 16,225; Estimate: 300 × 60 =

18,000 52 70,848; Estimate: 700 × 100 = 70,000

53 341,056; Estimate: 5,000 × 70 = 350,000 54 228,306;

Estimate: 9,000 × 30 = 270,000 55 356 56 19 57 17

58  15;

900 ÷ 60 = 900 ÷ 10 ÷ 6

= 90 ÷ 6 = 15

59 240;

96,000 ÷ 400 = 96,000 ÷ 100 ÷ 4

= 960 ÷ 4 = 240

60 56;

504,000 ÷ 9,000 = 504,000 ÷ 1,000 ÷ 9

= 504 ÷ 9 = 56

61 4,500 ÷ 50 = 90 62 6,000 ÷ 200 = 30 63 8,000 ÷ 4,000 = 2

64 4,200 ÷ 70 = 60 65 6 66 2 R 11 67 17 68 33 R 8

69 542 R 15 70 103 R 11

71 36

60 + 12 – 36 = 72 – 36

= 36

72 30

10 × 9 ÷ 3 = 90 ÷ 3

= 30

73 36

29 + 42 ÷ 6 = 29 + 7

= 36

74 35

(90 – 85) × 7 = 5 × 7

= 35

75 403

50 × 8 + 12 ÷ 4 = 400 + 12 ÷ 4

= 400 + 3 = 403

76 30

69 ÷ 3 – 3 + 10 = 23 – 3 + 10

= 20 + 10 = 30

77  Before

Cranberry bars

Walnut bars?

After

Cranberry bars

Walnut bars

66

3 units 66

1 unit 66 ÷ 3 = 22

8 units 8 × 22 = 176

He had 176 bars at first.

78  5 curtains: 5 × 3 = 15 yd

5 curtains + 1 cushion cover: 15 + 2 = 17 yd

20 – 17 = 3 yd

She has 3 yards of fabric left.

79  25 L = 25,000 mL

200 + 300 = 500 mL

25,000 ÷ 500 = 50

2 × 50 = 100 cups

The class sold 100 cups of orange juice.

80  Perimeter of rectangle = 12 + 15 + 12 + 15 = 54 in.

Perimeter of square = 4 × 19 = 76 in.

Perimeter of 2 triangles = 220 – 54 – 76 = 90 in.

Length of one side of each triangle = 90 ÷ 6 = 15 in.

The length of one side of each triangle is 15 inches.

81  260 ÷ 5 = 52

52 × $25 = $1,300

She bought the bags for $1,300.

260 ÷ 2 = 130

130 × $18 = $2,340

She sold them for $2,340.

$2,340 – $1,300 = $1,040

She made $1,040.

82  Correct Wrong Total Score

8 7 16 – 14 = 2

9 6 18 – 12 = 6

10 5 20 – 10 = 10

11 4 22 – 8 = 14

He answered 11 questions correctly.

ANSWER KEY MATH WORKBOOK

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83  20 × 14 = 280

280 ÷ 8 = 35

35 – 20 = 15

The container of pellets will last Lewis’ goldfish 15

more days.

84a  72 ÷ 9 = 8

Joan takes 8 hours to pick 72 pounds of strawberries.

84b  8 × 2 = 16

16 × $12 = $192

Joan earns $192.

85  2,488 + 160 = 2,648

2,648 + 2,488 = 5,136

5,136 ÷ 2 = 2,568

There are 2,568 students in Bellow School.

86  2 × 1,250 = 2,500 mL

1,250 + 2,500 = 3,750 mL

3,750 ÷ 15 = 250 mL

Each glass contains 250 milliliters of lemonade.

Chapter 3

Fractions and Mixed Numbers

Practice 3.1: Adding Unlike Fractions

Pages 93–98

1 Answers vary. ¾ = 6⁄8 = 9⁄12 2 2⁄5 = 4⁄10 = 6⁄15 3 5⁄6 = 10⁄12 =

15⁄18 4 1⁄7 = 2⁄14 = 3⁄21 5 ¾ 6 2⁄5 7 ⅔ 8 3⁄7 9 3⁄12; 5⁄12 10 1⁄10;

4⁄10 11 5⁄9; 6⁄9 12 6⁄16; 9⁄16 13 ⅔ = 4⁄6 = 6⁄9 = 8⁄12; ¾ = 6⁄8 =

9⁄12; 12 14 ¼ = 2⁄8 = 3⁄12; 5⁄6 = 10⁄12; 12 15 5⁄6 = 10⁄12 = 15⁄18 =

20⁄24; ⅜ = 6⁄16 = 9⁄24; 24

16  15

12

1⁄5 + ½ = 2⁄10 + 5⁄10

= 7⁄10

17  16

14

1⁄6 + ¼ = 2⁄12 + 3⁄12

= 5⁄12

18  15

23

1⁄5 + ⅔ = 3⁄15 + 10⁄15

= 13⁄15

19 

12 �

12 �

12

83 11

20 ¼ + ⅔ = 11⁄12

21  ⅓ + 1⁄9 = 3⁄9 + 1⁄9

= 4⁄9

22 ⅝ + 2⁄4 = ⅝ + 4⁄8

= 9⁄8

=1⅛

23 ½ + 6⁄7 = 7⁄14 + 12⁄14

= 19⁄14

=15⁄14

24 4⁄8 + 1⁄5 = 20⁄40 + 8⁄40

= 28⁄40

=7⁄10

25  ½ + 0 = ½ 26  1 + 0 + ½ = 1½

Practice 3.2:

Subtracting Unlike Fractions

Pages 99–101

1 

13 � 1

4 �

13 �

14 �

412

×4

×4

4

12

×3

×3

3

12

312

⅓ − ¼ = 4⁄12 − 3⁄12

= 1⁄12

2  7⁄12 − 2⁄4 = 7⁄12 − 6⁄12

= 1⁄12

3  4⁄5 − ⅓ = 12⁄15 − 5⁄15

= 7⁄15

4  1 − 5⁄6 − 1⁄12 = 12⁄12 − 10⁄12 − 1⁄12

= 1⁄12

5  7⁄9 − 1⁄6 = 14⁄18 − 3⁄18

= 11⁄18

6  1 – 0 = 1 7  ½ – 0 = ½

Math Journal

Page 102

4⁄5 = 8⁄10

8 out of 10 parts should be shaded instead.

½ = 5⁄10

5 out of 10 parts should be taken away.

45

12

?

Total = 10 parts

Remainder = 2 parts

Difference = 3⁄10

ANSWER KEY MATH WORKBOOK

C A L V E R T E D U C A T I O N

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Practice 3.3: Fractions, Mixed

Numbers, and Division Expressions

Pages 103–106

1 4 ÷ 5 = 4⁄5 2 5⁄7 3 3⁄10 4 4⁄9 5 2⁄11 6 5⁄12 = 5 ÷ 12

7 1⁄10 = 1 ÷ 10 8 6⁄7 = 6 ÷ 7 9 3 ÷ 2 = 3⁄2 = 1½

10 

7 � 4 �

�

�

� 1 �

�

7

4

4

4

3

4

3

4

3

41

11 35 � 11 �

�

�

� 3

�

�

35

11

33

11

2

2

11

2

113

11

12 7 � 2 � 3

1

2

32 7 6 1

13 

9 � 4 � 2 1

4

24 9 8 1

14 18 � 5 � 3

3

5

35 18 15 3

15 

18 � 4 �

�

�

18

4

9

2

1

24

16 22 � 6 �

�

�

22

6

11

3

2

33

Practice 3.4: Expressing Fractions,

Division Expressions, and

Mixed Numbers as Decimals

Pages 107–108

1 = 65⁄100 = 0.65 2 = 76⁄100 = 0.76 3 = 94⁄100 = 0.94

4 

4.2

3.5

2.25

4.04

415

312

214

4 125

5

6

7

8  = 20⁄5 + 2⁄5

= 4 + 2⁄5

= 4 + 0.4 = 4.4

9  = 40⁄20 + 7⁄20

= 2 + 7⁄20

= 2 + 0.35 = 2.35

10  = 25⁄25 + 7⁄25

= 1 + 7⁄25

= 1 + 0.28 = 1.28

11  = 603⁄25

= 243⁄25

= 24.12

Each piece is 243⁄25 or 24.12 feet long.

