Skill Intervention Workbook (Grade 8)

Skills Interventionfor AlgebraDiagnosis and Remediation

Student Workbook

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ISBN: 0-07-829949-7 Algebra InterventionStudent Workbook

1 2 3 4 5 6 7 8 9 10 009 09 08 07 06 05 04 03 02

Glencoe/McGraw-Hill

Glencoe/McGraw-Hill iii Algebra Intervention

Table of ContentsSkill Number and Operation1 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 Multiplication Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 Adding and Subtracting Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 Multiplying and Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75 Adding and Subtracting Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 Multiplying and Dividing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .138 Greatest Common Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159 Ratios as Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

10 Adding and Subtracting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1911 Adding and Subtracting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2112 Multiplying and Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2313 Multiplying and Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2514 Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2715 Percents as Fractions and Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2916 Percent of a Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3117 Percent Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3318 Percent of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Skill Algebra19 Solve Equations Involving Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . .3720 Solve Equations Involving Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . .3921 Solve Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4122 Use an Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4323 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4524 Proportional Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4725 Scale Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4926 Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5127 Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5328 Function Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5529 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5730 Solve Equations With Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5931 Graphing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6132 Slope of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6333 Graphing Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6534 Graphing Linear and Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

Skill Geometry35 Sums of Angles of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6936 Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7137 Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7338 Congruent Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7539 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7740 Dilations and Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7941 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81

Glencoe/McGraw-Hill iv Algebra Intervention

Skill Measurement42 Perimeter and Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8343 Area of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8544 Area of Rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8745 Area of Triangles and Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8946 Area of Irregular Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9147 Surface Area of Rectangular Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9348 Volume of Rectangular Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95

Skill Data Analysis and Probability49 Using Samples to Predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9750 Mean, Median, and Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9951 Make a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10152 Probability of Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10353 Expected Value of an Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10554 Theoretical and Experimental Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10755 Probability Using Area Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10956 Line Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11157 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11358 Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11559 Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11760 Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11961 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12162 Constructing and Interpreting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12363 Make a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12564 Interpreting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12765 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12966 Predicting Distribution of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13167 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13368 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

Skill Problem Solving69 Classify Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13770 Problem-Solving Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13971 Determine Reasonable Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14172 Work Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14373 Solve a Simpler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14574 Make a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14775 Make Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149

SKILL

1

Glencoe/McGraw-Hill 1 Algebra Intervention

Name ______________________________________ Date ___________

Order of OperationsWhen you evaluate an expression in mathematics, you must do the operations in a certain order. This order is called the order of operations.

Evaluate 56 � (17 � 9) � 7 � 3.

56 � (17 � 9) � 7 � 3 �

56 � 8 � 7 � 3 � Do all the operations within the grouping symbols.

7 � 21 � Do multiplication and division from left to right.

28 Do addition and subtraction from left to right.

Therefore, 56 � (17 � 9) � 7 � 3 � 28.

Evaluate each expression.

1. 2 � 9 � 5 � 3 33 2. (9 � 4) � 5 1

3. 10 � 4 � 1 7 4. 15 � 18 � 9 � 3 16

5. 30 � (12 � 6) � 4 9 6. (72 � 12) � 2 30

7. 2(16 � 9) � (5 � 1) 8 8. (43 � 23) � 2 � 5 10

9. 90 � 45 � 24 � 2 33 10. 81 � (13 � 4) 9

11. 7 � 8 � 2 � 8 40 12. 71 � (34 � 34) 71

13. 9 � 4 � 2 � 16 23 14. (24 � 10) � 3 � 3 5

EXERCISES

EXAMPLE


15. 4(22 � 18) � 3 � 5 1 16. 12(5 � 5) � 3 � 5 15

17. 18(4 � 3) � 3 � 3 9 18. (34 � 46) � 20 � 20 24

19. 92 � 66 � 12 � 4 23 20. (16 � 8) � 4 � 10 12

21. 60 � 12 � (4 � 1) 15 22. (100 � 25) � 2 � 25 175

23. 3 � 7 � 5 � 4 20 24. 9 � 4 � 2 � 10 8

25. 150 � 10 � 3 � 5 0 26. 5(35 �18) � 1 86

Use the price list at the right toanswer Exercises 27–29.

27. Alfred wants to buy 15 ping pong balls and 4 ping pong paddles. What is the cost of this purchase? $38

28. Ali plans to buy 6 softballs and 3 soccer balls for the teen club. If he has a coupon for $8 off his purchase, how much will he pay for the balls? $82

29. What is the cost of 20 ping pong balls, 2 ping pong paddles, 3 softballs, and 1 soccer ball? $59

30. Tickets for the play cost $12 for adults and $8 for children. How much would 3 adult tickets and 5 children tickets cost? $76

31. Use operation symbols, parentheses, and the numbers 1, 2, 3,and 4 to express the numbers from 1 to 15. For example, 2 � 3 � (4 � 1) � 1. See students’ work.

APPL ICAT IONS Sam’s Sporting SuppliesPrice List

Ping Pong Balls 5 for $2Ping Pong Paddles $8Softballs $5Soccer Balls $20

SKILL

2


Name ______________________________________ Date ___________

Multiplication PropertiesThe table shows the properties for multiplication.

Name the multiplicative inverse, or reciprocal, ofeach number.

1. 2. 3. 8 4. 91�9

1�8

3�19

19�3

11�6

6�11

EXERCISES

Property Examples

CommutativeThe product of two numbers is the same regardless of the order in which they are multiplied.

AssociativeThe product of three or more numbers is the same regardless of the way in which they are grouped.

IdentityThe product of a number and 1 is the number.

Inverse (Reciprocal)The product of a number and its reciprocal is 1.

DistributiveThe sum of two addends multiplied by a number is equal to the sum of the products of each addend and the number.

21 � 2 � 2 � 2142 � 42

5 � (3 � 6) � (5 � 3) � 65 � 18 � 15 � 6

90 � 90

81 � 1 � 81

� � 18�7

7�8

2 � (9 � 3) � (2 � 9) � (2 � 3)2 � 12 � 18 � 6

24 � 24


Name the property shown by each statement.

5. 67 � 89 � 89 � 67 6. 1 � 45 � 45commutative identity

7. � 1 � 8. � � � � � � � � �identity associative

9. � � � 10. � � � � � � � � � � �commutative distributive

11. � 4 � 1 12. 45(23 � 3) � (45 � 23) � (45 � 3)

inverse distributive

13. � � 1 14. � � �

inverse commutative

15. Jill runs for 1 as long as Eva. Find Jill’s running time if Eva

runs for 48 minutes.84 minutes

16. A chihuahua is 6 inches tall. The height of a German shepherd

is 3 the height of the chihuahua. Find the height of the

German shepherd.22 inches

2�3

3�4

APPL ICAT IONS

4�5

3�4

3�4

4�5

4�9

9�4

1�4

5�7

3�5

1�3

3�5

5�7

1�3

3�5

3�4

5�6

5�6

3�4

5�9

2�3

1�5

5�9

2�3

1�5

11�12

11�12

SKILL

3


Name ______________________________________ Date ___________

Adding and Subtracting DecimalsTo add decimals, line up the decimal points. Then add the same way youadd whole numbers.

4.76 � 3.62 12.8 � 3.467 � 8.56

4.76 12.800� 3.62 3.467 Annex zeros.

8.38 � 8.56024.827

The sum is 8.38. The sum is 24.827.

To subtract decimals, line up the decimal points. Then subtract the sameway you subtract whole numbers.

15.05 � 4.86 35 � 13.631

15.05 35.000 Annex zeros.� 4.86 � 13.63110.19 21.369

The difference is 10.19. The difference is 21.369.

Add or subtract.

1. 45.9 2. 6.83 3. 43.89� 12.7 � 3.77 � 56.3258.6 3.06 100.21

4. 205.7 5. 6.7 6. 18.75� 98.8 � 3.56 � 7.2106.9 10.26 11.55

7. 17.93 8. 77 9. 6.5� 33.5 � 12.66 � 7.547

51.43 64.34 14.047

EXERCISES

EXAMPLES

EXAMPLES


10. 4.7 � 0.89 11. 15.6 � 7.893.81 23.49

12. 25 � 4.76 13. 6.43 � 7.8 � 1320.24 27.23

14. 9.857 � 4.5 15. 65.8 � 15.75 � 7.8545.357 89.404

16. 408.7 � 56.78 17. 7.9 � 1.22 � 6.1 � 11351.92 26.22

18. 73.56 � 29 19. 11.444 � 5.9 � 13.9344.56 31.274

The results of the 1948 presidential election is given at the right. Use thisinformation to answerExercises 20–22.

20. What percent of the vote was cast for Truman orDewey? 94.62%

21. How many more percentage points did Truman receive than Dewey? 4.38 percentage points

22. What percent of the vote was not cast for Truman or Dewey? 5.38%

23. Albert had $284.73 in his checking account. He wrote checksfor $55.86 and $25.00. He deposited a check for $113.76. Whatis his new balance in his checking account? $317.63

24. For lunch, Connie buys a sandwich for $2.35 and a smalllemonade for $0.79. If she gives the cashier a five-dollar bill,how much change should she receive? $1.86

25. Tony drove 12.7 kilometers to the computer store. Then he drove 5.2 kilometers to the library, and finally 6.7 kilometers to his house. What was the total distance Tony drove? 24.6 km

APPL ICAT IONS Percent ofCandidate Popular Vote

Truman 49.5

Dewey 45.12

Thurmond 2.4

Wallace 2.38

Other 0.6

SKILL

4


Name ______________________________________ Date ___________

Multiplying and Dividing DecimalsMultiply 1.45 by 0.68.

1.45 2 decimal places� 0.68 2 decimal places

1160870

0.9860 4 decimal places

The product is 0.9860.

Divide 38.22 by 2.6.

1 4.72.6.�3�8�.2�.2�

2612 210 41 8 21 8 2

0

The quotient is 14.7.

Multiply.

1. 4.9 2. 53 3. 2.8� 35 � 3.7 � 3.5

171.5 196.1 9.80

4. 18.9 5. 0.014 6. 53.98� 3.7 � 0.65 � 71.2

69.93 0.0091 3,843.376

EXERCISES

EXAMPLE

EXAMPLE

The sum of the decimalplaces in the factors is 4, sothe product has 4 decimalplaces.

Change 2.6 to 26 by moving the decimalpoint one place to the right.

Move the decimal point in the dividend oneplace to the right.

Divide as with whole numbers, placing thedecimal point above the new point in thedividend.

��


7. 4.55 8. 0.133 9. 3.91� 41.8 � 4.2 � 8.5

190.19 0.5586 33.235

Divide.

1.42 2.8 0.3710. 6�8�.5�2� 11. 23�6�4�.4� 12. 53�1�9�.6�1�

84 73 16013. 1.6�1�3�4�.4� 14. 0.52�3�7�.9�6� 15. 0.23�3�6�.8�

0.022 650 5.1316. 1.7�0�.0�3�7�4� 17. 0.112�7�2�.8� 18. 7.4�3�7�.9�6�2�

The prices at Martha’s Meat Market are given at the right. Use this information to answerExercises 19–21.

19. What is the cost of a chicken that weighs 3.4 pounds?$3.91

20. Willy buys a package of ground beef for $6.84. How manypounds of ground beef did he buy?3.6 lb

21. A turkey breast costs $8.05. How much does the turkey breastweigh?4.6 lb

22. One centimeter on a map represents 56 kilometers. If a distance between two towns on the map is 3.2 centimeters,what is the actual distance between the towns?179.2 km

APPL ICAT IONS

Martha’s Meat Market

Specials of the Week

Ground Beef $1.90/lbChicken $1.15/lbTurkey Breast $1.75/lb

SKILL

5


Name ______________________________________ Date ___________

Adding and Subtracting IntegersWhen you add integers, remember:

The sum of two positive integers is positive.The sum of two negative integers is negative.The sum of a positive integer and a negative integer is:• positive if the positive integer has the greater absolute value.• negative if the negative integer has the greater absolute value.

To subtract an integer, add its opposite.

Solve j � –71 � 35. Solve w � –41 � (–73).

j � –71 � 35 w � –41 � (–73)

–71 � 35 w � –41 � 73

71 � 35 The sum is negative. w � 32

71 � 35 � 36 The solution is 32.

j � –36

The solution is –36.

Solve each equation.

1. –9 � (–5) � a 2. b � –3 � 9 3. c � 11 � (–14)–4 6 –3

4. d � –16 � 9 5. –11 � (–16) � e 6. f � –11 � (–12) � (–9)–7 5 –32

7. –10 � 8 � g 8. 14 � (–12) � h 9. j � –7 � (–9) � (–9)–18 26 –25

10. k � –56 � (–14) 11. m � –11 � 28 12. –37 � 11 � n–42 –39 –26

EXERCISES

EXAMPLES


13. p � –15 � (–36) 14. –12 � (–23) � q 15. –9 � (–23) � r21 11 32

16. 20 � (–13) � s 17. t � –5 � 23 � (–10) 18. w � –16 � (–35)7 8 –51

19. 19 � (–15) � x 20. –23 � 23 � y 21. (–10) � 36 � z34 0 26

Evaluate each expression if c � 4, x � –5, and h � 6.

