8 Grade Summer Math rod t, - Birmingham Covington Schoolbcsonline.info/Documents/Summer/2016-2017/2015_8th_grade_Math... · Grade Summer Math rod t, ... Writing expressions in exponential

Name

8 th Grade Summer Math rod t,

For Students who will be enrolled in Mrs. Kohrman's

8th Grade Math Class in the 2015/2016 school year.

e 0201.11.

Summer Math Packet Directions.

Please complete the following packet before the start of next school year. This packet will be turned in for a grade. All odd problems in this packet are required. All even problems are optional. All work for this packet should be completed on a separate piece of paper and saved to turn in with the final packet. Answers should be transferred from your work to the packet. Each page of this packet begins with an example to guide your practice. If you have questions while completing the packet, use what resources you have available to you (last year's online textbook, family, friends, the internet). All work and the original packet should be brought to class on the first day of school. I recommend using a folder or a binder to hold and organize the packet and your work.

In order to stay on top of the work, I recommend that you complete about 3 pages per week. Don't leave it all for the last minute! This packet is meant to review previously learned concepts that will be essential to your success this year. Remember, you get out of it what you put into it!

Good luck and have a great summer!

UNIT I

NAME CLASS

Parenfin s a[ e Order of C_ s

OBJECTIVE: Finding the value of an expression that contains parentheses

Often an expression contains parentheses. To find its value, you first perform any operations inside the parentheses. Remember to follow the correct order of the four basic operations.

EXAMPLE 1

Fiund the value of (9 + 3) X (17 — 8).

Solution

(9 + 3) X (17 — 8)

12 X 9 First work inside the parentheses.

108 Then multiply.

EXAMPLE 2 Find the value of 92 — 6(5 + 8).

Solution

92 — 6(5 + 8)

92 — 6(13) First work inside the parentheses.

92 — 78 Then multiply Note that 6(13) means 6 X 13.

14 Then subtract.

Find the value of each expression.

1.

3.

54 + (2 + 4) 2. (35 — 6) X 4

(12 — 4) + 2 4. (22 + 19) X 10

5. 36 X (12 + 6) 6. 45 — (9 + 27)

7. (18 — 3) X (4 + 7) 8. (25 — 9) + (3 + 5)

9. 38 — (14 — 2) + 9 10. 9 X (24 — 16) + 4

11. 27 + (3 + 14) X 6 12. 60 — (4 X 8) + 2

13. 6(33 — 29) 14. (41 + 39)2

15. 47 + 2(12 — 9) 16. 90 — 3(8 + 5)

17. 74 + (17 — 6) X (6 ± 2) 18. (49 — 7) ± (7 — 4) X 2

19. (16 — 2 X 4) + (64 ± 4 ± 2) 20. 36 + (9 X 2) + (6 + 18 + 2)

Unit 1 Order of Operations and Number Theory Making Sense of Numbers

DATE

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reserved.

NAME CLASS DATE

Using Exponents

OBJECTIVE: Writing expressions in exponential form and finding the value of exponential expressions

You can write some multiplications in a type of "mathematical shorthand" called exponential form.

Exponential Form

X3X3X3X3X3X 3,= 37

7 identical factors of 3

7 is the exponent.

3 is the base. 111118...._

r EXAMPLE 1

Write each expression in exponential form.

a. 3 X 3 X 3 X 3 b. 5 X 5 X 7 X7X7 c,2

Solution

a. 3 X 3 X 3 X 3 = 34 b. 5X 5 X 7X 7X7= 52 X 73 c. 2=2'

EXAMPLE 2 Find the value of each exponential expression.

a.53 b. 24 X 32 X 71

Solution

a. 53 = 5 X 5 X 5 b. 24 X 32 X 71 = 2X2 X2 X2X3X3X7

=125 = 16 X 9 X7

1008

Write each expression in exponential form.

1. 7X 7X7X7X7 2. 2 X2 X 2X2X2 X2X 2 X 2

3. 3 X3X5X5 X5X 5 4. 5 X5 X5X7X 7X7

5. 2X2X 3X5 X5X 5 6. 2 X5 X5X7X 7 X7

Find the value of each exponential expression.

7. 23 8. 32 9 . 2 5 10. 73

11. 131 12. 71 13. 27 14. 35

15. 22 X 34 16. 23 X 112 17. 23 x 32 x 52

18. 23 x 32 x 72 19. 25 X 32 X 51 20. 24 X 31 X 72

Making Sense of Numbers Unit 1 Order of Operations and Number Theory

5 5 X 2 10 7

6 6 X 2 = 2 = 1-12

NAME CLASS DATE

Adding41 3 Fractions: UnF°:e D hators UNIT 2

OBJECTIVE: Finding sums of two or more fractions with unlike denominators

To add fractions that have unlike denominators, you must first write equivalent fractions that have a common denominator. Then add using the method for fractions with like denominators.

3 1 3 5 Write each sum in lowest terms. a. + b. +

Solution

a. The LCM of 8 and 2 is 8. So the least common denominator is 8. 1

r Fi

erie' rot - Then add. wt 2 1 1 X 4 4 3 1 3 4 + = 3 + 4 7 2 2 X 4 8 8 2 8 8 8 8

b. The LCM of 4 and 6 is 12. So the least common denominator is 12.

First rewrite-3

and -5

Then add. 4 6.

3 3 X 3 9 3 5 9 10 9 + 10 19 + = + = 12 = 4 4 X 3 12 4 6 12 12 12

Write each sum in lowest terms.

6 9 - 10 6 '• 6, 8

3 11 5 11 10. 71 + Trz-i 11. —5 7

12 + 20 12. +

3 1 7 13. g + --4- +

1 1 1 17. + + 3

R 1 _4_ 1 1 — 6 4 14

2 2 5 19. + +

20. -5 8 4_ 3

9 + 15 10

34 Unit 2 Fractions Making Sense of Numbers

1; 7 1 3 8 2 4

2 7 5 • 16. + +

1 3 3 14. + 3 +

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NAME CLASS DATE

UNIT 2 Addini imbers: Unlike Dediuk.linators

OBJECTIVE: Finding sums of mixed numbers with unlike denominators

To add mixed numbers that have unlike denominators, you must first write equivalent mixed numbers that have a common denominator. Then add using the method for mixed numbers with like denominators.