Practice 3.5: Adding Mixed Numbers

Pages 109–112

1 � 1 � 2

� 3

8

12

3

11

12

12

2 2 15 � 3 1

2

� 2 � 3

� 5

2 5

7

10 10

10

3  = 34⁄14 + 25⁄14

= 59⁄14

4  = 57⁄12 + 33⁄12

= 810⁄12

= 85⁄6

5  = 4⅔ 0 + 19⁄30

= 51⅓ 0

6  = 122⁄18 + 915⁄18

= 2117⁄18

7 � 1 � 2

� 3

� 4

12 5

1515

17

15

2

15

8 

� 3 � 1

� 4

� 5

5 8

1212

13

12

1

12

9  = 215⁄20 + 38⁄20

= 523⁄20

= 63⁄20

10  = 210⁄18 + 115⁄18

= 325⁄18

= 47⁄18

11  = 73⅔ 6 + 915⁄36

= 1647⁄36

= 171⅓ 6

12  = 57⁄12 + 19⁄12

= 616⁄12

= 74⁄12

= 7⅓

13  10 + 7½ = 17½ 14  4½ + 10 = 14½

Practice 3.6:

Subtracting Mixed Numbers

Pages 113–116

1 � 4

89 � 3

� 1

3

9

5

9

2 

� 3 � 2

� 1

9

24

5

24

14

24

3 = 310⁄18 – 19⁄18

= 21⁄18

4  = 710⁄12 – 23⁄12

= 57⁄12

ANSWER KEY MATH WORKBOOK

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5  � 3 � 1 7

8

� �

�

2

8

210

81

7

8

13

8

6 � 5 � 3 5

12

� �

�

4

12

416

123

5

12

111

12

7  = 43⁄15 – 15⁄15

= 318⁄15 – 15⁄15

= 213⁄15

8  = 69⁄24 – 320⁄24

= 533⁄24 – 320⁄24

= 213⁄24

9  = 73⁄12 – 511⁄12

= 615⁄12 – 511⁄12

= 14⁄12

= 1⅓

10  = 84⁄12 – 49⁄12

= 716⁄12 – 49⁄12

= 37⁄12

11  12½ – 8½ = 4 12  20 – 5½ = 14½

Practice 3.7: Real-World Problems:

Fractions and Mixed Numbers

Pages 117–120

Practice 7

1 12 ÷ 5 = 12⁄5 = 22⁄5; Elena gives each friend 22⁄5 pieces of

banana bread.

2 2,001 ÷ 5 = 2,001⁄5 = 4001⁄5; The household used an

average of 4001⁄5 gallons of water each day.

3  50 – 5 = 45

45 ÷ 7 = 45⁄7 = 63⁄7

The length of each piece of string is 63⁄7 yards.

4  55 – 4 = 51

51 ÷ 6 = 51⁄6

= 83⁄6

= 8½

The weight of pears in each bag is 8½ pounds.

5  ⅜ + 1⁄6 = 9⁄24 + 4⁄24

= 13⁄24

13⁄24 quart of water is collected in the container in the two

hours.

6  8⁄9 – ¾ = 3⅔ 6 – 27⁄36

= 5⁄36

5⁄36 pound of ground turkey is left.

7  237⁄12 + 195⁄6 = 237⁄12 + 1910⁄12

= 4217⁄12

= 435⁄12

The snail is 435⁄12 inches from the bottom of the well

after 20 minutes.

8  3¼ − 1⅔ = 33⁄12 − 18⁄12

= 215⁄12 − 18⁄12

= 17⁄12

Johnny has 17⁄12 miles left to jog.

Pages 121–127

Practice 8

1  4 + 4 = 8; Susanne and Barry have 8 bagels altogether.

8 ÷ 5 = 13⁄5; Each person gets 13⁄5 bagels.

2  5 × 3 = 15; Maya cuts the 5 sheets of paper into 15

rectangles.

15 ÷ 6 = 23⁄6

= 2½

Each student gets 2½ rectangles.

3  2⁄7 + ⅓ = 6⁄21 + 7⁄21

= 13⁄21

Michael and Joel drink 13⁄21 quart of milk.

1 – 13⁄21 = 2½ 1 – 13⁄21

= 8⁄21

There is 8⁄21 quart of milk left.

4  5⁄9 + 1⁄12 = 20⁄36 + 3⁄36

= 23⁄36

She plants tomatoes and green beans on 23⁄36 of the land.

1 – 23⁄36 = 36⁄36 – 23⁄36

= 13⁄36

She plants potatoes on 13⁄36 of the land.

5  1⅔ + 25⁄6 = 14⁄6 + 25⁄6

= 39⁄6

= 4½

The total weight of the plain and wheat bagels is 4½

pounds.

5 – 4½ = ½

The weight of the sesame bagels is ½ pound.

6  2¼ – 1⅜ = 22⁄8 – 1⅜

= 110⁄8 – 1⅜

= ⅞

Jay walks ⅞ miles every morning.

2¼ + ⅞ = 22⁄8 + ⅞

= 29⁄8

= 3⅛

They walk 3⅛ miles every morning.

7  ¾ + 4⁄5 = 15⁄20 + 16⁄20

= 3½ 0

= 11½ 0

Becca uses 11½ 0 gallons of paint.

¾ + 11½ 0 = 15⁄20 + 11½ 0

= 126⁄20

= 26⁄20

= 23⁄10

They use 23⁄10 gallons of paint altogether.

8  33⁄5 + 4⅔ = 39⁄15 + 410⁄15

= 719⁄15

= 84⁄15

The monkey climbs 84⁄15 feet up the tree.

10 – 84⁄15 = 915⁄15 – 84⁄15

= 111⁄15

The monkey must climb 111⁄15 feet more to reach the top

of the tree.

ANSWER KEY MATH WORKBOOK

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45

Math Journal

Page 128

18

23

?

Step 1: Find the least common multiple of the

denominators, 8 and 3.

Step 2: Find the equivalent fractions of ⅛ and ⅔ using the

least common multiple of the denominators.

Step 3: Add the fractions.

⅛ + ⅔ = 3⁄24 + 16⁄24

= 19⁄24

The correct answer is 19⁄24.

Put On Your Thinking Cap!

Page 129

Answers vary.

8⁄25 + 1⁄5 + 1⁄5 = 8⁄25 + 5⁄25 + 5⁄25

= 18⁄25

1 – 18⁄25 = 7⁄25

Nate must place the tiles over 7⁄25 of the square grid.

Page 130

+ +

12

34

34

3¾ + 3¾ + ½ = 8

Paul has 8 kilograms of cement and sand mixture.

He does not have enough mixture.

10 – 8 = 2

Paul needs 2 kilograms more of the mixture.

Chapter 4

Multiplying and Dividing

Fractions and Mixed Numbers

Practice 4.1:

Multiplying Proper Fractions

Pages 131–132

1  1

3

8

3

2 4

2 3⁄16 3 35⁄96 4 7⁄66 5 1⁄6 6 � �

�

1

5

24

5

3 8

7 � �

�

2

18

77

9

7 11

8 � �

� �

� 1

�

�

�

1

7

25

2 7

5 10

7

5

7

55

5

9 � �

� �

� �

�

1

1

1

2

3

3 8

4 9

8

34

2

3

Practice 4.2: Real-World Problems:

Multiplying with Proper Fractions

Pages 133–138

1  waf es and scrambled eggs

waf es

Lena uses 2⁄5 of the total number of

eggs to make waffles.

2 

dress

56

yd

1 yd

jewelbox

The model shows that:

5 units 5⁄6 yd

1 unit 1⁄6 yd

Dawn uses 1⁄6 yard of lace for the

jewel box.