22. x � 5 � 9 � (–7) 23. –12 � c � (–3)2 –11

24. –6 � x � h 25. –12 � h � x � h–5 –5

26. When Jasmine went to bed, the temperature outside was 3°F.When she woke up the next morning, the temperature was–6°F. How many degrees did the temperature drop during thenight?9°F

27. Trackmaster Bike shop reported these profits and losses for thelast 5 years:

1988: profit of $25,0001989: loss of $47,0001990: loss of $30,0001991: profit of $13,0001992: profit of $34,000

a. How much more money was lost in 1989 than in 1990?$17,000

b. How much more were the total earnings in the last twoyears than in the first three years?$99,000

c. From 1988 to 1992, did the shop have a loss or a gain overall and how much?loss of $5,000

d. How much profit would be needed in 1993 for the bikeshop to break even (have total losses and profit be $0) forthe six years?$5,000

APPL ICAT IONS

SKILL

6


Name ______________________________________ Date ___________

Multiplying and Dividing IntegersIf two integers have the same sign, their product or quotient is positive.If two integers have different signs, their product or quotient is negative.If there is an even number of negative integers, their product is positive. If there is an odd number of negative integers, their product is negative.


d � (–5) � 7 One factor is negative and the other is positive.d � –35 The product is negative.


t � (–120) � (–6) Both integers are negative.t � 20 The quotient is positive.

The solution is 20.

k � (–6) � 7 � (–2) There is an even number of negative integers.k � 84 The product is positive.

The solution is 84.

Tell whether the product is positive or negative.Then find the product.

1. 9 � (–6) 2. (–3) � (–6)negative; –54 positive; 18

3. (–7) � (–7) 4. (–1) � 2 � (–1) � (–1)positive; 49 negative; –2

5. (–1) � (–2) � 3 6. (–5) � (–2) � (–1) � (–3)positive; 6 positive; 30


7. (–56) � (–4) � a 8. b � 26 � (–2)14 –52

9. c � 72 � (–12) 10. (–22) � (–3) � d–6 66

11. e � (–4) � 21 12. (–100) � 5 � f–84 –20

EXERCISES

EXAMPLES


13. 98 � (–7) � g 14. h � 13 � (–12)–14 –156

15. (–125) � (–25) � j 16. k � (–15) � (–7)5 105

17. 240 � (–16) � m 18. q � (–88) � 4–15 –22

19. y � (–6) � (–9) 20. t � (–90) � (–10)54 9

21. 12 � (–2) � 3 � n 22. (–2) � 5 � (–10) � p–72 100

23. (–12) � (–2) � (–2) � a 24. e � 10 � (–5) � 3–48 –150

25. f � (–9) � 2 � (–2) 26. (–2) � (–2) � (–2) � 2 � n36 –16

27. Find the value of each expression.

a. (–1)2 1 b. (–1)3 –1 c. (–1)4 1 d. (–1)5 –1

e (–1)6 1 f. (–1)10 1 g. (–1)25 –1 h. (–1)100 1

28. Write the rule for raising a negative number to a power. (Hint: Look for a pattern in Exercise 27.) A negative number raised to aneven power will produce a positive number, and a negativenumber raised to an odd power will produce a negative number.

29. On Monday, Tuesday, and Wednesday, the low temperatures inWisconsin were –20°F, –15°F, and –28°F. What is the averagelow temperature for the three days? –21°F

30. California had a record hot day in 1913 in Death Valley. Itreached 134°F. Its coldest day was in 1937 at Boca when it was–45°F. What is the average of these two record days? 44.5°F

31. The Los Angeles Raiders had four penalties during their lastgame. Each penalty was for 15 yards. What was the total lostyards due to penalties? 60 lost yards

APPL ICAT IONS

SKILL

7


Name ______________________________________ Date ___________

Prime FactorizationEvelyn has 105 books. She is trying to decide how to put them on theshelves of 3 separate bookcases.

How can she arrange the books if she wants to have the same number of books on each shelf?

To solve this problem, find the prime factorization of 105.

105 � 3 � 35

� 3 � 5 � 7

She can put 5 books on each of 7 shelves or 7 books on each of 5 shelves.

Find the prime factorization of each number.

1. 75 2. 36 3. 49 4. 723 � 5 � 5 2 � 2 � 3 � 3 7 � 7 2 � 2 � 2 � 3 � 3

5. 90 6. 42 7. 100 8. 1212 � 3 � 3 � 5 2 � 3 � 7 2 � 2 � 5 � 5 11 � 11

9. 275 10. 385 11. 210 12. 1475 � 5 � 11 5 � 7 � 11 2 � 3 � 5 � 7 3 � 7 � 7

13. 525 14. 66 15. 196 16. 5003 � 5 � 5 � 7 2 � 3 � 11 2 � 2 � 7 � 7 2 � 2 � 5 � 5 � 5

EXERCISES

EXAMPLE


17. 136 18. 495 19. 231 20. 1,0012 � 2 � 2 � 17 3 � 3 � 5 � 11 3 � 7 � 11 7 � 11 � 13

21. 234 22. 84 23. 255 24. 2522 � 3 � 3 � 13 2 � 2 � 3 � 7 3 � 5 � 17 2 � 2 � 3 � 3 � 7

Monty’s yard has dimensions of 35 feet by 35 feet. He wants to construct a rectangulargarden in his yard. Use this information toanswer Exercises 25–27.

25. Monty decides that the garden should have an area of 95 square feet. What are the whole number dimensions thatare possible for this garden? 19 ft and 5 ft

26. Monty changes his mind and decides that the garden shouldhave an area of 100 square feet. What are the whole numberdimensions that are possible for this garden? 25 ft by 4 ft, 10 ft by 10 ft, 20 ft by 5 ft

27. Monty’s neighbor asks Monty if he wants to construct a garden that they could share. One-half of the garden wouldbe in Monty’s yard and one-half would be in his neighbor’syard. His neighbor’s yard has dimensions 40 feet by 35 feet.They decide to construct a rectangular garden with an area of 250 feet. What are the whole number dimensions that arepossible for this garden? 50 ft by 5 ft, 25 ft by 10 ft

APPL ICAT IONS

SKILL

8


Name ______________________________________ Date ___________

Greatest Common Factor (GCF)Carlos is 20 years old and his brother Thomas is 24 years old. The greatestcommon factor (GCF) of their ages is the same as their niece Cristina’s age.

How old is Cristina?

To find Cristina’s age, find the GCF of 20 and 24. One way to find theGCF is to find the prime factorization of each number.

20 � 2 � 10 24 � 2 � 12� 2 � 2 � 5 � 2 � 4 � 3

� 2 � 2 � 2 � 3

Then find the common prime factors.

20 � 2 � 2 � 524 � 2 � 2 � 2 � 3

The common prime factors are 2 and 2. So, the greatest common factor of 20 and 24 is 2 � 2, or 4. So Cristina is 4 years old.

Find the GCF for each set of numbers.

1. 16, 24 2. 15, 18 3. 18, 36 4. 32, 488 3 18 16

5. 28, 70 6. 72, 96 7. 81, 48 8. 48, 3614 24 3 12

9. 40, 56 10. 14, 28 11. 30, 18 12. 84, 1548 14 6 14

EXERCISES

EXAMPLE


13. 24, 64 14. 35, 25 15. 100, 80 16. 75, 1208 5 20 15

17. 2, 4, 8 18. 8, 12, 16 19. 18, 30, 36 20. 15, 25, 302 4 6 5

21. 18, 12, 24 22. 24, 36, 48 23. 8, 16, 40 24. 12, 18, 726 12 8 6

Nita is making a baby quilt. She is usingstrips of material that are cut from piecesof material that are 36 inches wide and 48 inches wide. Use this information toanswer Exercises 25 and 26.

25. All of the strips are to be the same width and as wide as possible. How wide should the strips be? How many strips will Nita be able to cut from each piece of material?12 in. wide; 3 pieces from the 36-in. wide piece and 4 pieces from the 48-in. wide piece

26. Nita found another piece of material that she decided to usefor the quilt. The piece of material is 54 inches wide. If all ofthe strips from the three pieces of material are to be the samewidth and as wide as possible, how wide should the strips be?How many strips will Nita be able to cut from each of thethree pieces of material? 6 in. wide; 6 pieces from the 36-in. wide piece, 8 pieces from the 48-in. wide piece, and 9 pieces from the 54-in. wide piece

APPL ICAT IONS

SKILL

9


Name ______________________________________ Date ___________

Ratios as FractionsA ratio is a comparison of two numbers by division.

A store sold 360 newspapers last week and 440 this week. Write aratio in simplest form comparing last week’s sales to this week’s sales.

The ratio of last week’s sales to this week’s sales is

360 to 440, 360 : 440, or .

�

�

The ratio written as a fraction in simplest form is .

Two ratios are equivalent if the simplest form of the ratios are equal.

Are 12 : 18 and 14 : 21 equivalent ratios?

Express each ratio as a fraction in simplest form.

� � � �

Since � , the ratios are equivalent.

Express each ratio as a fraction in simplest form.

1. 8 cheese pizzas 2. 27 feet to 24 feetout of 14 pizzas

3. 35 sopranos in an 4. 14 hours to 3 days84-member chorus

7�36

5�12

9�8

4�7

EXERCISES

2�3

2�3

2�3

14 � 7�21 � 7

14�21

2�3

12 � 6�18 � 6

12�18

EXAMPLE

9�11

9�11

360�440

last week’s sales��this week’s sales

360�440

EXAMPLE


5. read 75 pages out of 90 6. 68 : 18

7. 6 pounds : 12 ounces 8. exercise 45 minutes out of 63

9. 15 : 50 10. 40 minutes per hour

Tell whether the ratios in each pair are equivalent. Show youranswer by simplifying.

11. 42 to 49 and 54 to 63 12. 18 : 42 and 20 : 44

� ; yes ; no

13. 144 : 36 and 72 : 32 14. 16 to 96 and 1 to 6

; no � ; yes

15. 3 lbs. : 12 ozs. and 16. 6 hours to 4 days and6 lbs : 24 ozs. 12 hours to 10 days

� ; yes ; no

17. Sound waves travel at about 740 miles per hour. In 1947,Chuck Yeager became the first person to fly a plane at a speedgreater than the speed of sound. If he had flown at the speedof sound, he would have flown at mach 1. The mach numberis the ratio of an object’s speed to the speed of sound. Findthe mach number of each of the following and express it as adecimal rounded to the nearest hundredth.a. a Bell X-15A2 rocket plane that flew at 4,520 miles per

hour in 1964 mach 6.11b. a Boeing 747 passenger plane that can fly at a speed of

625 miles per hour mach 0.84c. the space shuttle Columbia that has traveled through

space at over 16,600 miles per hour mach 22.43d. Carl Lewis who ran 100 meters in 9.86 seconds in 1991

(that is about 20 miles per hour) mach 0.03e. a Cessna passenger plane that can fly 176 miles per hour

mach 0.24

APPL ICAT ION

1�20

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SKILL

10


Name ______________________________________ Date ___________

Adding and Subtracting FractionsTo add fractions, you must have a common denominator.

Find each sum.

a. � b. �

�

� � � �

� 1

The sum . The sum is 1 .

To subtract fractions, you must have a common denominator.

Find each difference.

a. � b. �

�

� � � �

�

The difference is . The difference is .

Add or subtract. Write each answer in simplestform.

1. 2. 3.

� � �

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EXERCISES

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1�8

3�4

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1�2

2�12

5�8

5�8

11�12

1�2

5�8

2�12

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EXAMPLE

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10�12

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EXAMPLE


4. 5. 6.

� � �

1

7. 8. 9.

� � �

10. 11. 12.

� � �

1 1 1

13. Reginald planted of his garden with tomatoes and of his

garden with green beans. How much of his garden is planted with either tomatoes or green beans? How much of his garden is planted with other crops?

of the garden; of the garden

14. Tina rode her bicycle mile to the park and then mile to 1�2

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APPL ICAT IONS

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the library. Finally she rode her bicycle mile to her home.

How far did Tina ride her bike?

or 1 mi

15. In a survey, of the people said they preferred Brand A, and 2�7

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of the people said they preferred Brand B. What is the

difference between the fraction of people who prefer Brand A and the fraction of people who prefer Brand B?3

�35

1�5

SKILL

11


Name ______________________________________ Date ___________

Adding and Subtracting FractionsLina is making trail mix for a hiking trip. She has 2 cups of peanuts,

3 cups of raisins, and 2 cups of carob chips.

How many cups of trail mix will Lina have?

2 � 2

3 � 3

� 2 � � 2

7 �7 � �

� 7 � 1 �

� 8 �

� 8

Lina will have 8 cups of trail mix.

If Lina wants 15 cups of trail mix, how many more cups of trail mixdoes she have to make?

15 � 14 � 1 � 14 � � 14

15 � 14

� 8 � � 8

6

She needs to make another 6 cups of trail mix.

Add or subtract. Write each answer in simplest form.

1. � 2. � 3. � 1 1�3

5�9

7�9

3�5

3�10

9�10

3�4

2�12

7�12

EXERCISES

7�12

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5�12

5�12

12�12

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5�12

5�12

5�12

5�12

5�12

12�12

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1�2

EXAMPLES

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4. � 5. � 6. �

7. � 1 8. � 9. �

10. � 11. � 12. �

13. � 1 14. � 15. 7 � 2 9

16. 9 � 5 4 17. 5 � 2 8 18. 9 � 2 7

19. The route from Ramon’s house to city hall and then to the

school is mile. It is mile from city hall to the school.

What is the distance from Ramon’s house to city hall?