1 7 7 1 Write each sum in lowest terms. a. 8-5 + 3—b. 4-8 + 3-6 Siuhition

a. The LCM of 5 and 10 is 10. b. The LCM of 8 and 6 is 24. So the least common So the least common denominator is 10. denominator is 24.

1 2 7 21 8-5 8-10

7 7 1 4

9 25 1 1

11-10 7-24 = 7 + 1-24 = 8-24

Write each sum in lowest terms.

2 1. 6-3

3 4. 8-14

1 7. 2i

1 + 7-6

3 2. 9-8

3 5. 1 —4

11 8. im

1 + 3-2

1 5 3. 5-3 +2

-o

CC

C-

C

1 + 2-2

7 + g

+ 2 110

1 4 6. 3-6 + 1-15

7 + 1-9

1 5 9. 7-2 + —6

5 co

10. 7

12 _4_ R 2

11. 1

3-2 6

+ 2 4

12. -3- + 13 a,

13. 8

4-9

1 + —6 14.

7 —9 +

4 6 —15

3 5 15. 5-4 + 8-6

5

5 7 19 1 11 11 16. 3-8 + 1-10 17. 4 27) + 4E 18' 71' + 2 12

19. 1

1-4 1

+ 6-8 1

+ 8-4 20. 2

2-3 1

+ 4-6 2

+ 9-3

21. 4

1— 5 2

+ 3— 5 1

+ 22. 1

9-4 7

+ 6-8 3

+ 2-4 2

3 7 1 3 5 11 23. 4-5 + 2-10 + 9-2 24. 8-4 + —6 + 2-12


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NAME CLASS DATE

5-4 Subtracting Mixed Numbers UNIT 2

OBJECTIVE: Finding differences of mixed numbers without renaming the whole-number part

To subtract mixed numbers, you use the following general method.

Subtracting Mixed Numbers without Renaming

• 1. If necessary; write equivalent mixed numbers that have a common denominator.

2. Subtract the fractions. &• Subtract the whole .numbers, 4. If necessary, rewrite the difference in lowest terms.

7 1 3 1

Write each difference in lowest terms. a. 5-8 - 3-8 b. 5-4 - 1-3 EXAMPLE

Solution 3 9 Write equivalent 7 5-4 --> 5— a. 5- b. 8 12 mixed numl2ers

1 1 4 with common

- 3-8 - 1- -4 - 1— 3 12 denominators.

6 3 2-8 Rewrite the 5

= 4-12 2-4 difference in lowest terms.

Write each difference in lowest terms.

11 4 7 2 4-§- 2. 12-9 - 9 3 3. 711 - u 1. 6-i- - 5-13 5-

6 4. 9_ ! 7 1 6. 14-2 - 10 7 7 5. 5-8 - 2

4 1 14 7 9 1 9. 61-5 - 3-r)--0 7. 8-5 - 8-5 8. 1. 3 . - 1T;

8 2 5 3 11 1 12. 18E - 1-.1. 10. 15-9 - 4-9 11. 1-§ - -§

7 2 2 7 1 1 14. 11-3 - 2-12 15. 5-g - 1Th 13. 5m - 33

19 3 13 2 17. 10 -8- - 1 1

16. 7-gi - 6i T

Making Sense of Numbers Unit 2 Fractions 43

Without Ren. wing

UNIT 2

12.7

3 15. ,7

15 18. 28

7 8

9 10 9.

1

14 1

25

21. 16 15

9 14

Unit 2 Fractions 53

NAME CLASS DATE

plying Two Fractions

OBJECTIVE: Finding the product of two fractions

To multiply two fractions, you can use the following method.

Multiplying Fractions

1. Multiply the numerators.

2. Multiply the denominators.

3. If possible, divide both numerator and denominator by a

common factor. This will give you the final product in

lowest terms.

Write each product in lowest terms. a.

Solution 4

4 3 _ 6 b. x

1 a. 4

5 6

1 X 5 = 24

5 4 X 6

4 b* —11 4

3 11

1 X 3 3 —11 Divide each 4 by 4. 11 X 1 =

C. -§- = X 7

= 5 X 2 _ 10 Divide and 9 by 3. 3 X 7 — 21

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Write each product in lowest terms.

1 1 2. _ 1 1. X

2 8 3

4 2 5 4 7 3 9 5

7. 17 3

8. 4 12

1 4 5 11 6

2 3 5 3 13. — X 14. 9 8 18 20

25 4 21

16.X 17. 36 15 32

27 20 8 25 19. —32 21 20. 5 16

Making Sense of Numbers

NAME CLASS DATE

6-3 Multiplying a Fraction and UNIT 2

a Whole Number

OBJECTIVE: Finding and estimating products of fractions and whole numbers

To multiply a fraction and a whole number, write the whole number as a fraction with a denominator of 1. Then follow the procedure for multiplying two fractions.

Write the product 15 X hi lowest terms.

O' Solution 5 15 5 _ 5 5 X 5 25 1

15 X;=TX;- 1 X;=i-7- 2 =T=12-1

EXAMPLE I

Divide 15 and 612 y 3.

Copyright 0 by H

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rt and Winston. A

ll rights reserv ed.

You can estimate a product of a fraction and a whole number by rounding.

3 20 Estimate each product. a. Ti x 80 b. 47 X

0. Solution 3 20 Replace a. 17 X 80 b. 47 X F.

47 with 45

Replace t Stand 1 2

X 80 -3 1 with 48 x replace

20 , 2 -2--j about 20 about 32 with

EXAMPLE 2

Write each product in lowest terms.

1.1 1 2 3

x 40 2. 42 X 7 3. 15 X 3 4. X 48

1 1 9

5. -6 X 20 6. 30 X - 7 . -5 X 21 8. 24 X 9

Estimate each product.