3  4⁄5 × ¾ = 12⁄20

= 3⁄5

Megan took 3⁄5 hour to finish the job.

4  4⁄5 × ⅞ = 28⁄40

= 7⁄10

Lily pours 7⁄10 quart of milk into the bowl.

5  2⁄7 × ¾ = 6⁄28

= 3⁄14

Eduardo ran 3/14 mile.

ANSWER KEY MATH WORKBOOK

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6 

savedspent

⅓ of Jenny’s total paycheck is

saved.

7  people who do not wear

glassespeople who wear glasses

males

1⁄12 of the family are males who

do not wear glasses.

8 

frogs grasshoppers cranes

5⁄16 of the origami figures are grasshoppers.

9  1 − 5⁄12 = 7⁄12

7⁄12 × ⅔ = 7⁄18

Of the flowers in the garden, 7⁄18 are red roses.

10  1 − 2⁄5 = 3⁄5

3⁄5 × ¼ = 3⁄20

3⁄20 of the collection are foreign coins that are not from

Mexico.

Practice 4.3: Multiplying

Improper Fractions by Fractions

Pages 139–142

1 �

3

4

2 �

2

3

3 �

5

61

4  � 7 × 14 × 3

= 712

5 �

= 928

= 94

× 17

98 ÷ 2

× 2 ÷ 27

6  �

= 45

= 41

× 15

8 ÷ 23 ÷ 3

× 3 ÷ 310 ÷ 2

7 �

= 1 × 14

= 14

= 33

× 14

15 ÷ 59 ÷ 3

× 3 ÷ 320 ÷ 5

8 

= 1

= 1 × 21 × 2

= 14 ÷ 4

× 8 ÷ 42

3 ÷ 34

× 86 ÷ 3

9 

= 24

= 8 × 31 × 1

= 87 ÷ 7

× 21 ÷ 71

16 ÷ 27

× 212 ÷ 2

10 

= 2

= 3 × 23 × 1

= 312 ÷ 4

× 8 ÷ 41

15 ÷ 512

× 85 ÷ 5

11 

= 16

= 4 × 41 × 1

= 49 ÷ 9

× 36 ÷ 91

32 ÷ 89

× 368 ÷ 8

12 �

= 2120

= 1 120

= 7 × 34 × 5

78 ÷ 2

× 6 ÷ 25

13 �

= 779

= 8 59

= 11 × 73 × 3

1112 ÷ 4

× 28 ÷ 43

14  �

= 212

= 10 12

= 7 × 31 × 2

21 ÷ 35

× 156 ÷ 3

= 75 ÷ 5

× 15 ÷ 52

15 �

= 454

= 11 14

= 5 × 92 × 2

25 ÷ 54

× 1810 ÷ 5

= 54 ÷ 2

× 18 ÷ 22

16  �

= 353

= 11 23

= 5 × 73 × 1

30 ÷ 39 ÷ 3

× 72

= 10 ÷ 23

× 72 ÷ 2

17 �

= 3512

= 21112

= 7 × 54 × 3

14 ÷ 28 ÷ 2

× 53

Practice 4.4: Multiplying Mixed

Numbers and Whole Numbers

Pages 143–146

1 �

�

�

3

22

3

2 

�

�

�

14

7

36

3  � 215

× 15

= 21 × 155

= 3155

= 63

4  � 177

× 28

= 17 × 287

= 4767

= 68

5  � 24 × 116

= 24 × 116

= 2646

= 44

6  � 92

× 18

= 9 × 182

= 1622

= 81

7 � 114

× 16

= 11 × 164

= 1764

= 44

8  � 32 × 258

= 32 × 258

= 8008

= 100

9  � 4 × 259

= 1009

= 11 19

= 999

+ 19

10  � 5 × 177

= 857

= 12 17

= 847

+ 17

11  94

× 7

= 634

= 15 34

= 604

+ 34

12  354

× 2

= 704

= 17 12

= 684

+ 24

13  95

× 12

= 1085

= 21 35

= 1055

+ 35

14  12 × 198

= 2288

= 28 12

= 562

+ 12

= 572

15  21 × 239

= 4839

= 53 23

= 1593

+ 23

= 1613

16  26 × 76

= 1826

= 30 13

= 903

+ 13

= 913

Practice 4.5: Real-World Problems:

Multiplying with Mixed Numbers

Pages 147–148

1 8 guests � oranges

�

The 8 guests eat a total of oranges.

82 1

4

18

18

2  1 lb $3

8 ⅔ lb 8 ⅔ × $3

= $26

Jim pays $26 for the

chicken.

3  2 × 12⁄5 = 24⁄5

= 2.8

The area of the picture is 2.8 square yards.

ANSWER KEY MATH WORKBOOK

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4a 5 × 17⁄10 = 8½ ; She buys 8½ yards of fabric.

4b 8½ × $5 = $42½ = $42.50; She pays $42.50 for all

5 pieces of fabric.

5 1½ × 7 = 10½ ; Angela earns $10½ in a day.

10½ × 5 = 52½ ; Angela earns $52½ in a week.

52½ × 7 = 367½

= 367.50

Angela earns $367.50 in 7 weeks.

Practice 4.6: Dividing a

Fraction by a Whole Number

Pages 149–152

1 

is shaded.

16 � 3 �

18

1

118

16

2  45 � 2 �

25

45

3  67 � 3 � 2

767

4  34 � 2 �

38

34

5  25 � 3 � 2

1525

6 � 45

× 17

= 435

7 �

58

× 19

= 572

8 � 8

9 × 1

4

= 29

9  � 1011

× 15

= 211

10 

Each section is 1⁄10 of an acre.

11a  Method 1:

1 qt

49 qt

?

Method 2:

49 ÷ 4 = 4

9 × 1

4

= 19

The amount of milk in each mug is

1⁄9 quart.

11b  19 × 3 = 3

9

= 13

The amount of milk in 3 mugs is ⅓

quart.

12a  Method 1:

1 lb

35 lb

?

35 ÷ 6 = 3

5 × 1

6

= 110

The weight of 1 portion of beef is

1⁄10 pound.

12b  110

× 4 = 410

= 25

The weight of 4 portions of beef is

2⁄5 pound.

13 

5

56

÷ 4 =56

×14

=5

24

3 ×5

24 =

58

Method 21 km2

56

km2

?

Method 1

The total area of 3 smaller plots of land is ⅝ square

kilometer.

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Practice 4.7: Real-World Problems:

Multiplying and Dividing with Fractions

Pages 153–157

1 

morning

72 pages

afternoon evening

6 units 72 pages

1 unit 12 pages

5 units 60 pages

Evan typed 60 pages of notes in the morning and

afternoon.

2 

playing games

6 hours

studying talking

10 units 6 h = 360 min

1 unit 36 min

Jay spent 36 minutes talking with his friends.

3 

rent

$720

groceries and household

goods

6 units $720

1 unit $120

5 units $600

Joanne spends $600 on rent, groceries, and household

goods.

4  swam cycled run = 3,600 m

?

6 units 3,600 m

1 unit 600 m

20 units 12,000 m

= 12 km

The total distance of the triathlon is 12 kilometers.

5  Pizza 2⁄5 of the flour

Remaining flour 3⁄5 of the flour

1 – 3⁄10 = 7⁄10

Left 7⁄10 of 3⁄5

= 7⁄10 × 3⁄5

= 21⁄50

2 × 21⁄50 = 42⁄50

= 0.84

She has 0.84 pound of flour left.

6 

?

1 qt

67

qt

for houseplants used to clean bicycle

She uses 1⁄7 quart of water

for each houseplant.

7 

?

1 h

89

h

sport news and comics

world news

Ricardo spends ⅓

hour reading the

comics.

Math Journal

Page 158

Rachel did not solve the problem correctly. Rachel

subtracted ⅓ of the whole for Roberto’s share when he

actually poured ⅓ of the remainder. To find Roberto’s

share, she should have multiplied the remainder ⅔ by ⅓ .

Earl Roberto

As shown in the correct model, the fraction of juice left

should be 4⁄9.