20. To make a salad, Henry used pound of Boston lettuce and 3�4

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9�10

APPL ICAT IONS

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1�6

3�4

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take her 1 hours. How much time should she plan to spend

on these chores?

24. Mr. Vazquez wants to put a fence around his rectangular

vegetable garden. If the garden is 18 feet long and 10 feetwide, how much fence will he need?

1�2

3�4

1�2

pound of red lettuce. How much lettuce did he use?

21. Donna has 10 yards of ribbon. She needs 3 yards of ribbon

to make a bow. How much ribbon will she have after shemakes the bow?

22. Part of the daily diet of polar bears at the Bronx Zoo is

1 pounds of apples and a 1 -pound mixture of oats and

barley. What is the combined weight of these items?

23. Ani has two chores to do on Saturday. She has to wash the car

which will take her hour and rake the leaves which will 3�4

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mile

1 lb

7 yd

2 lb

2 hr

58 feet1�2

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1�4

5�12

3�5

SKILL

12


Name ______________________________________ Date ___________

Multiplying and Dividing FractionsTo multiply fractions, multiply the numerators and multiply the denominators.

What is the product of and ?

� �Multiply the numerators.Multiply the denominators.

� or Simplify.

The product is .

To divide by a fraction, multiply by its reciprocal.

What is the quotient of and ?

� � � Multiply by the reciprocal of .

�Multiply the numerators.Multiply the denominators.

� or 1 Simplify.

The quotient is or 1 .

Multiply or divide. Write each answer in simplestform.

1. � 2. � 3. �

4. � 5. � 6. � or 1

7. � 8. � or 1 9. �5�8

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6�7

5�6

5�7

2�15

1�6

4�5

3�4

2�3

1�2

1�3

2�3

1�2

EXERCISES

1�5

6�5

1�5

6�5

3 � 2�5 � 1

1�2

2�1

3�5

1�2

3�5

1�2

3�5EXAMPLE

1�4

1�4

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3 � 2�8 � 3

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3�8EXAMPLE


10. � 11. � 12. �

or 5

13. � 14. � 15. �

or 1

16. � 17. � 18. �

or 1 or 1

19. � 20. � 21. �

22. Of the 48 NBA World Championship Series from 1947 to 1994,

the Boston Celtics won of the championships. Two thirds

of the Celtics’ championships occurred before 1970. What fraction represents the championships that were won by theCeltics before 1970?

23. About of the land in the continental United States is in 1

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APPL ICAT IONS

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Texas. About of the land in Texas is used as rural

pastureland. What fraction of the land in the continentalUnited States is Texas pastureland?

24. Helen planted vegetables and flowers in her garden. Three

fourths of her garden is planted in flowers. If of the total

garden is planted in roses, what fraction of the flower gardenis planted in roses?

25. One third of the videos at Vinnie’s Video Store are appropriate

for young children. If of the children’s videos are cartoons,

what fraction of the videos in the store are children’s cartoons?

2�5

1�10

5�9

2�15

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SKILL

13


Name ______________________________________ Date ___________

Multiplying and Dividing FractionsA new industrial park is being developed. The ABC Manufacturing

Company owns a rectangular piece of property that is mile long and

mile wide.

What is the area of the property owned by the ABC Manufacturing Company?

To find the area of a rectangle, you multiply the length by the width.

� �Multiply the numerators.Multiply the denominators.

� or Simplify.

The ABC Manufacturing Company owns square mile of land.

The A to Z Distribution Company owns square mile of land in the

industrial park. If the land is in the shape of a rectangle and the

length of the land is mile, what is the width of their land?

To find the width, divide the area of the rectangle by the length.

� = � Multiply by the reciprocal of .

�Multiply the numerators.Multiply the denominators.

�

The width of the land owned by A to Z Distributing Company is mile.

Multiply or divide. Write each answer in simplest form.

1. � 2. � 3. �3

�141�2

3�7

5�8

2�5

1�4

1�6

1�4

2�3

EXERCISES

3�8

3�8

1 � 3�8 � 1

1�3

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1�8

1�3

1�8

1�3

1�8

1�10

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2 � 1�5 � 4

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EXAMPLES

1�4

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4. � 5. � 6. �

7. � 8. � 9. �

10. � 11. � 12. � or 1

13. � 14. � 2 15. �

16. � 17. � 18. � 3

19. � 20. � 21. �

22. About of the world’s population lives in Africa. About

of the population of Africa lives in Ethiopia. About what fraction of the world’s population lives in Ethiopia?

23. About of the world’s water supply is fresh water. If about

of Earth’s surface is covered with water, about what fraction of Earth is covered with fresh water?

24. Two thirds of Esma’s garden is planted in flowers. If of the

flowers are gladiolas, what fraction of the garden is planted in gladiolas?

25. One eighth of Jonas’ garden is planted in green beans. If

of his garden is planted in vegetables, what fraction of thevegetable garden is planted in green beans?

26. Three fourths of the books sold at Bernie’s Book Store are

paperbacks. If of the paperbacks sold are adventure

stories, what fraction of the books sold are paperback adventure books?

27. A honeybee can produce pound of honey in its lifetime. 1

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APPL ICAT IONS

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about

about

5 honeybees

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How many honeybees does it take to make pound of honey?

1�2

SKILL

14


Name ______________________________________ Date ___________

MultiplesBryan noticed that every time he spent $1 at the department store, he paid8¢ in sales tax. He decided to make a table of the amount of sales taxcharged on whole-dollar purchases.

Can you help him make the table?

The amount of sales tax charged on whole-dollar purchases can befound using multiples of 8. A multiple of a number is the product ofthat number and any whole number.

List the first four multiples of each number.

1. 10 2. 9 3. 1510, 20, 30, 40 9, 18, 27, 36 15, 30, 45, 60

4. 7 5. 18 6. 127, 14, 21, 28 18, 36, 54, 72 12, 24, 36, 48

7. 20 8. 25 9. 1620, 40, 60, 80 25, 50, 75, 100 16, 32, 48, 64

EXERCISES

EXAMPLE

Amount of Purchase Amount of Sales Tax

$1 8¢

$2 16¢

$3 24¢

$4 32¢

$5 40¢

$6 48¢

$7 56¢

$8 64¢

$9 72¢

$10 80¢


Determine whether the first number is a multiple of the secondnumber.

10. 56; 7 yes 11. 42; 14 yes 12. 81; 18 no

13. 45; 11 no 14. 100; 20 yes 15. 72; 36 yes

16. 95; 19 yes 17. 225; 25 yes 18. 110; 21 no

Kyle is planning a trip. He plans to drive 55 miles per hour. Use this information toanswer Exercises 19 and 20.

19. How far will Kyle travel in

a. 1 hour? 55 miles

b. 2 hours? 110 miles

c. 3 hours? 165 miles

d. 4 hours? 220 miles

e. 5 hours? 275 miles

f. 6 hours? 330 miles

20. Suppose after Kyle’s trip he determines that he actually averaged 60 miles per hour. How could you use your answersto Exercise 19 to determine the distance at this rate? Add successive multiples of 5 to each answer.

21. Tia is laying a pattern of tiles in rows. One row has tiles thatare 4 inches long, and the next row has tiles that are 5 incheslong. In how many inches will the ends of the two rows beeven and the pattern start to repeat?20 inches

APPL ICAT IONS

SKILL

15


Name ______________________________________ Date ___________

Percents as Fractions and DecimalsA stereo is on sale for 33 % off the original price.

Write this percent as a fraction in simplest form and as a decimal.

To express a percent as a fraction in simplest form, express the percent

in the form and simplify.

33 % �

� � Write 33 as and multiply by the reciprocal

of 100.

�

To express a percent as a decimal, express the percent in the form and then express the fraction as a decimal.

33 % � You found this in the example above.

� 0.33

Write each percent as a fraction in simplest formand as a decimal.

1. 40% 2. 8% 3. 29%

; 0.4 ; 0.08 ; 0.29

4. 55% 5. 25% 6. 81%

; 0.55 ; 0.25 ; 0.8181�100

1�4

11�20

29�100

2�25

2�5

EXERCISES

1�3

1�3

r�100

1�3

100�

31�3

1�100

100�

3

3313�

1001�3

r�100

EXAMPLE

1�3


7. 66 % 8. 98% 9. 16.5%

; 0.67 ; 0.98 ; 0.165

10. 30% 11. 240% 12. 0.05%

; 0.3 or 2 ; 2.4 ; 0.0005

Between 1980 and 1990, the population ofNew Hampshire increased by 20.5%. Usethis information to answer Exercises 13–17.

13. Write this percent as a fraction in simplest form.

14. Write this percent as a decimal.0.205

15. When is it best to use the percent instead of the fraction orthe decimal? Answers will vary.

16. When is it best to use the fraction instead of the percent orthe decimal? Answers will vary.

17. When is it best to use the decimal instead of the percent orthe fraction? Answers will vary.

18. Between 1975 and 1985, the disposable personal income inthe United States more than doubled. Does this mean theincome has increased by more than 200%? Explain. yes; When something doubles, it increases by a factor of 2 and 200% � � 2.200

�100

41�200

APPL ICAT IONS

1�2,000

2�5

12�5

3�10

33�200

49�50

2�3

2�3

SKILL

16


Name ______________________________________ Date ___________

Percent of a NumberTo find the percent of a number, you can either change the percent to afraction and then multiply, or change the percent to a decimal and thenmultiply.

In the Washington County championship basketball game, Lee made55% of his 20 attempted field goals. How many field goals did hemake?

Find 55% of 20.

Method 1 Method 2Change the percent to a fraction. Change the percent to a decimal.

55% � � 55% � or 0.55

� 20 � 11 0.55 � 20 � 11

Lee made 11 field goals.

Find the percent of each number.

1. 25% of 200 50 2. 30% of 55 16.5 3. 3% of 610 18.3

4. 5.5% of 25 1.375 5. 13% of 85 11.05 6. 97% of 12 11.64

7. 1% of 25 0.25 8. 140% of 125 175 9. 100% of 50 50

10. Which of the following does not belong? da. 25% of 80b. 80% of 25c. 2.5% of 800d. 8% of 2500

11. Hannah’s basketball team won 75% of their games this season.They played 28 games this year. How many games did theywin? 21 games

EXERCISES

11�20

55�100

11�20

55�100

EXAMPLE


Write a percent to represent the shaded area.

12. 13.

30% 35%14. 15.

57% 44%

16. Kleema owns 40 music CD’s. Fifteen of her CD’s are recordingsdone by rap groups. What percent of her CD collection is rapmusic? 37.5%

17. The Polletta’s went out to dinner, and the food bill was$35.00. The standard rate for tipping is 15%.

a. What is the decimal value of this percent? 0.15b. What should their tip be? $5.25c. What is their total food and tip bill? $40.25

18. Angie wants to put a winter coat in layaway at a store. To doso, she must pay the store 20% of the cost of the coat so theywill hold it. If the coat costs $48.99, about how much of adeposit does Angie need to pay the store? $10.00

19. Mrs. Saunders made $600 last week, and she put 15% of that amount into her savings account. How much did she save? $90

APPL ICAT IONS

SKILL

17


Name ______________________________________ Date ___________

Percent ProportionUse the percent proportion to solve problems dealing with percent.

� P � percentage B � base � rate

37.2 is what percent of 186? What number is 15% of 280?

� �

� �

(37.2)(100) � (186)(r) (P)(100) � (280)(15)

3,720 � 186r 100P � 4,200

20 � r P � 42

37.2 is 20% of 186. 42 is 15% of 280.

Tell whether each number is the percentage, base,or rate.

1. 12 is what percent of 30? 2. 6.25% of 190 is what number?percentage: 12, base: 30 base: 190, rate: 6.25%

3. What percent of 49 is 7? 4. 40% of what number is 82?percentage: 7, base: 49 percentage: 82, rate: 40%

Write a proportion for each problem. Then solve. Round answersto the nearest tenth.

5. What number is 10% of 230? 6. 25% of what number is 38?23 152

7. Find 15% of 160. 8. 24 is 20% of what number?24 120

9. 36 is 75% of what number? 10. 36% of what number is 18?48 50

EXERCISES

15�100

P�280

r�100

37.2�186

r�100

P�B

r�100

P�B

EXAMPLES

r�100

r�100

P�B


11. What percent of 224 is 28? 12. What number is 40% of 250?12.5% 100

13. 15% of 290 is what number? 14. 50% of what number is 74?43.5 148

15. Use a proportion to find 55 % of 66. Round to the nearest

tenth.36.6

16. Use a proportion to find 19 % of 45. Round to the nearest

tenth.8.7

17. In Juan’s math class, there are 16 boys and 9 girls. What percent of Juan’s class is girls?36%

18. To the nearest whole percent, 44% of the seventh-graders atKing Middle School are girls. There are 425 seventh-graders.What is the number of girls in the seventh grade?187 girls

19. If 69% of the 247 students in the seventh grade ride the busto school, about how many students do not ride the bus toschool?about 77 students

20. There are 20 students running for student council at Pine BluffHigh School. If the school will elect a president, vice president,treasurer, and secretary, what percent of the students runningwill win in the election?20%

21. There were 102,269 tickets available for a rock concert. If TheTicket Company sold 72.5% of the tickets available, about howmany tickets did they sell for the concert?about 74,145 tickets

APPL ICAT IONS

1�4

1�2

SKILL

18


Name ______________________________________ Date ___________

Percent of ChangeThe population of Iowa in 1980 was 2,913,808. The population in 1990 was 2,776,755.