1 1 2

9. 3 X 22 10. 30 X -8- 11. 3 X 19

7 4 11

12. 55 X -§- 13. —17 X 28 14. 54 X ;7

21 45 12 15. 35 X —51 16. X 36 17. E X 61


NAME CLASS • DATE

7-3 UNIT 2

Dividing With Fractions

OBJECTIVE: Finding quotients involving fractions

2 6 2 i 6 i In the division -3- ÷ 3 s the dividend, s the divisor, and the result of

this division is called a quotient. To divide by a fraction, you can use the following rule.

Dividing by a Fraction

1. Multiply the dividend by the reciprocal of the divisor.

2. If necessary, rewrite the result from Step 1 in lowest terms.

EXAMPLE 1 Write the quotient 3 + ,7 n lowest terms. L 2 6 i ,

Solution 1

2. 6 2 v 7 2X7_1X7_7 37— 3 — 6 = 3X — 3 X 3 — 9 3

When a division involves a whole number, write the whole number as a fraction with denominator 1.

1 8 Write each quotient in lowest terms. a. 3 + 3 b. 6+

So 1 uti on

1 1 3 = 1 1 1 X 1 1 a' 3 ±

b . 6— i9 _ § = — A

I = 1 X 4 —4- 4

Write each quotient in lowest terms.

3. 2.. 4 . 2 • 4 2.4 1. ±

3 2. . 7 5 5 • 9 3' 7 7 •7 • 7

2 . 2 2 , 2 1 . 1 1 1 3

5. Y 3

6. .. • 9 7. 3 ' 3 •8.4 . 7

15 . 3 24 . 4 4 . 8 9 .12 10. E 3 11. 9 . 13

3 . 5

13. —4

÷ 2 14. g + 6 15. ÷ —5

2 16. ± 7

3 2 1 3

17. —4

± 9 18. § ± 4 19. 6 ± 2O.8.


EXAMPLE 2

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NAME CLASS

Roundil ig cima s

OBJECTIVE: Rounding decimals

When you round a decimal, you replace the given decimal with a number that terminates at the specified decimal place.

ound each decimal to the specified decimal place. a. 24.567 to the nearest hundredth b. 18.42 to the nearest tenth c. 93.5 to the nearest whole number

Solution

a. Identify the digit in the hundredths place. Look at the digit to its right. This is the hundredths place.

24.56T1 Since the digit in the thousandths place is more than 5, replace 0 with 7.

Thus, 24.567 rounded to the nearest hundredth is 24.57.

b. Identify the digit in the tenths place. Look at the digit to its right. This is the tenths place.

18.4© Since the digit in the hundredths place is less than 5, leave 4 as 4.

Thus, 18.42 rounded to the nearest tenth is 18.4.

c. Identify the digit in the units place. Look at the digit to its right. This is the units place.

93.CD Since the digit in the tenths place is 5, replace 5 with 4.

Thus, 93.5 rounded to the nearest whole number is 94.

Round each decimal to the specified decimal place.

1. 123.451; nearest hundredth

2. 123.45; nearest tenth

3. 0.333; nearest tenth 4. 0.543; nearest hundredth

5. 19.95; nearest whole number 6. 8.09; nearest whole number

7. 3.141; nearest hundredth 8. 1.414; nearest tenth

9. 0.0045; nearest thousandth 10. 0.056; nearest tenth

11. 18.001; nearest whole number 12. 3.89; nearest whole number

Making Sense of Numbers Unit 3 Decimals 85

DATE

NAME CLASS DATE

Writing a Terminat: Decima UNIT 3

as a Fraction

OBJECTIVE: Writing a terminating decimal as a fraction or a mixed number

A terminating decimal is a decimal that has a finite number of nonzero digits to the right of the decimal point. For example, 12.345 is a terminating decimal. Terminating decimals can be written as fractions.

Write each decimal as a fraction in 1 west terms.

a. 0.7 b. 0.45

So°--tion 7 45 9 Divide numerator and a. 0.7 = b. 0.45 = = -

10 100 20 cleriorniriator by 5.

EXAMPLE 1

If a terminating decimal is greater than 1, then you can write the decimal as a mixed number.

Write each decimal as a mixed number with the fraction part in lowest terms.

EXAMPLE 2

a. 3.75

Solution

b. 7.125

75 1

Divi enumerator and 3 a. 3.75 = = 3

100 4 denominator by 25.

125 = 71

Divide numerator and b. 7.125 = 7 1000 8 denominator by 125.

Write each decimal as a fraction in lowest terms or as a mixed number with the fraction part in lowest terms.

1. 0.1 2. 0.6 3. 0.90 4. 04

5. 0.04 6. 0.25 7. 0.33 8. 0.78

9. 0.002 10. 0.100 11. 0.125 12. 0.375

13. 1.5 14. 1.7 15. 10.4 16. 99 9

17. 13.65 18. 10.250 19. 11.01 20. 35.76

21. 1.007 22. 4.555 23. 8.750 24. 6.222


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NAME CLASS DATE

Multipl g a Decimal by a Power of 10

OBJECTIVE: Multiplying a decimal by a power of 10

A power of 10 is any product of 10 with itself a finite number of times. Each of the products below is a power of 10.

10 - 10 = 102 (10 squared)

10 10 10 = 103 (10 cubed)

The exponent of 10 indicates the power of 10, or the number of times 10 is a factor in the product.

To multiply a decimal by a power of 10, move the decimal point to the right as many places as indicated by the number of zeros or the exponent in the power of 10.

Multi ly: a. 624345 X 100 b. 120.6 X 104

Solutio

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Multiply.

1. 12.225 X 102

4. 85.6 X 102

7. 0.435 X 103

10. 0.03 X 103

13. 1.006 X 103

16. 12,338.4 X 101 17. 17,229.05 X 101

a. Locate the decimal point.

6 2 4 3 4 5

2 places right

So, 624.345 X 100 = 62,434.5.

b. Locate the decimal point. Add three zeros.

120.6000

4 places right

So, 120.6 X 104 = 1,206,000.