Put On Your Thinking Cap!

Page 159

1 – ½ = ½ (remainder)

⅓ × ½ = 1⁄6 (group A)

½ + 1⁄6 = 3⁄6 + 1⁄6 = ⅔ (box and group A)

1 – ⅔ = ⅓ (group B)

⅓ ÷ 8 = ⅓ × ⅛ = ½ 4

?

kept in box group A group B

Each of the students in group B gets ½ 4 of the whole box of

markers.

Page 160sold 24

heads in the morning

sold in the afternoon

1

2of the total

number of heads left

3 units 24 heads

1 unit 8 heads

10 units 80 heads

Mimi’s Market had 80 heads of lettuce at the beginning of

the day.

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Cumulative Review for Chapters 3–4

Pages 161–174

1  13

35

⅓ + 3⁄5 = 5⁄15 + 9⁄15

= 14⁄15

2 ¾ + 1⁄12 = 9⁄12 + 1⁄12

= 10⁄12

= 5⁄6

3 3⁄5 + 2⁄7 = 2⅓ 5 + 10⁄35

= 3⅓ 5

4 8⁄9 + 2⁄5 1 + ½

= 1½

5 ⅛ + 6⁄7 + 1⁄6 0 + 1 + 0

= 1

6 

?

45

23

4⁄5 − ⅔ = 12⁄15 − 10⁄15

= 2⁄15

7 ¾ − 1⁄12 = 9⁄12 − 1⁄12

= 8⁄12

= ⅔

8 3⁄5 − 3⁄9 = 27⁄45 − 15⁄45

= 12⁄45

= 4⁄15

9 4⁄5 − ⅜ 1 − ½

= ½

10 7⁄12 − 5⁄9 ½ − ½

= 0

11 4⁄9 12 8⁄11 13 5⁄6 = 5 ÷ 6 14 7⁄12 = 7 ÷ 12

15 �

� �

� 1 �

�

7

5

5

5

2

5

2

5

2

51

16 �

� �

� 4 �

�

16

19

4

4

3

4

3

4

3

44

17 �

�

�

11

22

8

4

3

42

24 11 8 3

18 �

�

�

14

28

6

3

2

34

43 14 12 2

19 4⁄5 = 8⁄10

= 0.8

20 17⁄20 = 85⁄100

= 0.85

21 

Division expressionExpress division expression as

a mixed number a decimal

13 � 4

23 � 5

314

435

3.25

4.6

22

23 24

25 26

27 28

29 30

Problem Solving

31  67 � 58 � 15

28

67

�58

= 6 ÷ 23

� 5

8 ÷ 2

= 3 � 57 � 4

= 1528

32  45 � 10

12 � 23

45

�1012

= 4 ÷ 45

� 1012 ÷ 4

= 15 ÷ 5

� 10 ÷ 53

= 1 � 21 � 3

= 23

33  25 of 10

11 � 411

25

�1011

= 25 ÷ 5

�10 ÷ 5

7

= 2 � 2 1 � 11

= 411

34  89 of 5

12 � 1027

89

�512

= 8 ÷ 49

� 512 ÷ 4

= 2 � 59 � 3

= 1027

35  25 � 15

7 � 67

25

�157

= 25 ÷ 5

�15 ÷ 5

7

= 2 � 31 � 7

= 67

36  95 � 5

12 � 34

95

�512

= 9 ÷ 35

� 512 ÷ 3

= 35 ÷ 5

� 5 ÷ 54

= 3 � 11 � 4

= 34

2 2 _ 7 + 3 1

_ 2 = 5 11

__ 14

2 2

_ 7 + 3 1

_ 2 = 2 4

__ 14

+ 3 7 __ 14

= 5 11

__ 14

1 1 _ 2 + 1 5

_ 9 = 3 1

_ 8

1 1 _ 2 + 1 5

_ 9 = 1 9

__ 18

+ 1 10

__ 18

= 2 19

__ 18

= 3 1

__ 18

1 5 _ 8 + 1 1

_ 5

1 1 _ 2 + 1 = 2 1

_ 2

2 1 _ 6 + 3 4

_ 5

2 + 4 = 6

5 8 _ 9 − 3 5

_ 6 = 2 1

__ 18

= 5 16

__ 18

− 3 15

__ 18

= 2 1

__ 18

4 2 _ 7 − 2 7

_ 8 = 1 23

__ 56

= 4 16

__ 56

− 2 49

__ 56

= 3 77

__ 56

− 2 49

__ 56

= 1 23

__ 56

2 1 __ 10

− 1 4 _ 7

2 − 1 1 _ 2 = 1

_ 2

3 3 _ 8 − 1 7

_ 8

3 1 _ 2 − 1 1

_ 2 = 2

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37  159

43

76

= 4 ÷ 23

76 ÷ 2

= 2 73 3

= 149

= 159

38 

83

912

= 8 ÷ 43

912 ÷ 4

= 23 ÷ 3

9 ÷ 33

= 2 31 3

= 63 = 2

2

39  1 120

78

65

= 78 ÷ 2

6 ÷ 25

= 7 34 5

= 2120

= 1120

40  71316

254

108

= 254 ÷ 2

10 ÷ 28

= 25 52 8

= 12516

= 71316

41  36

2 14

16 = 94

16

= 1444

= 36

42  33

27 129

= 27

119

= 2979

= 33

43  231

5 36

42 = 336

42

= 1,3864

= 231

44  4212

2 56

15

= 17

6 15

= 2556

= 42 36

= 42 12

45  78

× 15

= 740

46  58

× 14

= 532

47  47

× 112

= 121

48  29

× 16

= 127

49  3⁄5 – 2⁄7 = 2⅓ 5 – 10⁄35

= 1⅓ 5

He used 1⅓ 5 pound more flour to bake bread than scones.

50  45⁄12 + 1⅔ = 45⁄12 + 18⁄12

= 513⁄12

= 61⁄12

They use 61⁄12 yards of wire altogether.

51  134

+ 313

= 1 912

+ 3 412

= 41312

= 5 112

Rosa made 5 112

quarts of mixed juice.

5 112

– 223

= 5 112

– 2812

= 41312

– 2 812

= 2 512

There were 2 512

quarts of mixed

juice left in the container.

52  1112

× 45

= 1115

He ran 1115

mile.

53  Method 11 – 1

4 = 3

4

1 – 19

= 89

89

× 34

= 23

Ashley has 23

of the packet of raisins left.

Method 21 – 1

4 = 3

419

× 34

= 336

= 112

34

– 112

= 912

– 112

= 812

= 23

54  1 pot 438

lb

12 pots 12 × 438

= 5212

lb

She used 52 12

pounds of meat altogether.

55  38

÷ 9 = 38

× 19

= 124

2 × 124

= 112

The volume of solution in two of these pails

is 112

gallon.

56  13

+ 25

= 515

+ 615

= 1115

1115

× 135 = 99

They sold 99 bottles of juice in the two hours.

57  6 units $840

1 unit $140

5 units 5 × $140

= $700

She spent $700 on the ticket and food altogether.

food

$840

ticket

58 

1 unit 800 ft

8 units 6,400 ft

The total distance he traveled was 6,400 feet.

walked 800 ft

bus jogged

59 

15

÷ 5 = 15

× 15

= 125

125

of the total amount of our was in each container.

cooking bread

packed into 5 containers

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60  67

× 78

= 68

= 34

34

× 70 = 5212

= 52.5

She used 52.5 gallons of fuel for the trip.

Chapter 5

Algebra

Practice 5.1: Using Letters as Numbers

Pages 175–182

1 10 + 6; Susan has (10 + 6) fruits. 2 x + 8 or 8 + x; Juan

has (x + 8) or (8 + x) fruits. 3 18 − 2; Henry has (18 − 2)

dollars left. 4 m − 5; Katie has (m − 5) dollars left.