Find the percent of decrease in the population.

To find the percent of decrease, you can follow these steps.

1. Subtract to find the amount of decrease.

2,913,808 � 2,776,755 � 137,053

2. Solve the percent proportion. Compare the amount of decrease tothe original amount.

�

137,053 � 100 � 2,913,808 � r

13,705,300 � 2,913,808r

�

5 � r

The population of Iowa decreased by about 5%.

Find the percent of change. Round to the nearestwhole percent.

1. old: $5 2. old: 45 studentsnew: $7 new: 50 students40% increase 11% increase

3. old: 32 dogs 4. old: $56new: 30 dogs new: $52 6% decrease 7% decrease

EXERCISES

2,913,808r��2,913,808

13,705,300��2,913,808

r�100

137,053��2,913,808

EXAMPLE


5. old: 345 adults 6. old: $648new: 450 adults new: $635 30% increase 2% decrease

7. old: 150 pounds 8. old: 9.5 hoursnew: 138 pounds new: 8 hours 8% decrease 16% decrease

Last year, the value of Paul’s used car was$19,990. Use this information to answerExercises 9–11.

9. This year, the value of his car is $11,994. What was the percentchange in the car’s value? 40% decrease

10. The year before last the value of his car was $24,500. Whatwas the percent change in the car’s value? How does thischange compare to the change from last year to this year? 18% decrease; It is much less of a change.

11. What was the total percent change in the car’s value over the two years? Can you find the answer to this question by simply adding the answers to Exercises 9 and 10? Why or why not? 51% decrease; no; 40% � 18% � 58% 51% which is the actual change.

12. A clothing store has a 65% markup on blazers. But, the blazers did not sell well at the listed price. So, the blazerswere put on sale at 65% off the listed price. Did the storebreak even, make a profit, or lose money? Explain. The store lost money because 65% of the original list price isgreater than 65% of the store’s cost.

APPL ICAT IONS

SKILL

19


Name ______________________________________ Date ___________

Solve Equations Involving Additionand Subtraction

Addition Property of Equality: If you add the same number to each side of anequation, the two sides remain equal.

Solve t � 57 � 46.

t � 57 � 46

t � 57 � 57 � 46 � 57 Add 57 to each side.

t � 103

Check: t � 57 � 46

103 � 57 46 Replace t with 103.

46 � 46 ✓

The solution is 103.

Subtraction Property of Equality: If you subtract the same number from eachside of an equation, the two sides remain equal.

Solve t � 24.4 � 25.1.

t � 24.4 � 25.1

t � 24.4 � 24.4 � 25.1 � 24.4 Subtract 24.4 from each side of

t � 0.7 the equation.

Check: t � 24.4 � 25.1

0.7 � 24.4 25.1 Replace t with 0.7.

25.1 � 25.1 ✓

The solution is 0.7.

Complete each statement.

1. y � 18 � 39 2. m � 23 � 17y � 18 � 18 � 39 � 18 m � 23 � 23 � 17 � 23

EXERCISES

?�

EXAMPLE

?�

EXAMPLE


Solve each equation. Check your solution.

3. w � 6 � 19 4. n � 4.7 � 8.4 5. m � 18 � 7813 13.1 60

6. 18.42 � t � 63 7. e � 0.9 � 17.4 8. b � 43 � 1844.58 18.3 61

9. h � 32 � 44 10. 947 � p � 43 11. 7.36 � w � 8.943�5

76 990 1.58

12. g � 6.3 � 9.5 13. r � 18 � 36 14. 2.17 � k � 4.1915.8 54 2.02

Each of Exercises 15–18 can be modeled byone of these equations:

n � 2 � 10 n � 2 � 10Choose the correct equation. Then solve theproblem.

15. Jameel loaned two tapes to a friend. He has ten tapes left. How many tapes did Jameel originally have? n � 2 � 10; 12 tapes

16. Ana needs $2 more to buy a $10 scarf. How much money doesshe already have? n � 2 � 10; $8

17. The width of the rectangle shown at the right is2 inches less than the length. What is the length?n � 2 � 10; 12 inches

18. In the figure at the right, the length of A�C� is10 centimeters. The length of B�C� is 2 centimeters. What is the length of A�B�? n � 2 � 10; 8 cm

A B C

10 in.

APPL ICAT IONS

3�5

SKILL

20


Name ______________________________________ Date ___________

Solve Equations InvolvingMultiplication and Division

Division Property of Equality: If you divide each side of an equation by thesame nonzero number, the two sides remain equal.

Solve 156 � 4r.

156 � 4r

� Divide each side by 4.

39 � r

Check: 156 � 4r

156 4 � 39 Replace r with 39.

156 � 156 ✓ The solution is 39.

Multiplication Property of Equality: If you multiply each side of an equation bythe same number, the two sides remain equal.

Solve = 4.2.

� 4.2

� 21 � 4.2 � 21 Multiply each side by 21.

w � 88.2

Check: � 4.2

4.2 Replace w with 88.2.

4.2 � 4.2 ✓ The solution is 88.2.

?�

88.2�21

w�21

w�21

w�21

w�21EXAMPLE

?�

4r�4

156�

4

EXAMPLE


Complete the solution of each equation.

1. 12h � 48 2. 34 �

� 34 � 3 � � 3

h � 4 102 � r

Solve each equation. Check your solution.

3. 3.6t � 11.52 3.2 4. � 15 60 5. w �

6. 1.4j � 0.7 0.5 7. 4.1m � 13.12 3.2 8. � 16 80

9. 1.3z � 3.9 3 10. � f 11. � 0.6 2.1

12. h � 12 = 4.8 57.6 13. 4.8g � 15.36 3.2 14. c � �

Each of Exercises 15–17 can be modeled byone of these equations:

2n � 10 � 10

Choose the correct equation. Then solve theproblem.

15. Chum earned $10 for working two hours. How much did heearn per hour?2n � 10; $5

16. Kathy and her brother won a contest and shared the prizeequally. Each received $10. What was the amount of theprize?

� 10; $20

17. In the triangle at the right, the length of P�Q� is twice the length of Q�R�. What is the length of Q�R�?2n � 10; 5 cm

n�2

n�2

APPL ICAT IONS

1�8

1�2

1�4

d�3.5

7�4

1�2

7�8

c�5

3�4

3�8

1�2

n�4

r�3

48�12

12h�12

r�3

EXERCISES

P R

Q

30°

60°10 cm

SKILL

21


Name ______________________________________ Date ___________

Solve Two-Step EquationsTo solve two-step equations, you need to add or subtract first. You also needto multiply or divide.


7v � 3 � 25

7v � 3 � 3 � 25 � 3 Add 3 to each side.

7v � 28


v � 4

The solution is 4.

(r � 3) � –5

6 � (r � 3) � 6 � –5 Multiply each side by 6.

r � 3 � –30

r � 3 � 3 � –30 � 3 Add 3 to each side.

r � –27


Name the first step in solving each equation. Thensolve each equation.

1. 6n � 2 � 22 2. (y � 3) � 12

Add 2 to each side; 4 Multiply each side by 2; 27

1�2

EXERCISES

1�6

1�6

28�7

7v�7

EXAMPLES



3. –5t � 5 � –5 4. 4x � 5 � 15 5. 24 � 17 � 2c0 5 –3.5

6. –5h � 6 � 24 7. 6 � 3b � –9 8. 12 � 4n � 4–6 5 2

9. 7 � � 9 10. (d � 20) � –10 11. (a � 18) � –6

8 –34 9

Translate each sentence into an equation. Then solve the equation.

12. Six less than a number divided by 3 is 12.

� 6 � 12; 54

13. The sum of a number and four, times 3, is negative twelve.3(n � 4) � –12; –8

14. Three times a number plus negative five is negative eleven.3n � (–5) � –11; –2

15. On a July day in Detroit, Michigan, the temperature rose to

80°F. Find this temperature in degrees Celsius. (F = C + 32)

about 26.7°C

16. Aardvark Taxis charge $1.50 for the first half mile and then$0.25 for each additional quarter of a mile. What would thecost be for a 2-mile trip?$3

17. Three pens cost $1.55 including $0.08 sales tax. How much dideach pen cost?$0.49

9�5

APPL ICAT IONS

n�3

2�3

5�7

k�4

SKILL

22


Name ______________________________________ Date ___________

Use an EquationLucy bought some stickers that cost $0.25 each and a sticker book for $3.50.She spent $6.00.

How many stickers did Lucy buy?

Let s equal the number of stickers. Write and solve an equation.

s stickers at $0.25 each plus a $3.50 book cost $6.00.s � $0.25 � $3.50 � $6.00

0.25s � 3.50 � 6.000.25s � 3.50 � 3.50 � 6.00 � 3.50 Subtract 3.50 from each side.

0.25s � 2.500.25s � 0.25 � 2.50 � 0.25 Divide each side by 0.25.

s � 10

Lucy bought 10 stickers.

Solve by using an equation.

1. A number increased by 14 is 27. Find the number.13

2. The product of a number and 5 is 80. Find the number.16

3. A number is divided by 7. Then 6 is added to the result. Theresult is 26. What is the number?140

4. Three times a number minus 17 is equal to 28. What is the number?15

EXERCISES

EXAMPLE


5. A number is multiplied by 12. Then 3 is added to the result. Ifthe answer is 51, what is the original number?4

6. Twelve less than 16 times a number is 2 less than the productof 10 and 15. What is the number?10

7. Ruiz earned $117. If his pay is $6.50 per hour, how many hoursdid he work?18 hr

8. There are 425 students at Dayville Elementary School. If 198 ofthe students are girls, how many students are boys?227 boys

9. Jason is driving to his grandmother’s house 635 miles away. Hedrives 230 miles the first day and 294 miles the second day.How many miles must he drive the third day to reach hisgrandmother’s house?111 miles

10. Pachee bought some baseballs for $4 each and a batting glovefor $10. She spent $26. How many baseballs did she buy?4 baseballs

11. Fred has saved $490 toward the purchase of an $825 clarinet.His aunt gave him $75 to be used toward the purchase. Howmuch more money must he save?$260

12. Cindy went to the hobby shop and bought 2 model sports carsat $8.95 each and some paints. If she spent $23.65, what wasthe cost of the paints?$5.75

13. Arlen drove for 3 hours at 52 miles per hour. How fast must hedrive during the next 2 hours in order to have traveled a totalof 254 miles?49 mph

14. Postage costs $0.29 for the first ounce and $0.23 for eachadditional ounce. Peter spent $1.44 to send a package. Howmuch did it weigh?6 oz

APPL ICAT IONS

SKILL

23


Name ______________________________________ Date ___________

ProportionsA proportion is an equation that shows that two ratios are equivalent. Thecross products of a proportion are equal.

Determine if the ratios and form a proportion.

Find the cross products of � .

2 � 18 � 36

3 � 12 � 36

So, � is a proportion.

If one term of a proportion is not known, you can use cross products to findthe term. This is called solving the proportion.

Solve � .

�

r � 8 � 24 � 7 Find the cross products.

8r � 168


r � 21

Therefore, r equals 21.

Solve each proportion.

1. � 2. � 3. �

4 15 9

k�15

12�20

m�24

5�8

5�10

2�n

EXERCISES

168�

88r�8

7�8

r�24

7�8

r�24EXAMPLE

12�18

2�3

12�18

2�3

12�18

2�3EXAMPLE


4. � 5. � 6. �

7 10 25

7. � 8. � 9. �

0.2 2 28

10. Holly was absent from school 8 out of 36 days. Juan wasabsent 9 out of 45 days. Do these ratios form a proportion?no

11. Denise needed 4 hours to paint 1,280 square feet of wallspace. How much time would she need to paint 1,600 squarefeet of space?5 hours

12. On a map, the scale is 1 inch:125 miles. What is the actual

distance if the map distance is 4 inches?

562.5 miles

13. If you spend 1.5 hours per day doing homework, how manyhours would you spend doing homework in 8 days?12 hours

14. Jenny got 3 hits in her first 8 at-bats this season. How manyhits must she get in her next 40 at-bats to maintain this ratio?15 hits

15. Josh spends 40 cents out of every dollar on snacks and 14 cents out of every dollar on school supplies. He puts therest in a savings account. If Josh earns $32.00 per week cuttinglawns, how much does he save per week?$14.72

1�2

APPL ICAT IONS

c�36

7�9

t�16

15�120

2�8

f�0.8

6�2

75�r

n�50

6�30

16�32

3.5�m

SKILL

24


Name ______________________________________ Date ___________

Proportional ReasoningThe park ranger stocks the fishing pond, keeping a ratio of 4 sunfish forevery 3 perch. The ranger has just added 296 sunfish.

How many perch should the ranger stock?

�

4p � 296 � 3 Cross multiply.

4p � 888


p � 222

The ranger should stock 222 perch.

Write a proportion to solve each problem. Then solve.

1. 40 nails hold 5 rafters. 96 nails hold r rafters. � ; 12 rafters

2. 2 quarts fill 8 cups. 5 quarts fill c cups. � ; 20 cups

3. 81 rivets on 3 panels. r rivets on 13 panels. � ; 351 rivets

4. 32 addresses are on 2 pages of the address book. � ; 144 addressesa addresses are on 9 pages.