2. 100.75 X 102

5. 35.25 X 103

8. 0.84 X 103

11. 0.007 X 103

14. 1.001 X 103

3. 88.55 X 102

6. 103.46 X 103

9. 0.01 X 103

12. 0.005 X 103

15. 0.0001 X 103

18. 88,888.25 X 102 19. 7500.08 X 103


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EXAMPLE 2 Multiply: 4.25 X 3.6

Solution

4.25 X3.6

255 1275

15.300

two decimal places one decimal place

three decimal places

Therefore, 4.25 X 3.6 = 15.300, or simply 15.3.

Multiply.

1. 12 X 8.5 2. 6.3 X 9 3. 12 X 3 6

4.125 X 4.8 5, 124X 3.1 6. 256 X 84

7. 1.5 X 2.5 8. 3.5 X 3.5 9. 4.4 X 3 2

10. 7.2 X 0.6 11. 0.3 X 11.6 12. 0.7 X 08

13. 17.2 X 3.65 14. 1.72 X 9.7 15. 8.68 X 1 7

16. 0.24 X 0.48 17. 0.65 X 0.65 18. 12.35 X 5.34

NAME CLASS DATE

10 2 Multiplying Decimals UNIT a

OBJECTIVE: Finding the product of two decimals

Multiplying two decimals is similar to multiplying whole numbers. The following examples will show you where to place the decimal point in the product.

r _miffisommito EXAMPLE 1 Multiply: 6.33 X 7

Soiution

6.33 two decimal places X 7

44.31 two decimal places

Thus, 6.33 X 7 = 44.31.

If both decimals have digits to the right of the decimal point, the product will have as many decimal places as the sum of the numbers of places in the given decimals.

102 Unit 3 Decimals Making Sense of Numbers

Dividing Decimals

Divide. a. 84.7 ± 7 b. 11.96 ± 2.3

5.2 23)119.6

115 b. 2.3)11.96


NAME CLASS DATE

OBJECTIVE: Dividing a decimal by another decimal

To divide with decimals, transform the decimal division into a whole-number division.

When dividing by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend.

When dividing by a decimal, move the decimal point in the divisor and the dividend enough places to make the divisor a whole number. Then divide as with a whole-number divisor.

= 12.1. So, 11.96 ± 2.3 = 5.2.

46 46

0

2. 72.18 ± 6 3. 213.3 ± 9

5. 1266.9 ± 3 6. 1863.5 ± 5

8. 55.5 ± 15 9. 223.2 ± 18

11. 235.3 ± 13 12. 355.3 ± 17

14. 32.66 ± 2.3 15. 33.92 ± 6 4

17. 91.98 ± 7.3 18. 129.05 ± 8 9

20. 204.96 ± 6.1 21. 509.74 ± 7 7

23. 452.6 ± 12.4 24. 1024.88 ± 18 4

Solution 12.1

a. 7)84.7 7 14 14

7 7 0

o, 84.7 ± 7

Divide. U,

1.

4.

7.

-7-a = 10.

2,. © - ,

13.

._ - r_ 16.

19.

22.

42.5 ± 5

530.4 ± 8

2.55 ± 1.5

35.88 ± 2.6

121.33 ± 1.1

240.45 ± 10.5

85.2 ± 12

124.8 ± 12

NAME CLASS DATE

UNIT 4

L Solving Proportions

OBJECTIVE: Solving a proportion for an unknown term

A proportion is an equality of two ratios. The four quantities in the proportion are called terms. If the ratios are equal, then the proportion is true.

4 2 Tell whether the proportion § -=- j. is true or false.

SoCution 4 2

Since 4 X 3 0 9 X 2, you may conclude that § 0

Therefore, the ratios are not equal. The proportion is false.

EXAMPLE 1

If a proportion involves an unknown, represented by a variable, you can find the number that makes the proportion true. To solve a proportion, find the cross products and set them equal to each other. Solve the equation that results.

If -a = ' -c

then ad = bc. b d

!Wm. --011 X 35 I EXAMPLE 2 Solve -8 =

Solution

To make the proportion true, the cross products must be equal. x 35 _ 8 40

40x = 8 X 35 Set cross products equal.

35 x

8 = 7 Divide each side by 40.

40 Therefore, x = 7.

Tell whether the given proportion is true or false.

1. 15 45 -6- =

13 2. 7 =

25 — 3. —

35 = —

15 178 14 9 4

Solve each proportion.

Ai x 30 5 Y a 18

1 _ z

5° -. =

R 3 t o 1 4

4 8 7 11. —n =

5 9 12. -i =

18 —m .§

128 Unit 4 Ratio, Proportion, and Percent Making Sense of Numbers

UNIT 4

NAME CLASS DATE

Solving Problems Involving Proportions

OBJECTIVE: Using proportions to solve real-world problems

When a constant rate is part of a real-world problem, you may be able to use a proportion to find the unknown quantity.

If an inspector examines 20 light bulbs in 18 minutes, how many light bulbs will the inspector examine in 72 minutes?

S'kiton

1. Write a proportion involving number of bulbs and time. Let x represent the number of bulbs examined in 72 minutes.

number of bulbs 20 _ x time 18 72

20 x 2. Solve the proportion for x. _

18 72 18x = 20 ° 72 Set cross products equal.

x -= 80 Divide each side 12y 15. The inspector will examine 80 bulbs in 72 minutes.

Solve each problem. Round answers as necessary.

1. If detergent costs $6.50 for 50 fluid ounces, what will be the cost of 120 fluid ounces?

2. If a motorist drives 180 miles in 3.2 hours, how far will the motorist drive in 4.5 hours?

3. If a student reads 18 pages of an assignment in 30 minutes, how many pages will the student read in 48 minutes?

4. If 420 students enter a stadium in 25 minutes, how many students will enter the stadium in 35 minutes?

5. How many students are in the reading club if the ratio of boys to girls is 5 to 4 and there are 35 boys in the club?

6. If 3 CDs cost $39, how much will 7 CDs cost?

7. If a motorist drives 220 miles in 3.8 hours, how long will it take the motorist to drive 550 miles?

8. How many sandwiches can be bought with $21 if three sandwiches cost $9?

Making Sense of Numbers Unit 4 Ratio, Proportion, and Percent 129

NAME CLASS DATE

UNIT 4 WrriAg Percents as Fractions

OBJECTIVE: Representing a percent as a fraction in lowest terms

You can write a percent less than 100% as a fraction in lowest terms.