5 20 − n; Hugo has (20 − n) dollars left. 6 11 + b or b + 11

7 c − 6 8 15 − p 9 d + 12 or 12 + d 10 g − 15

11 

ExpressionValue of the Expression

y � 25 y � 16

y � 5

y � 12

18 � y

35 � y

30 21

13

43

10

4

34

19

12

13

14 18m, m × 18, 18 groups of m, m groups of 18 (any

three) 15 75y, y × 75, 75 × y, y groups of 75 (any three)

16 12y, y × 12, 12 × y, 12 groups of y (any three) 17 4 × 12

or 12 × 4; Julio has (4 × 12) or (12 × 4) pencils.

18 10 × k = 10k or k × 10 = 10k; Tara has 10k pencils.

19 20 ÷ 4; Each tank contains (20 ÷ 4) gallons of

lemonade. 20 m ÷ 3 = m/3; Each person gets m/3 gallons of

lemonade. 21 f × 6 = 6f or 6 × f = 6f 22 m ÷ 3 = m⁄3

23 22 ÷ p = 22⁄p 24  t6 �

1566

= 26

25  16t � 16 × t= 16 × 156= 2,496

26  t13 � 156

13

= 12

27  x + 3

The pail contains (x + 3) gallons of water.

(x + 3) ÷ 4 = (x + 3) 4

There is (x + 3) 4

gallons of water in each container.

28 m × 2 = 2m; 15 − 2m; She has (15 − 2m) dollars left.

29  The 4 charities share (400 – g) food packages.

(400 – g) ÷ 4 = (400 – g)

4

Each charity got (400 – g)

4 food packages.

30  200 grams x eggs

100 grams x2

eggs

900 grams 9 × x2

= 9x2

eggs

Matt used 9x2

eggs.

31 14 + 3 × b = 14 + 3b

32 7 × d ÷ 5 = 7d⁄5

33  5x � 12 � 5 × 5 + 12= 25 + 12= 37

34  x10 � 2 � 5

10 + 2

= 12

+ 2

= 2 12

20 � 2x � 20 – 2 × 5= 20 –10= 10

6x5 � 12 � 6 × 5

5 + 12

= 6 + 12

= 18

35 36

37  3m 3m + 5 89

38 76 – m 24(76 – m)

2

m + 5 3(m + 5)11

4m 74m16

3m14

m14

+ 1

39 

40 

41 

42 51,457 43 3,481 44 9,430 45 234

Practice 5.2:

Simplifying Algebraic Expressions

Pages 183–186

1 9p 2 9b 3 7k 4 0 5 p 6 11a 7 0 8 7f 9 6x − 9

10 8m + 4 11 6p − 5 12 4 + k 13 10b + 1 14 3c + 8

15 12e + 3 16 8h + 6

17  8y – 3y – 7 = 5y – 7

(5y – 7) ÷ 4 = (5y – 7)

4

The length of each piece is (5y – 7)

4 yards.

18  2 × m = 2m

The mass of the 2 packages of flour is 2m pounds.

4m + 2m = 6m

Ling has 6m pounds of flour now.

19  5k − 2k = 3k

Linus had 3k paper cranes left on Monday.

3k + 4k + 5 = 7k + 5

Linus has (7k + 5) paper cranes now.

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20  An apple costs (b − 7) cents

4 pears cost 4b cents

4b + b − 7 = 5b − 7

Randy pays (5b − 7) cents.

Practice 5.3:

Inequalities and Equations

Pages 187–188

1  For y � 3, 6y 11. 2. For y � 6, 6y 36.

= 186y = 6 × 3 6y = 6 × 6

= 36

> =2

3  For y � 4, 6y 26. 4. For y � 5, 6y 24.

6y = 6 × 4= 24

6y = 6 × 5= 30

< >4

5  3x 20 6. 5x � 5 45

3x = 3 × 8= 24

5x+5 = 5 × 8 + 5= 40 + 5= 45

> =6

7  2x � 9 x � 1 8. 12 � x x � 2

12 – x = 12 – 8= 4

x ÷ 2 = 8 ÷ 2= 4

= =

2x – 9 = 2 × 8 – 9= 16 – 9

x – 1 = 8 – 1= 7

= 7

8

9 

a �

2a + 4 – 4 = 10 – 4 2a = 6 2a ÷ 2 = 6 ÷ 2 a = 3

3

b �

5b – 13 + 13 = 17 + 13 5b = 30 5b ÷ 5 = 30 ÷ 5

b = 6

6

10

11 

m �

2m – 3 + 3 = m + 3 2m = m + 3 2m – m = m + 3 – m m = 3

3

n �

12n + 7 – 7 = 8n + 15 – 7 12n = 8n + 8 12n – 8n = 8n + 8 – 8n 4n = 8 4n ÷ 4 = 8 ÷ 4 n = 2

2

12

13 

s �

2s + 16 + 6 = 4s – 6 + 6 2s + 22 = 4s2s + 22 – 2s = 4s – 2s 22 = 2s 22 ÷ 2 = 2s ÷ 2 11 = s

11

Practice 5.4:

Real-World Problems: Algebra

Pages 189–192

1a  5 × y = 5y

5y + 8

The total number of golf balls Raul has is (5y + 8).

1b I f y = 4, 5y + 8 = 5 × 4 + 8

= 20 + 8

= 28

Raul has 28 golf balls if y = 4.

2a  z × 9 = 9z

50 − 9z

The change Glenda received was (50 − 9z) dollars

2b  If z = 3, 50 − 9z = 50 − 9 × 3

= 50 − 27

= 23

Glenda received $23 as change if z = 3.

3a  w × 4 = 4w

4w + 3

Garrett’s father is (4w + 3) years old.

3b  If w = 9, 4w + 3 = 4 × 9 + 3

= 36 + 3

= 39

Garrett’s father is 39 years old if w = 9.

4a  16 × m − 16m

16m − 10

(16m − 10) pens were left in the supply room.

4b  If m = 5, 16m − 10 = 16 × 5 − 10

= 80 − 10

= 70

70 pens were left in the supply room if m = 5.

5a Sarah has (x + 4) ribbons.

5b  x + 4 = 12

x + 4 − 4 = 12 − 4

x = 8

Sarah and Jill will have the same number of ribbons for

x = 8.

6a  If y = 6,

2y + 4 = (2 × 6) + 4 3y – 9 = (3 × 6) – 9

= 12 + 4 = 18 – 9

= 16 = 9

16 > 9

If y = 6, Henry would have made more paper cranes.

6b  2y + 4 = 3y – 9

2y + 4 + 9 = 3y – 9 + 9

2y + 13 = 3y

2y + 13 – 2y = 3y – 2y

13 = y

They will have made the same number of paper cranes

for y = 13.

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7a  (y – 2) ÷ 5 = (y – 2)

5(y – 2)

5 yards of fabric was used to make each jacket.

7b  If y = 17, (y – 2)

5 = 17 – 2

5

= 155

= 33 yards of cloth was used to make each jacket if y = 17.

8a  p ÷ 2 = p2

p2

+ 2

The pen costs (p2

+ 2) dollars.

8b  If p = 5, p2

+ 2 = 52

+ 2

= 4.50If the book costs $5, the pen costs $4.50.

Math Journal

Pages 193–194

1 He should not have subtracted 10 from the numerical

value in 16w; Correct solution: 4w + 12w − 10 = 16w − 10

2 He did not follow the order of operations. He added first

before subtracting. He should have followed the left to

right rule, and subtracted before adding; Correct solution:

20p − 2p + 4p = 18p + 4p = 22p 3 He reversed the

numerator and denominator; Correct solution: 6 ÷ q = 6⁄q

4 He calculated the cost of the 3 cartons in cents but gave

his answer for the change received in dollars. He should

have converted $10 to cents first.

Correct solution:

3 × y = 3y

3 cartons of milk cost 3 y cents.

$1 = 10 ten cents

= 100 cents

$10 = 10 × 100 cents

= 1,000 cents

1,000 − 3y; Clarissa received (1,000 − 3y) cents as change.

Put On Your Thinking Cap!

Page 195

a  The cost of 7 bags is (100 − g) dollars.