5. 60 sliced mushrooms on 4 pizzas. m sliced mushrooms on 15 pizzas. � ; 225 mushrooms

6. 98 beats a minute. y beats per hour. � ; 5,880 beatsy�60

98�1

m�15

60�4

a�9

32�2

r�13

81�3

5�c

2�8

96�r

40�5

EXERCISES

888�

44p�4

sunfishperch

←←

296�

p4�3

→→

sunfishperch

EXAMPLE


Solve by using proportional reasoning.

7. Naturalists can determine the number of fish in a pond byusing the capture/recapture procedure. To simulate this procedure, put an unknown quantity of cut pieces of paper (at least 50) in a bag. Take out a small handful of pieces andmark them with an X. Place these pieces back in the bag and mix up the pieces.

Take out another small handful. This is the recapture. Recordthe number of recaptured pieces and the number recapturedwith an X. Return the pieces and repeat the recapture 9 times.

Find the sum of the recaptured pieces and the sum of therecaptured ones with an X. Use the following proportion todetermine the total number of pieces in the bag.

� Answers will vary.

8. A shop produces 47 surf boards in 6 days. How long will ittake them to make 423 surf boards? 54 days

9. Cole can pick 2 rows of beans in 30 minutes. How long will it take him to pick 5 rows if he works at the same rate? 75 min

10. Suppose 4 kilograms of meat will serve 20 people. How many kilograms of the meat are needed to serve 110 people? 22 kg

total recaptured with an X��

total recapturedoriginal number captured��

number in the bag

APPL ICAT IONS

SKILL

25


Name ______________________________________ Date ___________

Scale DrawingsChuck has a scale drawing of Detroit’s Tiger Stadium. The scale of the

drawing is inch equals 25 feet. On the drawing, the home-run distance

What is the actual home-run distance from home plate to right field?

Think of inch as 0.25 inch and 3 inches as

3.25 inches. Use the scale 0.25 inch equals 25 feet and write a proportion to find the actual distance.

�

0.25x � 25 � 3.25 Cross multiply.

0.25x � 81.25

� Divide each side by 0.25.

x � 325

The actual distance is 325 feet.

On a map, the scale is 1 inch equals 150 miles. For each map distance, find the actual distance.

1. 3 inches 2. 8 inches 3. inch

450 miles 1,200 miles 75 miles

4. 5 inches 5. 1 inches 6. 4 inches

750 miles 225 miles 675 miles

1�2

1�2

1�2

EXERCISES

81.25�0.25

0.25x�0.25

drawing��actual distance

←←

3.25�

x0.25�25

→→

drawing��actual distance

1�4

1�4

EXAMPLE

1�4

from home plate to right field is 3 inches.1�4


On a scale drawing of a floor plan for a new building, the scale is inch equals 1 foot. Find the actual dimensions of the rooms if themeasurements from the drawing are given.

7. 5 inches by 3 inches 8. 2 inches by 4 inches

20 ft by 12 ft 8 ft by 16 ft





An igloo is a domed structure built of snowblocks by Eskimos. Sometimes several families built a cluster of igloos connected by passageways. Use the scale drawing ofsuch a cluster to answer Exercises 13–17.

13. What is the actual diameter ofthe living chambers? 8 ft

14. What is the actual diameter ofthe entry chamber? 6 ft

15. What is the actual diameter ofthe recreation area? 12 ft

16. What is the actual diameter ofthe storage area? 10 ft

17. Estimate the actual distancefrom the entry chamber to theback of the storage chamber.about 28 ft

APPL ICAT IONS

1�4

3�4

1�2

1�4

1�2

1�2

1�2

1�4

Recreation

1 in.1–2

in.3–4

StorageScale_____1 inch = 8 feet

1 in.1–4

Living

1 in.

Living

1 in.

Living

1 in.

Living

1 in.

Entry

SKILL

26


Name ______________________________________ Date ___________

Square RootsIf a2 � b, then a is the square root of b.

Joanna wants to buy a house. The realtor told her that the familyroom in a certain house has a floor area of 144 square feet. What isthe length of a side of the room if all four sides of the room are thesame length?

If all four sides of the room are the same length, then the room isshaped like a square. The area of a square is given by the formula A � s2. Use this formula to find the length of the sides of the room.

A � s2

144 � s2

�1�4�4� � �s2� To solve this equation, find the square root of each side.

12 � s The square root of 144 is 12.

The length of a side of the room is 12 feet.

Find each square root.

1. �9� 2. �2�5� 3. �8�1� 4. �1�6�9�3 5 9 13

5. �3�6� 6. �1�6� 7. �6�4� 8. �1�2�1�6 4 8 11

9. �1�0�0� 10. �4�0�0� 11. �9�0�0� 12. �1�0�,0�0�0�10 20 30 100

13. �1�9�6� 14. �0�.0�9� 15. �0�.8�1� 16. �1�.4�4�14 0.3 0.9 1.2

EXERCISES

EXAMPLE


17. �0�.4�9� 18. �0�.0�4� 19. �2�.2�5� 20. �0�.1�6�0.7 0.2 1.5 0.4

21. �� 22. �� 23. �� 24. ��

The area of a square picture is 64 squareinches. Use this information to answerExercises 25–27.

25. What is the length of each side of the picture?8 inches

26. What is the length of each side of a picture frame for the picture if the area of the picture and the frame is 121 squareinches? 11 inches

27. Will a square mat with an area of 81 square inches be largeenough on which to mount the picture? Why or why not? Yes, the length of each side of the square mat is 9 inches,which is greater than the side of the picture.

28. A square dog run with an area of 289 square feet is fenced inon all sides. What is the length of the fencing along one side? 17 feet

29. What is the diameter of a pizza that has an area of 254square inches? about 18 inches

30. The area of the bottom of a pizza box is 100 square inches. If a circular pizza fits in the box with the pizza touching thesides of the box at their midpoints, what is the diameter ofthe pizza? 10 inches

APPL ICAT IONS

5�6

25�36

7�10

49�100

4�5

16�25

2�3

4�9

SKILL

27


Name ______________________________________ Date ___________

Ordered PairsA horizontal number line and a vertical number line meet at their zeropoints to form a coordinate system. The horizontal line is the x-axis. The vertical line is the y-axis. The location of a point in the coordinate system can be named using an ordered pair of numbers.

(x, y)

x-coordinate y-coordinate

Name the ordered pair for point P.

Start at O. Move along the x-axis untilyou are above point P. Then move downuntil you reach point P. Since you moved4 units to the right and 3 units down, theordered pair for point P is (4, –3).

Graph point (–2, 4).

Start at O. Move 2 units left on the x-axis.Then move 4 units up parallel to the y-axis to locate the point.

Name the ordered pair for each point.

1. G (–1, 4) 2. H (5, 3)

3. J (–3, –4) 4. K (3, –3)

5. M (5, –2) 6. N (2, 0)

EXERCISES

EXAMPLES

54321

-1-2-3-4-5

-5 -4 -3 -2 -1O

1 2 3 4 5

y

P

x

(-2, 4)

4

-2 4

-3

54321

-1-2-3-4-5

-5 -4 -3 -2 -1O

1 2 3 4 5

y

KJ

N

HG

M

x


Graph and label each point.

7. A(–5, 5) 8. B(2, 4)

9. C(0, 5) 10. D(–4, 0)

11. E(2, 2) 12. F(4, –3)

A botanist is interested in what part of acertain leaf is being infested by an insectthat leaves black spots. She places a clearcoordinate plane over several leaves thatare about the same size and shape.Complete each of the following.

13. Find the coordinates of the black spotson the leaf at the right. (0, 2), (2, –2)(7, 3), (8, –3)

(–5 , 1), (–3, –5)

14. Draw and label the spots having the following coordinates on the leaf at the right.

A(2, –3) B(3, –2) C(0, –4) D(–4, 0)E(–5, 3) F(10, 2) G(2, 7) H(0, 5)

1�2

APPL ICAT IONS

54321

-1-2-3-4-5

-5 -4 -3 -2 -1O

1 2 3 4 5

y

x

SKILL

28


Name ______________________________________ Date ___________

Function TablesThe data at the right shows the shipping and handling charged by a catalog company.

Complete the table.

First look for a pattern in thedata that is already given. Each entry in the shipping andhandling column is $3 greaterthan the previous entry. So, tocomplete the table, add $3 toeach entry in the second columnto get the next entry. The entries for the last 3 rows of the table are given below.

Complete each table.

1. 2.

EXERCISES

EXAMPLE

Maximum Shipping and Purchase Handling (dollars) (dollars)

50 6.95100 9.95150 12.95200 15.95250300350

Maximum Shipping and Purchase Handling

250 18.95300 21.95350 24.95

Principal Interest(dollars) (dollars)

1,000 10

1,500 15

2,000 20

2,500 25

3,000 303,500 354,000 404,500 45

Distance Time(feet) (seconds)

5 7.5

10 15

15 22.5

20 30

25 37.5

30 4535 52.540 60


3. 4.

The table at the rightshows the amount ofFederal individual incometax for 1993 for differentamounts of adjusted grossincome between $22,100and $53,500 for single taxpayers. Use the data toanswer Exercises 5–7.

5. Complete the table. See table above for answer.6. Make a new table that includes 27,500, 32,500, 37,500, 42,500, 47,500,

and 52,500 in the adjusted gross income column. Explain how you found the income tax for these amounts. The new entries in the second col-umn would be 7,700, 9,100, 10,500, 11,900, 13,300, and 14,700.

7. Do you think it would be useful to have a table that containsmore data? Why or why not? How can you add more data tothe table? Answers will vary.

8. The rate for single taxpayers with an adjusted gross income between $53,500 and $115,000 is 31%. Make a table using adjusted gross incomes of $55,000, $60,000, $65,000, $70,000, $75,000, $80,000, $85,000, and $90,000. The entries in the second column would be17,050, 18,600, 20,150, 21,700, 23,250, 24,800, 26,350, and 27,900.

9. Extend the table you made in Exercise 8 to include any additional data you think would be useful. Explain why you included the data you did. Answers will vary.

APPL ICAT IONS

Purchase Tax(dollars) (dollars)

10 0.60

20 1.20

30 1.80

40 2.40

50 3.0060 3.6070 4.2080 4.80

Length of call Cost(minutes) (dollars)

1 1.00

2 1.35

3 1.70

4 2.05

5 2.406 2.757 3.108 3.45

Adjusted Income Gross Income Tax

(dollars) (dollars)

25,000 7,000

30,000 8,400

35,000 9,800

40,000 11,20045,000 12,60050,000 14,000

SKILL

29


Name ______________________________________ Date ___________

Graphing FunctionsCarin’s motor home averages about 25 miles on twogallons of gasoline. The function table at the rightshows this relationship.

Graph the function.

To graph the function, first label the axes and graph the points named by the data.Then connect the points as shown in thegraph at the right.

Graph each function.

1.

EXERCISES

0 8 12 16Gallons of Gasoline

4

50

100

150

200

Mile

s

EXAMPLE

Gallons ofGasoline Miles

2 25

4 50

6 75

8 100

10 125

12 150

14 175

16 200

Length of AreaSide (cm) (sq cm)

1 1

2 4

3 9

4 16

5 25

6 36

7 49


2.

The function table at theright shows the apparenttemperature for the givenroom temperatures for arelative humidity of 80%.Use the data to answerExercises 3–5.

3. Graph the function.

4. If this pattern continues, what would you expect the apparent temperature to be for a room temperature of 68°F? 69°F

5. Where does a change in the pattern of the function occur? Why do you think this change occurs? between the room temperaturesof 70°F and 71°F; Answers will vary.

APPL ICAT IONS

Time Savings(years) (dollars)

1 100

2 250

3 150

4 300

5 600

6 550

7 650

Room ApparentTemperature Temperature

(in °F) (in °F)

69 70

70 71

71 73

72 74

73 75

74 76

75 77

SKILL

30


Name ______________________________________ Date ___________

Solve Equations With Two VariablesAn ordered pair that makes an equation true is a solution for the equation.

Find four solutions for the equation y � 5x � 1.

Four solutions are (–4, –21), (–2, –11), (0, –1), and (2, 9).

Complete the table for each equation. Then usethe results to write four solutions for each equation. Write the solutions as ordered pairs.

1. y � 3x � 2 2. y � 4x 3. y � –3x � 4

(1, 5), (2, 8), (–1, –4), (0, 0), (–1, –1), (0, –4),(3, 11), (4, 14) (1, 4), (2, 8) (1, –7), (2, –10)

Find four solutions for each equation. Write your solutions asordered pairs. Answers will vary. Samples are given.

4. y � x � 4 5. y � 3x � 1 6. y � –3(0, –4), (1, –3), (–1, –2), (0, 1), (0, –3), (1, –3),(2, –2), (3, –1) (1, 4), (2, 7) (2, –3), (3, –3)

EXERCISES

EXAMPLE

Choose values for x. Calculate y values. Write ordered pairs.