Write each percent as a fracti n in lowest terms.

a. 42% b. 0.5%

Solution

Write each percent as a fraction with denominator 100. 42 21 0.5 5 1

a. 42% -> b. 0.5% -> 100 50 100 1000 200

EXAMPLE 1

You can represent a percent greater than 100% as a mixed number with the fraction part in lowest terms.

MAMPLE 2 Write 125% as a mixed number with the fraction in lowest terms.

Solution

Write 125% as a fraction with denominator 100. 125 5 1

125% = 1- 100 4 4

Write each percent as a fraction in lowest terms or as a mixed number with the fraction in lowest terms.

1. 1% 2. 5% 3. 20% 4. 25%

5. 60% 6. 75% 7. 80% 8. 100%

9. 150% ' 10. 140% 11. 160% 12. 350%

13. 1000% 14. 101% 15. 110% 16. 105%

17. 0.6% 18. 1.5% 19. 130.4% 20. 112.5%

21, 1 22. 1 37% 23. 3 62-5% 24. 2 49-5%

25.

121%

1 26. 3 25-8% 27. 44-116% 28. 7

50-8

%

29.

8-8

%

1 34% 30. 66-2 % 31. 14-2 % 32. 11-1 %

3 7 9


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NAME CLASS DATE

_ UNIT

Writing Fractions as Percents

OBJECTIVE: Writing a fraction as a percent

You can use equivalent fractions to write a given fraction as a percent.

Write each fraction or mixed number as a percent. 3 3 a. b. 1-5

Solution

3 3 X 25 75 = a° 4 4 X 25 = 00 75%

Sometimes you need to use division to write a fraction as a percent.

EXAMPLE 1

3 8 8 X 20 160 b. 1 = = _ = 160% 5 5 5 X 20 100

Write each fraction as a percent. 5

b 1 a. g 3

Solution 0.625 0.3

5 1 a.

-> 8)5.00 b. -3 ---> 3)1.00

EXAMPLE 2

5 - 1 -8 = 0.625, or 62.5% 1 -3 = 0.3, or 33-3%

Write each fraction or mixed number as a percent.

1. 7

2. L3 3°

7 3 id

5. 17

10

6. 12.5

TO

7. 18.45

4.25

8. 16.25 25

a 27

50

10. 32

50

7 11. zi

25

12. 9 20

1 13. 2-5

25

3 14. 1-4

4 15. 3-5

3

16. 4-3

17. 5 18. 7 5 19. 316

20

11 20. 416 12

1 21. 3-8

8

7 22. 1-8

2 23. 1-3

5 24. 2-6

25. 7 26. 2- 1 27. 2-

3 1

16 16 28. 10-6


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reserved.

NAME CLASS DATE

1/Vr:ng Percents as Decimals

OBJECTIVE: Writing a percent as a decimal

Recall that when you divide a number by 100, you move the decimal point two places to the left.

EXAMPLE 1

Write each ercent as a decimal. a. 6% b. 7.3% c. 136.4%

Solution

a. 6% = 6 ± 100 = 0.06 b. 7.3% = 7.3 ± 100 = 0.073 c. 136.4% = 136.4 ± 100 =- 1.364

Sometimes you need to change a fraction to a decimal before writing the given percent as a decimal.

r. EXAMPLE 2 1

Write each percent as a decimal. a. 47:% b. 33-3%

SokK:on

a. 47-3% = 47.6% = 47.6 ± 100 = 0.476

• 5

1 1 - b. 33% = 33.3% = 0.333 = 0.3 Recall -that = 0.5.

-d

> 2

Write

1.

5.

each percent as a decimal.

21% 2. 2.1% 3. 98.5% 4. 53.3%

0.3% 6. 0.04% 7. 0.9% 8. 0.003%

5 9. 0.04% 10. 0.001% 11. 0.999% 12. 0.101%

13. 129% 14, 108% 15. 118.5% 16. 206.1%

> ..

_.-- 17.

1 -2°/o 18.

1 -4% 19. -1

% 20. -3% 40 5

L. E 21.

1 50-2% 22,

1 1-4% 23.

3 28-4% 24.

4 34-5%

5 3 1 7 25. 26. 5-8% 27. 124-% 28. 200-12% 98-

8%

29. 66-2% 30.

2 5-

3% 31.

1 120-

3% 32.

5 130-

6%

3


UNIT 4

NAME CLASS DATE

iiizing Equivalent Fractims, Decimals, and Percents

OBJECTIVE: Writing equivalent fractions, decimals, and percents

The table below shows equivalence of frequently used fractions, decimals, and percents. Having worked with these, you should be able to change from any form given to an equivalent one.

Equivalence of Commonly Used Perc ts, Deci als, and Fractions

20% = 0.2 = -5 15% = 0.25 = -4 12-2% = 0.125 = -8 16-3% = 0.16 = -6

40% = 0.4 = 2 -5 50% = 0.5 = 1 -2 1 37-2% = 0.375 = 3 -8

1 33-3% = 0.3 = -1 3

60% = 0.6 = 3 -5 75% = 0.75 = 3 -4

1 62-2% = 0.625 = 5 -8 6640 3 = 0.6 = 2 3

80% = 0.8 = 4 -5 100% = 1.00 = 1 1 87-2% = 0.875 = 7 -8 831% 3 = 0.83 = 5 6

EXAMPLE a. Write the equivalent decimal and fraction for 62 -1 °/0.

b. Write the equivalent percent and decimal for ;

Solution

a. From the table above, 6*/0 = 0.625 = ;.

b. From the table above, -7 = 87-

1% = 0.875. 8 2

Given each fraction, decimal, or percent, write the other two equivalent forms.

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6. -4 2

9. 0.8 10. 0.125

13. 0.83 14. 0.6

17. 1

62-2% 18. 1 33-3%

21. 1

162-2% 22. 1

133-3%

7.

3. 1 4

5 8 .