(100 – g) ÷ 7 = 100 – g

7

The cost of each bag is 100 – g

7 dollars.

b  If g = 1, 100 – g

7 = 100 – 1

7

= 14.14

The cost of each bag will be $14.14 if g = 1. g cannot be 1

because the cost of each bag is a whole number.

If g = 2, 100 – g

7 = 100 – 2

7

= 14

The cost of each bag will be $14 if g = 2.

The least possible value of g is 2.

Page 196

a 

x

Girls

Boys

40

(40 – x) ÷ 2 = 40 – x2

There are 40 – x2

boys.

b  If x = 4, 40 – x2

= 40 – 42

= 362

= 18

There are 18 boys if x = 4.

Chapter 6

Area of a Triangle

Practice 6.1: Area of a Triangle

Pages 197–198

1 VW; TU 2 BC; AZ 3 ST; UV 4 AC; BD

5 base

base

basebase

base

base

heightheight

heightheight

height

height

6

8

10

7 

9 

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Practice 6.2:

Finding the Area of a Triangle

Pages 199–200

1  �

� 221 m2

12

× 26 × 17�

� 855 ft2

12

× 38 × 45�

� 1,944 cm2

12

× 72 × 54 2 3

4  �

� 252 in.2

12

× 18 × 28�

� 12 12

cm2

12

× 5 × 5 5 6�

� 24 in.2

12

× 4 × 12

7 �

� 14 in.2

12

× 7 × 4�

� 105 in.2

12

× 14 × 158

Math Journal

Pages 201–202

1  Zach: He did not multiply by ½ .

Preeti: She used the wrong height.

Brian: He used the wrong base.

James: He used the wrong base.

The area of the shaded triangle is: ½ × 4 × 4 = 8 cm2

2  Area of related rectangle = 2 × area of triangle

= 2 × 15

= 30 cm2

3 The lengths of the bases are the same, and they have the

same height. So, their areas are the same.

Put On Your Thinking Cap!

Pages 203–206

1  BE = 10 ÷ 2 = 5 cm

Area of shaded triangle = ½ × 5 × 10

= 25 cm2

2  FB = 8 ÷ 2 = 4 cm

AE = 18 ÷ 2 = 9 cm

Area of shaded triangle = ½ × 4 × 9

= 18 cm2

3a  Length = 48 ÷ 4 3b Area of triangle = ½ × 4 × 4

= 12 in. = 8 in.2

DF = 12 ÷ 3 = 4 in.

4  Area of shaded region, ABED

= area of rectangle ABCD − area of triangle DEC

= (12 × 5 ) − (½ × 8 × 5)

= 60 − 20 = 40 cm2

5  Area of triangle EDC = Area of triangle BCF

= ½ × 8 × 4 = 16 cm2

Area of triangle AEF = ½ × 4 × 4 = 8 cm2

Area of triangle CEF = (8 × 8) − (2 × 16) − 8

= 64 − 32 − 8 = 24 cm2

6  Width = 1 unit

Length = 3 units

Perimeter = 1 + 3 + 1 + 3 = 8 units

8 units 256 in.

1 unit 32 in.

3 units 96 in.

Area of triangle = ½ × 96 × 32 = 1,536 in.2

7a  ED = BF = 8 cm

ED = ⅔ AD

So, AD = 12 cm

Width = 72 ÷ 12

= 6 cm

7b  A

B

DE

F C

A

B

DE

F C

Area of shaded region, EBFD

= ED × DC

= 8 × 6

= 48 cm2

Pages 207–208

1  ½ × 32 × 32 = 512 cm2; Triangle 8

Area of Triangle 6 = ½ × 64 × 64 = 2,048 cm2

Area of Triangle 7 = ½ × 128 × 128 = 8,192 cm2

Area of Triangle 8 = ½ × 256 × 256 = 32,768 cm2

2  A

X Z

W D

B Y C

S

S

Area of square ABCD = 20 × 20 = 400 cm2

Area of triangle WDC = ½ × 10 × 20 = 100 cm2

Area of triangle S = ½ × 5 × 10 = 25 cm2

Area of unshaded parts = 100 + (2 × 25) = 150 cm2

Total area of the shaded parts = 400 – 150 = 250 cm2

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Chapter 7

Ratio

Practice 7.1: Finding Ratio

Pages 209–214

1 27 2 p

The ratio of ... Ratio

the number of points Yolanda has to the number of points Vanna has is

the number of points Norita has to the number of points Sue has is

the number of points Sue has to the number of points Norita has is

the number of points Yolanda has to the total number of points is

the total number of points to the number of points Vanna has is

8 : 11

5 : 3

3 : 5

8 : 27

27 : 11

3 4; 9

4 9; 4 5 9; 13 6 31 qt; 32 gal 7 9 : 5 8 7 : 6 9 31 : 9 10 10 : 32

11 3; 7 12 7; 10 13 3; 20 14 7; 4 15 4; 3 16 4; 14

17  18

19a 

$15 − $7 = $8

Dianne got $8.

? $7

$15 19b  The ratio of the amount of

money Linda got to the amount

of money Dianne got from

Grandma is 7 : 8.

20a 

25 − 8 = 17

Amelia has 17 postcards left.

? 8

25 20b  The ratio of the

number of

postcards Amelia

has left to the

number of

postcards she had at

first is 17 : 25.

21a 2 × 16 = 32

Clark has 32 ounces of corn.

32 − 18 = 14

Clark used 14 ounces of corn to make the casserole.

18 ?

32

21b The ratio of the amount of corn Clark used to make

the casserole to the amount of corn he had at first is 14 : 32.

22a 8 × 2 = 16; The least possible weight of red peppers is

16 pounds. 22b 13 × 2 = 26; The least possible weight of

green peppers is 26 pounds

23  Number of Counters Taken Out

Number of Counters Left in the Bag Ratio

1 5 1 : 52 4 2 : 43 3 3 : 34 2 4 : 25 1 5 : 1

Practice 7.2: Equivalent Ratios

Pages 215–216

1 4; 8 2 1; 2 3 4 : 8 = 1 : 2 in simplest form. 4 18; 27

5 6; 9 6 2; 3 7 3 8 6 9 4

10 3

9

3 4

28

411

12 18 13 24 14 20 15 49 16 72 17 81

18 6

2

6 3

7

319

20 2 21 2 22 4 23 9 24 3; 8 25 3; 7 26 1; 12 27 8; 3

Practice 7.3:

Real-World Problems: Ratios

Pages 217–220

1  24 + 18 = 42; Ms. Grande bought 42 fruits altogether.

24 : 42 = 4 : 7; The ratio of the number of apples to the

total number of fruits Ms. Grande bought is 4 : 7.

2  44 − 12 = 32; There are 32 fish filets.

12 : 32 = 3 : 8; The ratio of the number of chicken filets

to the number of fish filets in the freezer is 3 : 8.

3  12 + 3 = 15; There are 15 boys now.

18 − 2 = 16; There are 16 girls now.

The ratio of the number of boys to the number of girls in

the class now is 15 : 16.

4  $42 − $6 = $36; Monica has $36 in the end.

$18 + $6 = $24; Naomi has $24 in the end.

36 : 24 = 6 : 4

= 3 : 2

The ratio of the amount of money Monica has to the

amount of money Naomi has in the end is 3 : 2.

5  Mark

Julia

36

?

3 units 36

1 unit 12

7 units 84

Mark and Julia collected 84 tickets altogether.

6 

Roger

Calvin

18

?

3 units 18

1 unit 6

10 units 60

They have 60 stamps altogether.

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7 

A

B

260 gal

?

13 units 260 gal

1 unit 20 gal

18 units 360 gal

The total amount of water used by Households

A and B on that Saturday is 360 gallons.

8 

Water

Cleaningsolution

1,200 mL

15 units 1,200 mL

1 unit 80 mL

19 units 1,520 mL

The total volume of the mixture is 1,520 milliliters.