Let x � –4. y � 5(–4) � 1 � –21 (–4, –21)Let x � –2. y � 5(–2) � 1 � –11 (–2, –11)Let x � 0. y � 5(0) � 1 � –1 (0, –1)Let x � 2. y � 5(2) �1 � 9 (2, 9)

x 4x y

–1 4(–1) –40 4(0) 01 4(1) 42 4(2) 8

x –3x � 4 y

–1 –3(–1) � 4 –10 –3(0) � 4 –41 –3(1) � 4 –72 –3(2) � 4 –10

x 3x � 2 y

1 3(1) � 2 52 3(2) � 2 83 3(3) � 2 114 3(4) � 2 14


7. y � –2x �2 8. y � 2.5x 9. y � –2x � 4(–2, 2), (–1, 0), (–2, –5), (–1, –2.5), (0, 4), (1, 2),(0, –2), (2, –6) (0, 0), (1, 2.5) (2, 0), (3, –2)

10. y � – x � 4 11. y � x � 1 12. y � x � 3

(–2, –3), (0, –4), (0, 1), (3, 2), (–2, 2), (0, 3),(2, –5), (4, –6) (6, 3), (9, 4) (2, 4), (4, 5)

13. One number is three more than half another number.Determine which ordered pairs in the set {(0, 3), (–2, 2), (4, –1),

(1, 3 )} are solutions for the two numbers.

(0, 3), (–2, 2), (1, 3 )

14. An organization donates one third of all the money it raisesfor housing the homeless. How much will it donate if it raises$6,000?

y � x; $2,000

15. You can show the distance in feet it takes a car to stop whentraveling at a certain speed on a dry, concrete surface by usingthe formula d � 0.042 s2 � 1.1s. Complete the table to find thedistance for each speed. Round the distances to the nearestfoot.

1�3

1�2

1�2

APPL ICAT IONS

1�2

1�3

1�2

speed in mph (s) 30 35 40 45 50 55 60 65 70 75

distance in feet (d) 71 90 111 135 160 188 217 249 283 319

SKILL

31


Name ______________________________________ Date ___________

Graphing EquationsGraph the equation y � 2x � 2.

Make a function table for y � 2x � 2.Then graph each ordered pair and complete the graph.

y � 2x � 2

Complete each function table. Then graph theequation.

1. y � x � 1

2. y � 5 � x

-2 4O

y

x2 6

-2

6

4

2

6

4

2

-2 4O

y

x2 6

-2

EXERCISES

8

6

4

2

-2 4O 8

y

x2 6

-2

EXAMPLE

x 2x � 2 y (x, y)

0 2(0) � 2 2 (0, 2)1 2(1) � 2 4 (1, 4)2 2(2) � 2 6 (2, 6)3 2(3) � 2 8 (3, 8)

x 5 � x y (x, y)

0 5 � 0 5 (0, 5)1 5 � 1 4 (1, 4)2 5 � 2 3 (2, 3)3 5 � 3 2 (3, 2)4 5 � 4 1 (4, 1)

x x � 1 y (x, y)

1 1 � 1 0 (1, 0)2 2 � 1 1 (2, 1)3 3 � 1 2 (3, 2)4 4 � 1 3 (4, 3)5 5 � 1 4 (5, 4)


Graph each equation.

3. y � x � 2 4. y � 3x 5. y � x � 1

6. An electrician charges an initial fee of $40, plus $50 forevery hour she works. Let x represent the number of hoursshe works and y represent the total fee. Write an equationto represent the total fee. Graph the equation.

y � 50x � 40

7. A blizzard at the Slippery Ski Area deposited foot of

snow per hour atop a 3-foot snow base. Let x represent thenumber of hours and y represent the total amount of snow.Write an equation to represent the total amount of snow.Graph the equation.

y � x � 3

8. Yukari averages 40 miles per hour when she drives from Los Angles to San Francisco. Let x represent the number ofhours and y represent the distance traveled. Write an equa-tion to represent the distance traveled. Graph the equation.

y � 40x

1�2

1�2

APPL ICAT IONS

-2

-2 4O

y

x2 6

6

4

2

-2

-2 4O

y

x2 6

6

4

2

-2 4O

y

x2 6

-2

6

4

2

1�2

2O

y

x1 3

120

80

40

4O

y

x2 6

6

4

2

2O

y

x1 3

120

80

40

SKILL

32


Name ______________________________________ Date ___________

Slope of a LineThe graph of a line is shown below.

Find the slope of the line.Follow these steps to find the slope.

1. Choose any two points on the line. The points chosen at the right havecoordinates (3, 4) and (–2, 8).

2. Draw a vertical line and then a horizontal line to connect the two points.

3. Find the length of the vertical line to find the rise. The rise is 4 units up or 4.

4. Find the length of the horizontal line to find the run. The run is 5 units to the left or –5.

5. slope � �4

�–5

rise�run

O

y

x

run = –5

rise = 4

(–2,8)

(3,4)

EXAMPLE

O

y

x


Find the slope of each line shown.

1. –1 2.

3. 4. 4

Paula works as a sales representativefor a computer manufacturer. Sheearns a base pay of $1,000 eachmonth. She also earns a commissionbased on her sales. The graph at theright shows her possible monthlyearnings. Use the graph to answerExercises 5–8.

5. What is the slope of the line?

6. What information is given by the slope of the line?

The rate of commission Paula earns is or 20% of her sales.

7. If Paula’s base pay changed to $1,100, would it changea. the graph? Why or why not?

Yes, the entire line would move up 100 units.b. the slope? Why or why not?

No, the rate of commission would not change.8. If Paula’s rate of commission changed to 25%, would it change

the graph? Why or why not? Yes, the slope would be .1�4

1�5

1�5

0

1,000

3,000

5,000 10,000

Paula’s Monthly EarningsPa

y

15,000Sales

2,000

4,000

6,000

5,000

APPL ICAT IONS

O

y

x

–3�4

O

y

x

1�3

O

y

xO

y

x

EXERCISES

SKILL

33


Name ______________________________________ Date ___________

Graphing Exponential EquationsJamie conducted an experiment that began with 400 bacteria. He foundthat the number of bacteria, y, after x hours was given by the equation y � 400(2x).

Use a graphing calculator to graph this equation.

Follow the steps below to graph the equation.

1. Press the key. Then enter the equation by pressing 400 2

.

2. Press to view the current boundaries of the viewing window of the calculator. Set the boundaries at Xmin � 0, Xmax � 10, Xscl � 1, Ymin � 0, Ymax � 500000, and Yscl � 50000.

3. Press to draw the graph shown below.

Use a graphing calculator to graph each equation.Make a sketch of each screen.

See students’ work. Graphs will vary.1. y � 5x 2. y � 0.8x

EXERCISES

GRAPH

WINDOW

X,T,�

�

EXAMPLE


3. y � � �x 4. y � 22x

5. y � 30(0.5x) 6. y � 500(0.25x)

Carbon-14 has a half-life of 5,730 years.Manford has a sample that contains 200 gof carbon-14. The equation for the grams of carbon-14 in the sample, y, after x 5,730-year intervals is given by the equation y � 200(0.5x).

7. Use a graphing calculator to graph this equation. See students’ work. Graphs will vary.

8. How would you use the information shown on this graph? See students’ work. Answers will vary.

9. Do you think this graph is the best way to display this information? Why or why not? See students’ work. Answers will vary.

10. Jaunita conducted an experiment that began with 200 bacteria. She found that the number of bacteria, y, after xhours was given by the equation y � 200(3x). Use a graphingcalculator to graph this equation. How would you use theinformation shown on this graph?See students’ work. Answers will vary.

APPL ICAT IONS

1�8

SKILL

34


Name ______________________________________ Date ___________

Graphing Linear and Exponential Equations

The value of manufacturing equipment with an initial value of $25,000depreciates at a rate of 10% a year. The value, V, of the equipment after nyears is given by the equation V � 25,000(1 � 0.10)n.

Use a graphing calculator to graph this equation.

Before you graph the equation on a graphing calculator, you mustrewrite the equation using Y for V and X for n. So the equation youwill use is Y � 25,000(1 � 0.10)X. Then follow the steps below to graphthe equation.

1. Press the key. Use the key to delete any equationsfrom the Y � list. Enter the equation by pressing 25000 1 1 .

2. Press to view the current boundaries of the viewing window of the calculator. Set the boundaries at Xmin � 0, Xmax � 50, Xscl � 10, Ymin � 0, Ymax � 25000, and Yscl � 1000.

3. Press to draw the graph shown below.GRAPH

WINDOW

X,T,�).�( �

EXAMPLE


Use a graphing calculator to graph each equation.Make a sketch of each screen.

See students’ work. Graphs will vary.1. y � 25x � 12 2. y � 16x � 15

3. y � –0.7x � 19 4. y � 3x

5. y � 4.5x 6. y � 452x

A $500 deposit is made into an account thatearns 3.5% interest and is compoundedmonthly. If no deposits or withdrawals aremade, the amount of money, A, in thisaccount after t years is given by the equation

A � 500�1 � �12t.

7. Change A and t to y and x and use a graphing calculator tograph this equation. See students’ work. Graphs will vary.

8. Change the boundaries of the viewing window and graph the equation again. Do you think these new boundaries giveyou a better graph than your original boundaries? Why orwhy not? See students’ work. Answers will vary.

9. Change the boundaries several more times until you find a graph that you like best. Why do you think this graph is the best? See students’ work. Answers will vary.

10. The 1993 Indianapolis 500 winner completed the race with anaverage speed of 157.2 mph. Write an equation that can beused to compute the distance, d, traveled after t hours. Use agraphing calculator to graph your equation. d � 157.2t; Graphs will vary.

0.035�

12

APPL ICAT IONS

EXERCISES

SKILL

35


Name ______________________________________ Date ___________

Sums of Angles of PolygonsA convex polygon is a closed figure in a plane that

• has at least three sides, all of which are segments,• has sides that meet only at a vertex,• has exactly two sides meeting at each vertex, and• has diagonals that lie entirely within the polygon.

If n is the number of sides of a polygon, then 180(n � 2) expresses the sumof the measures of the angles of any polygon.

Find the sum of the measures of theangles of the figure at the right.

There are 7 sides and 7 vertices.Substitute 7 for n in the expression 180(n � 2).

180(7 � 2) � 180(5) or 900

The sum of the measures of the angles ofthe figure is 900°.

Pick one vertex and draw all the diagonals possible from that vertex. How many diagonalscan be drawn from that one vertex in each figurebelow?

1. 2.

2 33. 4.

1 0

EXERCISES

EXAMPLE


Find the sum of the measures of the angles of each of the following polygons.

5. quadrilateral 6. triangle360 180

7. 8.

720 1,080

9. What is the fewest number of sides that a polygon can have?3

10. What is the shape of home plate on a baseball field? What isthe sum of the measures of the angles of home plate?pentagon, 540°

11. Find the value of x in quadrilateral ABCD if m�A � 108°,m�B � 72°, m�C � 108°, and m�D = x.x � 72°

12. Find the value of x in pentagon ABCDE if m�A � 120°, m�B � 60°, m�C � 210°, m�D � 55° and m�E � x.x � 95°

APPL ICAT IONS

STOP

SKILL

36


Name ______________________________________ Date ___________

Similar FiguresIf two or more figures are the same shape, they are similar. Similar figuresmay differ in size.

Determine if each pair of figures is similar.

The figures are the same The figures are not theshape. Therefore, the same shape. Therefore,figures are similar. the figures are not similar.

In Exercises 1–6, determine if each pair of figuresis similar.

1. yes 2. yes

3. no 4. yes

5. no 6. yes

EXERCISES

EXAMPLE


7. List the pairs or groups of similar figures.

a, d, and e; c and f

8. List three real-world examples of similar figures.Sample answers: T-shirt in different sizes, a photograph and anenlargement of the photo, a model of a car and the actual car

9. Use toothpicks to make the figure at the right. Use toothpicks to make a similar figure that isnot the same size as the original figure.See students’ work.

10. Use toothpicks to make a figure. Then maketwo more figures that are similar to the original. The figures should not be the same sizes.See students’ work.

11. Use grid paper to draw a figure that is similar to the figure at the right. See students’ work.

APPL ICAT IONS

a.

d.

b.

e.

c.

f.

SKILL

37


Name ______________________________________ Date ___________

Similar TrianglesThe triangles below are similar.

Measure each side of the triangles to the nearest centimeter. Write the ratios of the corresponding sides of the similar triangles.What do you notice about the ratios of the corresponding sides?

The measures of the sides are marked next to the triangles.

� � �

The ratios of the corresponding sides all equal .

Use the similar triangles below to answerExercises 1–3.

1. Measure each side of each triangle to the nearest centimeter.

EXERCISES

1�2

1�2

4�8

1�2

2�4

1�2

3�6

side of the first triangle��side of the second triangle

EXAMPLE

8 cm

4 cm

3 cm

4 cm

2 cm

6 cm


2. Find the ratios of the corresponding sides. � , � , �

3. What do you notice about the ratios of the correspondingsides?All the ratios equal .

Determine if each pair of triangles is similar.

4. 5.

no yes

Find the value of x in each pair of similar triangles.

6. 7.

12 ft 15 m

8. A lamppost casts a shadow 16 feet. A girl standing nearby casts a shadow of 4 feet. The two triangles formed aresimilar. If the girl is 5 feet tall, how tall is the lamppost?20 ft

9. Use similar triangles to find the distance across the pond.32 m

8 m ?

10 m

40 m

10 m

40 m

4 ft16 ft

?

5 ft

APPL ICAT IONS

3 m

9 m

5 m

4 m

12 m

x

5 ft10 ft

24 ft

13 ft26 ft

x

16 cm12 cm

9 cm

20 cm15 cm

12 cm

9 in.6 in.

8 in.

18 in.12 in.

11 in.

3�2

3�2

6�4

3�2

9�6

3�2

9�6

SKILL

38


Name ______________________________________ Date ___________

Congruent FiguresTwo or more figures that are the same shape and size are called congruent figures.