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-3

11. 0.4 12. 0.375

15. 0.3 16. 0.16

19. 1

37-2% 1

20. 83fo

23. 1

1371% 1 24. 183-3%


&9N IT 4

NAME CLASS DATE

Ca culatirg Ti7s and Total Cost

OBJECTIVE: Finding an amount to leave for a tip and the total cost of a meal

For service rendered, a customer may leave a tip when paying for a meal. A rule of thumb is to leave a tip that is 15% of the cost of the meal when service is done well.

EXAMPLE 1

Find the amount of the tip for a meal that costs $34.22. Assume that the tip is 15% of the cost.

SolOon

Method 1: Calculate 15% of 34.22. 15% of 34.22 = 0.15 X 34.22 = 5.133

1 Method 2: Use —

7 and 35 as estimates of 15% and 34.22. 1 1 —7

of 35 = —7 X 35 = 5

The value of the tip is $5.13, or about $5.

The total cost of a meal is the sum of the tip and the meal cost.

Find the total cost of the meal in Example 1.

S lution

Method 1: Add $5.13 and $34.22. Method 2: Add $5 and $35.00.

5.13 + 34.22 = 39.35 5 + 35 = 40

EXAMPLE 2

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The total cost is $39.35, or about $40.00.

Assume the tip is 15%. Use 0.15 to find the actual amount of tip and total cost. 1

Then use —7 to estimate the tip and total cost.

1. $21.95

2. $98.75

3. $50.30

4. $110.46

5. $13.80

6. $235.68

7. 7.05


NAME CLASS

DATE

411 Calcule a S Tax arid Total Cost UNIT 4

OBJECTIVE: Finding the amount of sales tax and total cost including the amount of tax

In many states, consumers pay a tax on certain purchases. The tax is a fixed percent of the purchase price. In one state, the sales tax may be 5% of the cost of an item.

rEXAMPLE 1

If the sales tax is 5%, find the amount of tax on a table that costs $495.50.

Solution

To find the amount of tax, calculate 5% of S495.50. 5% of S495.50 0.05 X 495.5 = 24.775

The amount of tax is $24.78.

The total cost of an item is the sum of the purchase price and the amount of tax. There are two ways to find total cost.

EXAMPLE 2

Find the total cost of the table in Example 1.

Solution

Method 1: To find the total cost, add the purchase price and the amount of tax. $495.50 + $24.78 = $520.28

Method 2: Multiply the cost of the table by 105%. 105% of $495.50 1.05 X 495.5 = 520.275

By either method, the total cost is $520.28.

Find the amount of tax and total cost.

1. sales tax 6%; purchase price: $102.45


3. sales tax 6.5%; purchase price: S39.95




7. sales tax 5.5%; purchase price: $6885.44


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8. 50%; $1000.00 0

9. 3y1

/o; $330 10. 12 1 %. $560 2 '

Co

11. 16/o; $360 12. 66-2

'%• $150

3 2

3 1 13. 14%; $420.70 14. 83- '%. $720

3

15. 20.5%; 268.50 16. 18.5%; $337.80

1. 20%; $120

3. 40%; $480

5. 10%; $380.30

7. 30%; $111.10

2. 25%; $250

4. 50%; 12.50

6. 10%; $18.42

NAME CLASS DATE

Solving 9iscount Problems

OBJECTIVE: Finding the amount of discount and reduced sale price

To encourage consumers to buy a product, sellers may offer a discount, a reduction of the original price to a lower sale price.

EXAMPLE 1

A coat ordinarily sells for $238.45. If the price is reduced by 20%, find the amount of the discount.

Solution

20% of 238.45 0.2 X 238.45 = 47.69 The amount of the discount is $47.69.

To find the new sale price after discount, apply either Method 1 or Method 2 below.

EXAMPLE 2

Find the reduced sale price of the coat in Example 1.

Solution

Method 1: Subtract dollar amounts. $238.45 - $47.69 = 190.76

Method 2: Calculate 80% of the original price. 80% of 238.45 0.8 X 238.45 =- 190.76

By either method, the reduced sale price is $190.76.

Use any method to find discount and reduced sale price.

Making Sense of Numbers

Unit 4 Ratio, Proportion, and Percent 151

UNIT 4

NAME CLASS DATE

Solving IVhrk p Problems

OBJECTIVE: Finding the amount of markup and the new selling price

To pay expenses and make a profit, a seller will buy a product for one price, mark the price up, and sell the product for a higher price.

A store owner buys a product for $12.50 and marks the price up 60%. Find the amount of markup.

Soluti n 3

Calculate 60% of 12.50. 0.6 X 12.5 = 7.5 or —5 X 12.5 ---- 7.5

By either method, the amount of markup is S7.50.

EXAM LE 1

To find the selling price after markup, add the amount of markup to the original price or multiply the original price by 100% plus the percent of markup.

EXAMPLE 2 Find the final selling price of the product in Example 1.

Solution

Method 1: Add markup and original price. S7.50 + S12.50 = $20.00

Method 2: Multiply original price by 160%. 160% of S12.50 = 1.6 X 12.5 = 20

By either method, the final selling price is S20.00.

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Find the amount of markup and final selling price.

1. original price: 35.60; markup: 80%

2. original price: $64.20; markup: 100%





7. original price: $278.66; markup: 133-1%


EXAMPLE 2

NAME CLASS DATE

UNIT 5

OBJECTI E. Simplifying an expression involving both addition and subtraction

of integers

To find the value of an expression involving both addition and subtraction of integers, follow the order of operations.

A Th ig and S iltracting With Integers

Simplify —17 + 6 + (-7) + 13.

Solution

Method 1: Use the order of operations. —17 + 6 + (-7) + 13 = —11 + (-7) + 13 Add: —17+ 6 = —11

= —18+13 Add: —11 + (-7) = —10

=-5

Method 2: Group —17 and —7. Group 6 and 13. —17 + 6 + (-7) + 13 --= —17 + (-7) + 6 + 13

= —24 + 19 = —5

Remember to simplify any expressions within parentheses first.

Simplify —[12 + (-15)] — [7 — (12)].