Practice 7.4: Ratio in Fraction Form

Pages 221–226

1 C 2 ⅜ 3 ⅜ 4 8⁄3 5 8⁄3 6 4⁄3 7 ¾ 8 3; 7 9 3⁄7 10 4⁄7

11a  18 − 6 = 12

Jack played 12 tennis matches in that week.

11b  Total number of matches played = 18 + 12

= 30Number of matches Pete played

Total number of matches

18

30

3

5= =

The ratio of the number of matches Pete played to the

total number of matches both boys played is 3

5.

11c  Number of matches Jack played

Number of matches Pete played

12

18

2

3= =

The number of matches Jack played is times the

number of matches Pete played.

12a 

Kenny’s weight

Melvin’s weight6

7=

Kenny

Melvin

The ratio of Kenny’s weight to Melvin’s weight is 6

7.

12b  Melvin’s weight

Total weight7

13=

The ratio of Melvin’s weight to the total weight of the

two boys is 7

13.

12c  Kenny’s weight

Total weight6

13=

Kenny’s weight is times the total weight of the two

boys.

13a  Kimberly’s age

Halley’s age3

1=

The ratio of Kimberly’s age to Halley’s age is .

13b  Halley’s age

Total age1

4=

The ratio of Halley’s age to their total age is .

13c Halley’s age is times Kimberly’s age.

13d Kimberly’s age is times their total age.

14a  Number of nonfiction books

Number of fiction books

4

1=

The ratio of the number of nonfiction books to the

number of fiction books is 4 : 1.

14b  The number of fiction books is times the number

of nonfiction books.

14c  Fiction

Non ction

Number of nonfiction books

Total number of books

7

9=

The ratio of the number of nonfiction books to the

total number of books in the library would be 7 : 9.

Practice 7.5:

Comparing Three Quantities

Pages 227–228

1 

60,000 � 1,000 �

000 � 100 �

8 000 � 1 000 �

Set of Numbers Greatest Common Factor

2, 6 and 8 25, 10 and 20

3, 9 and 15

6, 24 and 27

533

2 3 

4  7,700 � 100 �

360,000 � 1,000 �

2,000 � 100 �

415,000 � 1,000 �

which class of animals does the salamander belong?

� : :

�

16 : 12 : 8 5.

� : :

�

21 : 15 : 18

� : :

�

20 : 30 : 45 7.

� : :

�

7 : 21 : 35

� � � �

� � � � 4

234

3

657

5

964

7

531

4 3 3

5 5 7 7

4

5

76

8 2; 8; 9 9 9; 4; 7 10 8; 2; 5 11 7; 2; 3

2

3

6

13

3

1

1

41

3

3

4

1

4

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12 ete.

600 � 300 20. 1,600 � 400

� (600 � ) � � (1,600 � ) �

� � � �

� �

81,000 � 900 22. 31,500 � 500

� (81,000 � ) � � (31,500 � ) �

ample

100 3

6 3

2

� : 15 :

�

2 : 5 : 7

� �

13.

� 12 : :

�

3 : 7 : 11

� �

� : : 6

�

20 : 15 : 30

� �

15.

� 8 : :

�

32 : 20 : 28

� �

4

4428

3

5

4

3

216

4

75

33 44

55 44

13

1514 

16 1 : 2 : 5 = 3 : 6 : 15 17 7 : 4 : 3 = 28 : 16 : 12

18  4 : 5 : 9 = 20 : 25 : 45 19 16 : 14 : 6 = 8 : 7 : 3

20 18 : 24 : 30 = 3 : 4 : 5 21 35 : 42 : 56 = 5 : 6 : 8

Practice 7.6:

Real-World Problems: More Ratios

Pages 229–234

1  4 : 10 : 8

= 2 : 5 : 4

The ratio of the number of bottles of orange juice to the

number of bottles of fruit punch to the number of bottles

of apple juice Lolita’s parents donated was 2 : 5 : 4.

2  Amount Charity C received = $900 – $200 – $400

= $300

200 : 400 : 300

= 2 : 4 : 3

The ratio of the amount Charity A received to the

amount Charity B received to the amount Charity C

received is 2 : 4 : 3.

3   ?

35 cm

5 units 35 cm

1 unit 7 cm

2 units 14 cm

The shortest part is 14 centimeters long.

4 

?

Dave

Randy

7 years

Martin

1 unit 7

6 units 42

The total age of all three brothers is 42 years.

5a  7 units 21 dolls

1 unit 3 dolls

6 units 18 dolls

Lisa has 18 dolls.

5b  17 × 3 = 51

The total number of dolls that the three girls have is 51.

6  Amin

Barb ?

Curt

98

7 units 98

1 unit 14

29 units 29 × 14 = 406

They collected 406 seashells altogether.

7a  The ratio of Kieran’s savings to Simon’s savings to their

total savings is 9 : 2 : 11.

7b   Kieran’s savings

Total savings9

11=

Kieran’s savings is times the total amount of money

saved.

7c   Simon’s savings

Total savings2

11=

Simon’s savings is times the total amount of money

saved.

7d  9 − 2 = 7

7 units $28

1 unit $4

11 units 11 × $4 = $44

Both of them save $44 altogether.

8  Number of words Lita typed : Number of words Kala typed

= 2 : 1 = 8 : 4

Number of words Lita typed : Number of words Rose typed

= 8 : 1

1 unit 48 words

8 units 8 × 48 = 384 words

Lita typed 384 words.

9a  Vanilla

Chocolate

Strawberry

The ratio of the amount of vanilla-flavored milk to the

amount of chocolate-flavored milk to the amount of

strawberry-flavored milk it produces in a day is 6 : 3 : 1.

9b  Total number of units = 6 + 3 + 1

= 10

 

Amount of vanilla-flavored milk produced

Total amount of milk produced

6

10=

3

5=

The amount of vanilla-flavored milk produced is

times the total amount of milk produced.

9

11

2

11

3

5

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Math Journal

Pages 235–236

Andy drew the wrong number of units for chicken and

beef; Clara misunderstood the question to be asking for the

total weight of meat from the butcher and fish from the

fish market.

Chicken

Beef

Fish

?

10 lbs

5 units 10 lb

1 unit 10 ÷ 5 = 2 lb

4 units 4 × 2 = 8 lb

He bought 8 pounds of meat from the butcher.

Put On Your Thinking Cap!

Page 237

1  6 × 6 = 36

36 − 16 = 20

16 : 20 = 4 : 5

The ratio of the area of the small square to the area of

the remaining part of the larger square is 4 : 5.

2a  4 units 16 cm

1 unit 4 cm

2 units 8 cm

The perimeter of the smaller square is 8 centimeters.

2b 8 ÷ 4 = 2; The length of one side of the smaller square

is 2 centimeters.

Page 238

1a  2 units 16 plants

1 unit 8 plants

7 units 56 plants

Trish and Sarah bought 56 plants altogether.

1b  56 × $17 = $952

The total cost of the plants Trish and Sarah bought is $952.

2a  5 units 60

1 unit 12

13 units 156

The total number of boys and girls at the fair is 156.

2b  156 × $3 = $468

The total admission fees for the boys and girls is $468.