Determine if each pair of figures is congruent.

The figures are the The figures are the The figures are thesame shape, but not same size, but not same shape and thethe same size. The the same shape. same size. The figuresfigures are not The figures are are congruent.congruent. not congruent.

In Exercises 1–3, determine whether each pair offigures is congruent.

1. 2. 3.

no yes yes

EXERCISES

EXAMPLE

4. List the congruent figures. a, b, and f; c and d

In Exercises 5–8, use the grid to draw a figure that is congruent tothe given figure.

5. 6.

7. 8.

9. An architect wants to use ceramic tiles on the floor of a building she is designing. She wants to create a design usingfour-by-four squares of tiles. She plans to divide each four-by-four square into two congruent halves. There are six ways todivide the squares in this manner. One way is shown at theright. Show the five other ways below.

10. Draw a six-by-six tile pattern. Show at least three ways it canbe divided into congruent halves. See students’ work.

APPL ICAT IONS

a. b. c. d. e. f.


Name ______________________________________ Date ___________

ReflectionsIn a transformation, every point in an image corresponds to exactly onepoint on the figure. Reflections are one type of transformation.

Use the grid to reflect, or flip, the figure over the given line.

For each vertex on the figure, find the point that is exactly the same distance from the line of reflection, but on the other side of the line. Draw the completed image.

Reflect each figure over the given line.

1. 2.

EXERCISES

Figure Reflection

EXAMPLE

SK ILL

39



3. 4.

Use your reflections to answer Exercises 5–8.

5. Are the reflections in Exercises 1–4 smaller, larger, or the samesize as the original figures? the same

6. In Exercise 2, are the arrows pointing in the same direction?Do you think that direction is the same for a figure and itsreflection? no; no

7. In Exercise 3, the x’s and the dot are in a straight line. In thereflection, are the x’s and the dot in a straight line? yes

8. In Exercise 3, the dot is between the two x’s. In the reflection,is the dot between the two x’s? yes

M. C. Escher used transformations such as reflections to createinteresting art. A simple example of his type of art starts with a square. A simple change is made and this change is reflectedover the dashed line. Other reflections are made over otherdashed lines as shown.

Make a drawing using reflections, squares, and the changes indicated.

9. 10.

11. Make your own design using reflections. See students’ work.

APPL ICAT IONS

x

x

SKILL

40


Name ______________________________________ Date ___________

Dilations and RotationsIn mathematics, there are several ways that a figure may be moved orchanged. Two of these ways are dilations and rotations.

Draw the image of the triangle ABC for a dilation with a scale factorof 2.

Draw a dashed line from the origin ofthe coordinate plane to point A. Extendthe dashed line so that its length is twiceas long as the distance from the origin topoint A. This is one vertex of the dilatedtriangle. Repeat the procedure for theother two vertices and draw the dilatedtriangle.

Draw three rotated images of triangle DEF. Rotate the image aroundthe origin of the coordinate plane using 90° as the angle for each successive rotation.

Visualize point E rotating around the origin clockwise 90°. Remember that theimage point must be the same distancefrom the origin as the original point. Inthis case the image of (0, 3) is (3, 0). Find the image points for the other twovertices and draw the rotated triangle.Rotate the image two more times.

EXAMPLES

O

y

x

AB

C

O

y

x

E D

F


Draw a dilation for the given scale drawing.

1. Scale factor: 3 2. Scale factor:

Draw three images using 90° rotations around the origin.

3. 4.

Answer each of the following.

5. Does a dilation form similar or congruent figures? similar

6. Does a rotation form similar or congruent figures? congruent

7. Does the movement of a Ferris wheel represent a dilation or arotation? rotation

8. Does an enlargement of a photograph represent a dilation ora rotation? dilation

9. Make a design using rotations. See students’ work.

10. Make a design using dilations. See students’ work.

APPL ICAT IONS

1�2

EXERCISES

SKILL

41


Name ______________________________________ Date ___________

TranslationsA translation is a slide or movement of a figure from one place to another.

Translate triangle ABC 5 units to the right and 3 units down.

Move point A 5 units to the right and 3 units down. Move point B 5 units to the right and 3 units down. Finally, move point C 5 units to the right and 3 units down and draw the new triangle.

Translate each figure as indicated.

1. 7 units to the left 2. 8 units to the right and 2 units down

EXERCISES

A1

123

2 3 4 5

C

B

EXAMPLE


3. 5 units to the right and 4. 2 units to the left and2 units up 1 unit down

5. 5 units to the left and 6. 6 units to the right and2 units down 4 units up

Answer each question.

7. Are the translated figures congruent or similar to the originalfigures? congruent

8. In Exercise 5, are the arrows pointing in the same direction? Isdirection the same for a figure and its translation? yes; yes

9. In Exercise 6, the x’s and the dot are in a straight line. In thetranslation, are the x’s and the dot in a straight line? yes

10. In Exercise 6, the dot is between the two x’s. In the translation,is the dot between the two x’s? yes

11. Describe the dive from A to B in terms of a translation.4 units to the right and 6 units down

12. Describe a translation from your house to a friend’s house. See students’ work.

B

A

APPL ICAT IONS

SKILL

42


Name ______________________________________ Date ___________

Perimeter and AreaTova Albert wants to make a garden with a perimeter of 54 feetbecause that is the amount of fence that she has. She wants the leastarea possible because she doesn’t have that much space in her yard.What should be the dimensions of her garden?

Notice that the perimeter stays 54 feet but the area continues toincrease. Therefore, the least area with a perimeter of 54 feet is a garden with dimensions 1 foot by 26 feet.

Find the perimeter and area of each figure.

1. 2. 3.

P � 18 units P � 16 units P � 16 unitsA � 18 units2 A � 16 units2 A � 12 units2

4. 5. 6.

P � 20 units P � 18 units P � 18 unitsA � 24 units2 A � 12 units2 A � 14 units2

EXERCISES

EXAMPLE

Dimensions Perimeter Area

1 � 26 54 262 � 25 54 503 � 24 54 724 � 23 54 92


7. A cardboard tube has a circumference of 7 inches and a lengthof 15 inches. When it is cut straight down its length, itbecomes a rectangle. How much cardboard is used to makethis tube?105 in2

8. Ryan Allaire wants to build a deck onto the back of his house.He wants the area to be at least 240 square feet. There isspace for the length to be up to 20 feet, but the width cannotbe more than 15 feet.

a. Will he have room to build the size deck that he wants?yes

b. What is the largest deck that he can build?300 ft2

c. If he wants the deck to be exactly 240 square feet, what are the whole number dimensions that are possible for him?15 ft � 16 ft; 20 ft � 12 ft

9. Using the large square below, show how to cut it into two pieces (cuts must be made along the grid lines) that can be rearranged to form a rectangle with a perimeter of 26 centimeters.

10. Bovinet Candy Company needs to have a box designed so thatthe bottom has an area of 96 square inches but has the leastperimeter possible. What would be the whole number dimensions of the bottom of the box?8 in. � 12 in.

APPL ICAT IONS

SKILL

43


Name ______________________________________ Date ___________

Area of CirclesThe parts of a circle are illustrated at the right. Notice that the radius is one-half of the diameter.

The area (A) of a circle equals the product of pi (�) and the square of the radius (r).

A � �r2

The value of � is approximately 3.14.

Find the area of the circle.

The diameter of the circle is 14 meters. The radius of the

circle is (14) or 7 meters.

A � �r2

A � �(7)2

A � �(49)A � 3.14(49) Use 3.14 for �.A � 153.86

The area is about 153.86 square meters.

Find the area of each circle. Use 3.14 for �.

1. 2. 3.

113.04 m2 379.94 in2 200.96 in2

8 inches

22 inches

6 meters

EXERCISES

1�2

14 meters

EXAMPLE

center

radiu

s

diameter


4. 5. 6.

153.86 m2 803.84 ft2 28.26 cm2

7. 8. 9.

200.96 km2 63.585 km2 1,808.64 ft2

10. The Astrodome covers an area in the shape of a circle with a diameter of 214 yards. What area does the Astrodome cover? about 35,950 yd2

11. Find the floor of a ring in a circus tent if the diameter is 12 yards. about 113 yd2

12. The world’s largest cylindrical sundial is at Walt Disney World in Orlando, Florida. Arata Isozaki of Tokyo, Japan designed it. The face of the sundial has a diameter of 122 feet. What is the area of the face? about 11,684 ft2

13. Find the area of a 12-inch pizza. about 113 in2

14. The largest pizza ever baked was 21 feet across. What was itsarea? about 346 ft2

15. A California earthquake in 1989 sent horizontal shock wavesabout 60 miles from its epicenter. Find the area affected bythe earthquake. about 11,304 mi2

16. The stage of a theater is a semicircle. If the radius of the stageis 28 feet, what is the area of the stage? about 1231 ft2

APPL ICAT IONS

24 feet9 kilometers16 kilometers

3 centimeters

16 feet14 meters

SKILL

44


Name ______________________________________ Date ___________

Area of RectanglesArea is the number of square units needed to cover a surface. The area ofa rectangle is the product of its length (�) and its width (w).

A � �w

Find the area of the rectangle at the right.

A � �wA � 9 � 4A � 36

The area of the rectangle is 36 square meters.

Find the area of each rectangle.

1. 2. 3.

36 ft2 15 cm2 150 yd2

4. 5. 6.

54 m2 21 in2 360 cm2

7. 8. 9.

65 in2 18 m2 9 ft23 ft

3 ft

2 m

9 m

13 in.

5 in.

20 cm

18 cm7 in.

3 in.

9 m

6 m

15 yd

10 yd

5 cm

3 cm12 ft

3 ft

EXERCISES

9 m

4 m

EXAMPLE


Find the area of each playing field.

10. volleyball court 11. polo field

1,800 ft2 48,000 yd2

12. four-wall handball court 13. squash court

800 ft2 592 ft2

The maximum and minimum sizes of a soccer field are given at the right. Use this information to answerExercises 14–16.

14. What is the maximum area of a soccer field?81,000 ft2

15. What is the minimum area of a soccer field?64,350 ft2

16. What is the difference between the maximum area of a soccerfield and the minimum area of a soccer field?16,650 ft2

17. Henry wants to carpet a rectangular room that is 6 yards by 5 yards. If the carpet costs $29.50 a square yard, how muchwill it cost to carpet the room?$885.00

Soccer Field Size

Maximum 225 ft by 360 ftMinimum 195 ft by 330 ft

18' 6"32'

20'

20'40'

12'

300 YD160 Y

DCENTERO

F FIELD

60 YD MARK

40 YD MARK

30 YD MARK

60 YD MARK

40 YD MARK

30 YD MARK

MARKS ONGUARD BOARDS

MARKS ONGUARD BOARDS

15 YD

5 YD

GOAL PO

STS

GOAL LINE

TU

RF

60'7' 7'

30'

8'

APPL ICAT IONS

SKILL

45


Name ______________________________________ Date ___________

Area of Triangles and TrapezoidsA triangle is a polygon that has three sides. The area of a triangle is equal toone-half the product of its base and height.

Find the area of the triangle shown at the right.

A � bh

A � � 12 � 5

A � 30

The area of the triangle is 30 square meters.

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The area ofa trapezoid is equal to the product of half the height and the sum of the bases.

Find the area of the trapezoidshown at the right.

A � h(a � b)

A � (8)(15 � 12)

A � (8)(27)

A � 108

The area of the trapezoid is 108 square centimeters.

Find the area of each triangle or trapezoid.

1. 2. 3.

6 m2 18 cm2 240 in224 in.

20 in.7 cm

3 cm

5 cm

3 m

4 m

EXERCISES

1�2

1�2

1�2

15 cm

12 cm

8 cm

EXAMPLE

1�2

1�2 12 m

5 m

EXAMPLE


4. 5. 6.

240 ft2 160 mm2 20 m2

7. 8. 9.

5.92 cm2 91 yd2 20 in2

10. 11. 12.

84 ft2 67.5 cm2 20.615 m2

13. A rose garden is in the shape of a trapezoid. The bases of the trapezoid are 4 meters and 5 meters long, and the height of the trapezoid is 2 meters. Each rose plant needs 0.5 square meters of space. How many roses can be planted in the garden? 18 plants

14. The shape of the state of Delaware resembles a triangle with the base of 39 miles and a height of 96 miles. Find the approximate area of Delaware. about 1,872 mi2

15. The shape of the state of Wyoming is approximately a trapezoid with bases of 362 miles and 349 miles and height of 275 miles. Find the approximate area of Wyoming. about 97,762.5 mi2

16. About how many times larger is Wyoming than Delaware? about 52 times

17. A wastebasket has four congruent sides that are in the shape of trapezoids. If the bases of each trapezoid are 8 inches and 14 inches long and the height of each trapezoid is 15 inches, what is the area of the sides of the wastebasket? 660 in2

APPL ICAT IONS

8.6 m

3.1 m

4.7 m

15 cm

9 cm7 ft

8 ft

16 ft

8 in.

5 in.

8 yd

18 yd

7 yd3.2 cm3.7 cm

6 m

4 m

4 m40 mm

8 mm

12 ft

15 ft

20 ft

SKILL

46


Name ______________________________________ Date ___________

Area of Irregular ShapesOne way to estimate the area of an irregular figure is to find the mean ofthe inner measure and the outer measure of the figure.