Solution

—[12 + (-15)] — (7— 12) = —(-3) — (7— 12) 12 + (-15) = —5

= — ( -3) — (-5) 7— 12 = — 5

= 3 + 5 — (-3)= 5 =8

L —5 + 6 — (-7) 2. 10 — 13 — [2 +(— 4)]

3. — [-3 +(-24) — 30] 4. —5 + 6 — (2 — 7)

5. —9 — (-9) — (-3 + 7) 6. 10 — (-10) — 18 + 1

7. —5 + 9 + 5 — 9 8. —(4 — 7) — (-2 + 3)

9. 7 + (-8) — (-5 — 1) 10. 12 — [-2 — (-3)]

11. —[-7— (-7)] 12. 3 + (-11) + 11

13. 10— [2 + (-5)] 14. —[3 — (-5)] — (-3)

170 Unit 5 Integers Making Sense of Numbers

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NAME CLASS DATE

UNIT 5

Mu tiplying Integers

OBJECTIVE: Finding the product of two integers

When finding the product of two integers, there are three possibilities.

• The factors have like signs. The product will be positive. o The factors have unlike signs. The product will be negative. • The product of 0 and any integer is 0.

To multiply, find the product as if both factors were positive. Then decide whether the product is positive, negative, or zero.

TEXAMPLE 1 Multiply: a. — 7 X 4 b. (-7)(-9) c. 0 X 24

Solution

a. 7 X 4 =- 28 unlike signs Therefore, —7 X 4 = — 28. b. 7 X 9 = 63 like signs Therefore, ( —7)( —9) = 63. c. 0 X 24 = 0 the product of zero and any integer is zero

You can multiply three or more integers using the rules above.

EXAMPLE 2 Multiply: 5( — 2) ( — 3)

Solution

5(-2)(-3) = (-10)(-3) 5(-2) =30

= —10

1.

Multiply.

7 X 8 2. 1 X 8 3. 11 X 0

4. 0 X (-20) 5. — 7 X (-7) 6. —6 X (-12)

7. — 10 X 20 8. —6 X 13 9. 9 X (-9)

10. 2 X (-46) 11. (-11)2 12. (-12)2

13. — 4 X 4 14. 4 X (-4) 15. — (-5)2

16. (-2) X 5 X (-3) 17. (-4) X (-4) X (-5)

18. 6 X (-2) X (-3) 19. (-6) X 1 X (-7)

20. (-3) X (-2)>< 11 21. (-3)2(-2)2

Making Sense of Numbers Unit 5 Integers 171

8th

Grade Answer Ke%

Compare your solutions to the answers as you work

through the packet to check your understanding.

MATHEMATICS is not about

numbers, equations, computations, or

alorithms: it is about

UNDERSTANDING.

Akeiketm,Pt7humitsoil

1. 133

1 2. *,or 1-8

7 1 3. 3, or 1-g

4 4 . 3

5. 12 1? U. -3-5-

7 5 . Tg

8. 14_

5.

29 5 9. Tri, or

43 15 10. or 1F3

23 11. -30

29 12' -45 3 1

13. -2, or 1-2

31 11 14' - ' or 1- 20 20

17 1 15.

25 16. or 2+2.

31 1 17' - ' or 1- 30 30

41 18. 84

19. Ti19), or 1-11-9)

20. m25, or lri7

1. 13-6 13. 51 18

CHAPTER 1 Order of Operations

1. 9 1. 75 2. 116 2. 28 3. 4 3. 32 X 54

4. 410 4. 53 X 73 5. 72 5. 22 x 31 X 53 6. 9 6. 71 X 52 X 73

7. 165 7. 8 8. 2 8. 9 9. 35 9. 25

10. 18 10. 343 11. 129 11. 13 12. 44 12. 7 13. 24 13. 128 14. 160 14. 243 15. 53 15. 324 16. 51 16. 968 17. 107 17. 1800 18. 28 18. 3528 19. 16 19. 1440 20. 17 20. 2352

CHAPTER 4 Addition of Fractions and Mixed Numbers

2.12k 14.7k

3. 7-3 4 15. 14 7 1-72-

4. 10-5 7 16. 5-4130

5. 3_17 20

17. 9-1 30

6. 4-13 30 18. 10 13 -

20

7. 1 3-8 19. 151

8. 2 3-178- 20. 1 161

9. 8-1 3

21. 5-07

10. 6-41 22. 18 8

11. 4-5 14 23. 16-4 5

12' 7

2-15 24. 1 12-2

21 12. - 50

13. 1Ti

14 1 . 1-4-

2 15. - 5 5 16. -§ 7 '. -4-6- 6 18.

1 9 45 . Tg 5 1 20. or 21

44 40 19 or 1E-

4 5.

6.

3 9. 5

2 10. 33

10 'it 33

CHAPTER 5 Subtraction of Fractions and Mixed Numbers

CHAPTER 6 Multiplication of Fractions and Mixed Numbers

1 1. 8 5. -3-, or 33

15 1 2. 6 6.

35 2 3. 10 7. -3-, or 11-3

4. 18 8. 108 3 --7-, or 157

9-17. Estimates may vary. Accept any reasonable answer.

9. about 7

14. about 9 111 about 4

15. about 14 11. about 8

16. about 24 12 about 42

17. about 30 13. about 7

• 10

1 1 1. 8.5 2. 12.03 3. 23.7 4. 66.3 5. 422.3 6. 372.7 7. 7.1 8. 3.7 • 9. 12.4

10. 10.4 11. 18.1 12. 20.9 13. 1.7 14. 14.2 15. 5.3 16. 13.8 17. 12.6 18. 14.5 19. 110.3 20. 33.6