Cumulative Review for Chapters 5 to 7

Pages 239–264

1 = 5 + 9 = 14 2 = 16 – 5 = 11 3 = 4 × 5 = 20 4 = 5⁄5 = 1

5 = 4y 6 = 2a − 2 7 = 6b 8 = 7c + 5

9  d � 7 15 10. 3d � 10 11

2d � 6 3d � 2 12. (35 � d) � 5 d

< =

>>

d + 7 = 7 + 7

= 14

3d – 10 = 3 × 7 – 10

= 21 – 10

=11

2d + 6 = 2 × 7 + 6 = 14 + 6 =203d – 2 = 3 × 7 – 2 = 21 – 2 = 19

(35 ÷ d) + 5 = (35 ÷ 7) + 5= 5 + 5

=10

10

1211

 13  2e � 8 14. 3f � 3 � 18

e � f �

6g � 5 � 2g � 3 16. 4h � 11 � h � 16

g � h �

4 5

2 9

2e ÷ 2 = 8 ÷ 2e = 4

3f + 3 – 3 = 18 – 3 3f = 15 3f ÷ 3 = 15 ÷ 3 f = 5

6g – 5 + 5 = 2g + 3 + 5 6g = 2g + 8 6g – 2g = 2g + 8 –2g

4g = 8 g = 2

4h – 11 + 11 = h + 16 + 11 4h = h + 27

4h – h = h + 27 –h3h = 27

h = 9

14

1615 

17 BC; AC 18 DG; EF 19 BC; AD 20 PQ; RU

21  × 150 × 27 = 2,025; Area = 2,025 cm2

22 × 20 × 16 = 160; Area = 160 in.2

23   × 62 × 125 = 3,875; Area = 3,875 ft2

24 × 4 × 12 = 24; Area = 24 yd2

25 × 40 × 35 = 700; Area = 700 in.2

26 × 45 × 142 = 3,195; Area = 3,195 cm2

27 

Area � 120 cm2

24 × 10 = 240

240 – 120 = 120

× 24 × 10 = 12012

28 5; 11 29 7; 5 30 11; 23 31 11; 23

32 2; 5 33 8; 20; 2; 5 34 12 35 35 36 7 37 9 38 5⁄9

39 5⁄9 40 9⁄14 41 2; 3; 6 42 7; 3; 12 43 36; 54 44 12; 14

45a Jay scores (b – 3) points.

45b  Kareem scores 2b points.

b + 2b + b − 3 = 4b − 3

The three players scored (4b − 3) points.

46a  3x + 6 = (3 × 7) + 6 4x − 4 = (4 × 7) − 4

= 21 + 6 = 28 − 4

= 27 = 24

27 > 24

If x = 7, David’s book has more pages.

12

12

12

12

12

12

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46b  3x + 6 = 4x − 4

3x + 6 + 4 = 4x − 4 + 4

3x + 10 = 4x

3x + 10 – 3x = 4x − 3x

10 = x

When x = 10, the two books will have the same

number of pages.

47  AD = CD

= 2 × 18 = 36 cm

Area of shaded triangle ABC = × 18 × 36

= 324 cm2

48  AF = BF

= 12 ÷ 2

= 6 cm

Area of Δ AEF = × 6 × 12 = 36 cm2

AD = 2 × 12 = 24 cm

DE = 24 − 12 = 12 cm

Area of ΔCDE = ×12 × 12 = 72 cm2

Area of ΔBCF = × 6 × 24 = 72 cm2

Area of rectangle ABCD = 12 × 24 = 288 cm2

Area of shaded triangle CEF = 288 − 36 − 72 − 72

= 108 cm2

49a  The ratio of the number of pennies in Container A to

that in Container B at first is 45 : 79.

49b  45 − 7 = 38

79 + 7 = 86

38 : 86 = 19 : 43

The ratio of the number of pennies in Container A to

that in Container B in the end is 19 : 43.

50  Peggy

Dakota

The ratio of the distance that Peggy cycles to the

distance that Dakota cycles is .

50b  Total number of units = 3 + 1 = 4

The distance Peggy cycles is times the combined

distance.

51a  7 units $5,096

1 unit $728

3 units $2,184

It donates $2,184 to Charity A in a year.

51b  3 + 7 + 9 = 19

19 × $728 = $13,832

It donates $13,832 to all three charities in a year.

52a  3 units 24

1 unit 8

7 units 56

There are 56 girls in the camp.

52b  56 × $50 = $2,800

The total amount of fees the girls pay is $2,800.

Mid-Year Review Test Prep

Pages 251–264

Multiple Choice

1 A 2 C 3 C 4 B 5 C 6 B 7 B 8 A 9 B 10 A 11 C

12 A 13 D 14 D 15 A 16 D

Short Answer

17 7,000 18 916,236, 164,239, 35,982, 35,928 19 31,500

20 952,000 21 215 × $17 = $3,655; $3,655

22  $45,900 − $5,300 = $40,600

$40,600 ÷ 14 = $2,900; $2,900

23 

47

(2 + 4) × 7 – 6 + 11 = 6 × 7 – 6 +11

= 42 – 6 + 11

= 36 + 11

= 47

24 

38 ÷ 6 = 386

= 193

= 613

193

613

,

25  24

12

+ 338

= 2778

2778 ounces

26 

2414

– 1512

= 8 34

= 8.75 8.75

27 

16.35 kilometers

73

10 + 1

34

= 91

20

91

20 + 7

310

= 167

20

= 16.35

28 

52

12

706

× 184

= 351

× 32

= 105

2

= 5212

29a  a.

b.

1 25

× 3 = 75

× 3 = 215

= 4 15

415

× 8 = 215

× 8 = 168

5 = 33

35

4 15

miles

33 35

miles29b

30 

910

÷ 3 = 9

10 ×

13

= 3

10

31 

$11r

5r × 2 = 10r2r ÷ 2 = r10r + r = 11r

12

12

1212

31

34

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32 

a = 4

4a – 8 = a + 44a – 8 + 8 = a + 4 + 8 4a = a + 12 4a – = a + 12 – 3a = 12 a = 4

33 A

C B

base

height

34 

56 square centimeters

12

× 16 × 7 = 56 cm2

35  (56 × 50) – (12

× 56 × 50)

= 2,800 – 1,400= 1,40012 × 50 × 14 = 350

1,400 + 350 = 1,7501,750 square centimeters

36 

1,920 grams

7 units → 1,120 g 1 unit → 1,120 ÷ 7 = 160 g12 units → 12 × 160 = 1,920 g

37 2⁄7 38  Total number of units = 3 + 4 + 5 = 12

12 units 156 cm

1 unit 156 ÷ 12 = 13 cm

2 units 2 × 13 = 26 cm

26 centimeters

39a 16 39b 13

40  1 km = 1,000 m

6 km = 6,000 m

6,000 ÷ 400 = 15

There are 15 trees altogether.

15 + 1 = 16

There are 16 poles.

41 Number Between

70 and 85÷ 4 ÷ 6

75 18 R 3 12 R 3

79 19 R 3 13 R 1

The number is 79.

42  $8,250 ÷ 11 = $750

$750 + $1,250 = $2,000

$2,000 − $525 = $1,475

Andrew earns $1,475 each month.

12 × $1,475 = $17,700

He earns $17,700 in a year.

43  725 – 4

58 = 2

3140

2 3140

pounds of the sh were mackerel.

23140 – 1

78 =

3640

= 910

= 0.9He had 0.9 pounds of mackerel left.

44  2 45 + 1

25 = 3

65 = 4

15

There were 4 15 quarts of milk in Container A in the end.

10 – 4 15 = 5

45

There were 5 45 quarts of milk in Container B in the end.

5 45 – 1

25 = 4

25 = 4.4

There were 4.4 quarts of milk in Container B at rst.

45  2 units → 40 pages 1 unit → 20 pages6 units → 120 pages

There are 120 pages in the book.

1st day

left2nd day

40 pages

46a  9y – 4y = 5y

12y – 3y = 9y

18 – 15 = 3

5y + 9y + 3 = 14y + 3

The total number of vehicles remaining is (14y + 3).

46b  Trucks: Cars and vans:

12 × 7 = 84 (9 × 7) + 18 = 81

84 > 81

There are more trucks at first.

47 

K L14 in.

P

Area of triangle = 12

× 14 × 14

= 98 in.2

Area of triangle = 14

× 14 × 14

= 49 in.2

Total area of shaded parts = 98 – 49 = 49

The total area of the shaded parts is 49 square inches.

48 

10 units → 110 1 unit → 11 8 units → 88David and Gary have 88 comic books in total.

Freddie

David

Gary

110

?

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49a  3 units 900 mL

1 unit 300 mL

8 units 2,400 mL

The volume of water in Container C is 2,400

milliliters.

49b  2 + 3 + 8 = 13

13 × 300 = 3,900 mL

The total volume of water in the three containers is

3,900 milliliters.