21. 66.2 22. 22.9 23. 36.5 24. 55.7

1. 1222.5 1. 102

2. 10,075 2. 56.7

3. 8855 3. 43.2

4. 8560 4. 600

5. 35,250 5. 384.4

6. 103,460 6. 2150.4

7. 435 7. 3.75

8. 840 8. 12.25

9. 10 9. 14.08

10. 30 10. 4.32

11. 7 11. 3.48 12. 5 12. 0.56

13. 1006 13. 62.78

14. 1001 14. 16.684

15. 0.1 15. 14.756

16. 123,384 16. 0.1152

17. 172,290.5 17. 0.4225

18. 8,888,825 18. 65.949

19. 7,500,080

CHAPTER 7 Division of Fractions and Mixed Numbers

CHAPTER 8 Decimal Fundamentals

1. 123.45 2. 123.5 3. 0.3 4. 0.54 5. 20 6. 8 7. 3.14 8. 1.4 9. 0.005

10. 0.1 11. 18 12. 4

3 7 14. 1- 10 9

15. 10 -2 10 5

4. 2 16. 99-9 5 10

5. T5-1 • 17.13 1-1

6. 1 18. 10-21° 4 4 33

19. 11- 1 100 100

8. 39 20. 35-19 50 25

9. 1 500 21.1j 0700

10. 1 22. 111 10 4200

11. -18- 23. 8- 3 4

3 111 12. 1.-3 24. 6500

CHAPTER 10 Multiplication of Decimals

CHAPTER 11 Division of Decimals

1. true 2. false 3. false 4. x = 15 5. y = 25 6. a = 54

7. z = -

8. t=i-7,or6/

9. a = 52

• 5 1 10' c=-'or2- 2 2 63 3 11* n = ' 5 - or 12- 5

12. m = 38

4 1 .

3 5. 3

3 6.

7 4 I. 3

8. 1

9. 11

CHAPTER 12 Ratio and Proportion

1. $15.60 2. 253.125, or

1 253a- miles

3. 28.8 pages 4. 588 students 5. 63 students 6. $91 7. 9.5 hours 8. 7 sandwiches

_ CHAPTER 13 Percent Fundamentals

12. 3-2

13. 10

1 14. 1j-

15. 11 io

16. 1-1 20 3

500 3

200 38 19. 1-125 1 20. 1-8

2 1

10. 1-5 21. 8

3 3

11.1k 22. 8

313 23. 500 247 500 13

160 203 800 141 320 407 800 1 3 2 30. 3

31.

32.

1. 70% 2. 130% 3. 35% 4. 12% 5. 68% 6. 25% 7. 36.9% 8. 65% 9. 135%

10. 128% 11. 175% 12. 180% 13. 220% 14. 175% 15. 380% 16. 415%

2 17. 4130/0

18. 87-% 2

1 19' 331-% 4 3 20' 468-% 4 1

21. 312-% 2 1 22. 187-% 2 2 23. 166-% 3 1 24. 283-% 3

• 1 25. 131-% 4 3 26. 243-% 4 1 27' 233-% 3

28. 2 1016-% 3

17.

18.

24.

25.

26.

27.

28.

29.

1 7 1 9

CHAPTER 13 Percent Fund entth

1. 0.21 2. 0.021 3. 0.985 4. 0.533 5. 0.003 6. 0.0004 7. 0.009 8. 0.00003 9. 0.0004

10. 0.00001 11. 0.00999 12. 0.00101 13. 1.29 14. 1.08 15. 1.185 16. 2.061 17. 0.005 18. 0.0025 19. 0.00025 20. 0.006

21. 0.505 22. 0.0125 23. 0.2875 24. 0.348 25. 0.98625 26. 0.05375 27. 1.24125 28. 2.005873 29, 0.6 30. 0.05-6- 31. 1.203 32. 1.308-3-

1. 0.4 and 40% 2. 0.375 and 37.5%

3. 0.3 and 331% 3

4. 0.8 and 80% 5. 0.5 and 50°/o 6. 0.75 and 75%

7. 0.85 and 831% 3

8. 0.6 and 661231%

4 -5 and 80°/o

1 1 10. -

8 and 12-2°/o

11. -2

and 40% 5

12. -3

and 37-2%

8 1

13. -5g and 83-k%

2 14. 3 and 66-3-%

1 15.

1 -3

and 33-7,%

1 2 16. -

6 and 16-3°/o

17. -5 and 0.625 8 1

18. -3 and 0.3

19. -3 and 0.375 8

20. -5 and 0.85' 6

5 21. 1-8

and 1.625

1 . 17, and 1.5.

3 . 1-8

and 1.375

24. 1-5 and 1.85 6

$3.29 and $25.24; -1- $3 and $24 7'

1 $14.81 and $113'56' •-•

' $14 and $112 7

1 $57.85., $7 and $57 $7.54 and . 7

1.

2.

3.

15%:

15%:

15%:

4. 15%: $16.57 and $127.03., 1 . $16 and $128 7

5. 15%: $2.07 and . $15.87' - -1 $2 and $16 7°

6. 15%: $35.35 and $271.03.' 1 $34 and $272 7 '

7. 15%; 1

$1.06 and $8.11; .7: $1 and $8

CHAPTFR 15 Applications I Percent

1. $6.15; $108.60 1. $24; $96

2. $13.20; $343.10 2. $62.50; $187.50

3. $2.60; $42.55 3. $192; $288

4. $60.03; $1260.69 4. $6.25; $6.25

5. $261.42; $4618.41 5. $38.03; $342.27

6. $219.55; $5708.30 6. $1.84; $16.58

7. $378.70; $7264.14 7. $33.33; $77.77

8. $500; $500

9. $110; $220

10. $70; $490

11. $60; $300

12. $100; $50

13. $61.42; $359.28

14. $600; $120 15. $55.04; $213.46

16. $62.49; $275.31

1. $28.48; $64.08

2. $64.20;$128.40

3. $94.05; $193.05

4. $496.63; $1087.84

5. $94.74; $173.69

6. $49.65; $82.75

7. $371.55; $650.21

CHAPTER 18 Subtraction of Integers

1. 8 2. -1

3. 57 4. 6 5. -4 6. 3 7. 0 8. 2 9. 5

10. 11

11. 0 12. 3 13. 13

14. -5

CHAPTER 19 Multiplication of Integers

1. 56 2. 8 3.0

16. 30 17. -80 18.36

4. 0 19. 42 5. 49 20. 66 6. 72 21. 36 7. -200 8. -78 9. -81

10. -92 11. 121 12. 144 13. -16 14. -16 15. -25