Shaun had his friend draw the outline of his bodyon a piece of paper. The diagram at the rightshows this outline on a piece of grid paper inwhich each square represents 25 square inches.Estimate the area of his body shape.

inner measure: 8 � 25 � 200 square inches

outer measure: 49 � 25 � 1,225 square inches

mean: � 712.5 square inches

An estimate of the area of Shaun’s body is 712.5 square inches.

Estimate the area of each figure.

1. 2. 3.

� 20.5 units2 � 27.5 units2 � 12 units2

4. 5. 6.

� 15.5 units2 � 26.5 units2 � 12 units25 � 19�2

14 � 39�2

7 � 24�2

5 � 19�2

17 � 38�2

11 � 30�2

EXERCISES

200 � 1,225��

2

EXAMPLE


7. Estimate the area of the two leaves below.

19.5 units2 20 units2

8. Refer to the following grids of letters.

a. Which letter appears to have the greatest area? See students’ work.b. Which letter appears to have the least area? See students’ work.c. Estimate, in order, the areas of the letters from greatest to

least. s � 13 units2, h � 10 units2, f � 9.5 units2

9. Use the grid at the right to answer the following:a. Estimate the area of the figure using the

mean of the inner and outer measure.

� 49.5 units2

b. Draw a line from A to C on the grid and findthe actual area of the figure using the triangle and trapezoid formulas for area.49.5 units2

c. Was the estimate greater or less than the actual area and why?It was the same because it is not an irregular-shaped figure.

10. Keith had a countertop custom-made. He designed it so that it would not be more than 60 square feet. The company altered one of the measurements by mistake. The grid at the right shows the new size of the countertop. Is it still under the designed area?

yes: � 58 ft250 � 66�2

39 � 60�2

A

B

C

E D

APPL ICAT IONS

a. b.

SKILL

47


Name ______________________________________ Date ___________

Surface Area of RectangularPrisms

The surface area of a prism is the sum of the areas of all of the faces of the prism. The surface area of a rectangular prism can be found using theformula A � 2(�h � �w � wh).

Find the surface area of the math book shown at the right.

� � 11 in., w � 9.25 in., h � 1.5 in.

A � 2(11 � 1.5 � 11 � 9.25 � 9.25 � 1.5)A � 2(16.5 � 101.75 � 13.875)A � 2(132.125)A � 264.25

The surface area of the math book is 264.25 square inches.

Find the surface area of each prism shown ordescribed below. Round answers to the nearesttenth.

1. 2. 3.

606 in2 410.5 cm2 143 cm2

4. length, 15 m 5. length, 9.6 cm 6. length, 100 in.width, 12 m width, 7.5 cm width, 100 in.height, 9 m height, 7.7 cm height, 50 in.846 m2 407.3 cm2 40,000 in2

EXERCISES

EXAMPLE

11 in.

9.25 in.1.5 in.

7 in.

15 in.9 in.

10 cm

6 cm1–2

8 cm1–2

8.5 cm

3 cm4 cm


7. Each face of a cube has an area of 12 square inches. What isthe surface area of the cube?72 in2

8. A cube has a surface area of 108 square feet. What is the areaof one face?18 ft2

9. The surface area of a cube is 486 square inches. What is thelength of one side of the cube?9 in.

10. Jesse is making enclosed storage cubes for his room. The sides

of the cubes will each be 1 feet. He has three 32-square-foot

sheets of plywood. How many storage containers can hemake?5 storage cubes

11. Twenty-seven cubes are used to make a large cube that isthree cubes long by three cubes wide by three cubes high. Theoutside of the large cube is painted red. How many of thesmall cubes will be red on one side only?6 cubes

12. Mr. Thomas’ eighth grade class is working on a service project.This project consists of painting the walls and ceiling of asenior citizens’ activity room. This room is 12 feet long, 16 feetwide, and 9 feet high. There are two windows that are 3 feetby 5 feet each and a door that is 2 feet by 6.5 feet in thisroom.

a. How much area will they have to paint?653 ft2

b. How many gallons of paint will be needed if a gallon ofpaint covers 400 square feet?2 gallons

c. How much will it cost to buy the paint if each gallon costs$17.99 and the sales tax is 6.5%?$38.32

3�4

APPL ICAT IONS

SKILL

48


Name ______________________________________ Date ___________

Volume of Rectangular Prisms

The volume (V) of a rectangular prism is found by multiplying the length (�), the width (w), and the height (h).

V � �wh

Nicholas has been working with his dadin the evenings and on weekends in hisdad’s repair shop. For Nicholas’ birthday,his dad bought him a new toolbox andsome of the starting tools he wouldneed. What is the volume of Nicholas’toolbox if it is 18 inches long, 8 inches tall, and 7.5 inches deep?

V � �whV � 18 � 7.5 � 8V � 1,080 The volume of the toolbox is 1,080 cubic inches.

Find the volume of each rectangular prism shownor described below. Round decimal answers to thenearest tenth.

1. 2. 3.

936 ft3 116.1 in3 69.3 mm3

4. 5. 6.

EXERCISES

EXAMPLE

cube:side, 9.2 cm778.7 cm3

length, 4 feet

width, 3 feet

height, 5 feet

81 ft31�4

3�4

1�3length, 14 meters

width, 23 metersheight, 18 meters5,796 m3

9 ft

8 ft13 ft

3.8 in.6.5 in.

4.7 in.

9.7 mm

3.4 mm2.1 mm


7. Draw and label a rectangular prism whose length is 6 centimeters, width is 4 centimeters, and height is 10 centimeters. Find its volume.See students’ work.; 240 cm3

8. How many different rectangular prisms can be formed with 18 cubes?4

9. The surface area of a cube is 486 square inches. What is thevolume of the cube?729 in3

10. A cube has a volume of 1,000 cubic inches. What is the surfacearea of the cube?600 in2

11. What is the height of a rectangular prism if the volume is2,112 cubic yards, the length is 48 feet, and the width is 36 feet?33 ft or 11 yd

12. A rectangular prism has a volume of 36 cubic centimeters.Make a list showing all the possible whole-number dimensionsof the prism.1�1�36; 1�2�18; 1�3�12; 1�4�9; 1�6�6; 2�2�9; 2�3�6;3�3�4

13. A bar of soap has the dimensions 2 � 4 � 1.5 inches. A bathtub has the inside dimensions of 21 � 50 � 15 inches.How many bars of soap would it take to fill the bathtub?1,312.5 bars

14. An aquarium is 3 feet long and 1 feet wide. It is filled with

water to a height of 1 foot. How many gallons of water are inthe aquarium? (Hint: 1 cubic foot � 7.5 gallons.)about 33.75 gal

1�2

APPL ICAT IONS

SKILL

49


Name ______________________________________ Date ___________

Using Samples to PredictIn a community of 35,000 people, 45 out of 100 randomly selectedpeople responded that they prefer vanilla ice cream to chocolate. Howmany people in the community can be expected to prefer vanilla?

The ratio, 45 out of 100, is 45%.

To answer the question, find 45% of 35,000.

45% of 35,000 � 0.45 � 35,000� 15,750

You can predict that about 15,750 people in the town prefer vanilla ice cream.

Use the sample information to answer each question.

1. Two hundred people from a town of 28,000 people were chosen at random and asked if the town needed more bicyclepaths. Seventy-eight of those surveyed said yes. How many people in the town can be expected to think that the townneeds more bicycle paths?about 10,920 people

2. Mr. Tata surveyed his class about their favorite foods. Ten of the30 students surveyed said their favorite food was pizza. Howmany students out of the 250 students in the school would youexpect to like pizza best?about 83 students

EXERCISES

EXAMPLE


3. In a recent survey of radio listeners, 125 out of the 500 peoplesurveyed said they actually listen to the commercials. How many people out of 10,000 would you expect to listen to thecommercials?about 2,500 people

4. Six out of 20 families surveyed said they own a video camera.How many families out of 150 would you expect to own a videocamera?about 45 families

Of the TV households surveyed by theNielsen Media Research Company, the top 5television programs of 1992–1993 are listedin the table below.

5. How many households in a town with 40,000 households wouldyou expect to have watched 60 Minutes?about 8,760 households

6. How many households in a town with 80,000 households wouldyou expect to have watched the show in fourth place?about 14,320 households

7. How many households in a town with 100,000 householdswould you expect did not watch Home Improvement?about 80,800 households

8. Suppose that 2,124 of the households responding to this surveywatched Murder, She Wrote. How many households were surveyed?about 12,000 households

APPL ICAT IONS

Show % of TV Households

1. 60 Minutes 21.9

2. Roseanne 20.7

3. Home Improvement 19.2

4. Murphy Brown 17.9

5. Murder, She Wrote 17.7

SKILL

50


Name ______________________________________ Date ___________

Mean, Median, ModeYou can analyze a set of data by using three measures of center: mean,median, and mode.

Hakeem Olajuwon, 1994’s Most Valuable Player in the NationalBasketball Association, helped the Houston Rockets win the NBAchampionship. In winning the 7-game series, Olajuwon scored 28, 25,21, 32, 27, 30, and 25 points. Find the mean, median, and mode of hisscores.

Mean: � 26.857

The mean is about 27 points.

Median: 21, 25, 25, 27, 28, 30, 32↑

median

The median is 27.

Mode: The mode is 25 since it is the number that appears the most times.

Find the mean, median, and mode for each set of data.

1. 5, 4, 7, 2, 2, 1, 4, 3 mean � 3.5; median � 3.5; mode � 2 and 4

2. 25, 18, 14, 27, 25, 16, 18, 25 mean � 21; median � 21.5; mode � 25

3. 13, 11, 7, 9, 12, 5 mean � 9.5; median � 10; mode � none

4. 234, 163, 634, 267, 545, 874 mean � 452.8; median � 406; mode � none

5. 23, 36, 48, 95, 36, 28, 24 mean � 41.4; median � 36; mode � 36

6. 299, 100, 237, 492, 333, 263, 295 mean � 288.4; median � 295; mode � none

EXERCISES

28 � 25� 21 � 32 � 27 � 30 � 25��

7

EXAMPLE


7. 2,500, 2,366, 1,939, 1,933, 1,835, 2,498, 2,943 mean � 2,287.7; median � 2,366; mode � none

8. 9, 2, 5, 7, 8, 9, 4, 4, 6, 4 mean � 5.8; median � 5.5; mode � 4

9. 29, 48, 20, 43, 33, 20, 40, 69, 48 mean � 38.9; median � 40; mode � 20 and 48

10. 7,899, 4,395, 9,090, 9,588, 4,880, 9,587, 4,756 mean � 7,170.7; median � 7,899; mode � none

The data at the rightshows the record high temperatures forseveral states in theU.S. Use the data toanswer Exercises 11–15.

11. What is the mode? 112°F12. What is the median? 112°F13. What is the mean? 110.5°F14. If each of the high temperatures increased by 1°F, would it change

a. the mode? Why or why not? Yes, the mode would now be 113°F,because the numbers that occur most often increased.b. the median? Why or why not? Yes, the median would now be113°F, because the middle numbers both increased.c. the mean? Why or why not? Yes, the mean would now be 111.5°F,because all of the numbers increased, but they were still divided by 6 to find the mean.

15. If the high temperature for Vermont increased to 112°F, wouldit changea. the mode? Why or why not? No, 112°F would still be the numberoccurring most often.b. the median? Why or why not? No, 112°F is still the middle number.c. the mean? Why or why not? Yes, the sum of the numbers will begreater but will still be divided by the same number.

16. Find the hand spans of ten people. Ask each person to spreadapart the little finger and thumb of his or her right hand asfar as possible. Then measure and record the distance from tipto tip to the nearest centimeter. Find the mean, median, andmode for the data you collected. Answers will vary. See students’ work.

APPL ICAT IONS State Record High Temperature (°F)

Alabama 112

Alaska 100

Michigan 112

Oklahoma 120

Vermont 105Wyoming 114

SKILL

51


Name ______________________________________ Date ___________

Make a ListPat’s Pizza offers 7 different toppings: pepperoni, sausage, bacon, green peppers, onions, mushrooms, and anchovies. The Davis family wants to order a 3-topping pizza. Tommy Davis does not like anchovies.

How many different pizzas can the Davis family order if they want to satisfy all members of the family?

Let P � pepperoni, S � sausage, B � bacon, G � green peppers, O � onions, M � mushrooms, and A � anchovies. List the possible combinations that do not include anchovies.

PSB PSG PSO PSMPBO PBM PGO PGMSBG SBO SBM SGOSOM BGO BGM BOMPBG POM SGM GOM

There are 20 different pizzas the Davis family can order.

Solve by making a list.

1. How many different ways can a triangle, a square, and a circlebe arranged in a row?6 ways

2. How many different four-digit numbers can be formed from thenumbers 4, 5, 6, and 7 if all the digits must be different?24 numbers

3. How many different three-digit numbers can be formed from the numbers 4, 5, 6, and 7 if all the digits must be different?24 numbers

EXERCISES

EXAMPLE


4. How many different two-digit numbers can be formed from thenumbers 4, 5, 6, and 7 if both the digits must be different?12 numbers

5. How many numbers between 77 and 103 are divisible by 3?9 numbers

6. A vendor at a rock concert sells T-shirts in three colors: red,blue, and yellow. The T-shirts come in 4 sizes: small, medium,large, and extra large. How many different T-shirts are available to the customers?12 T-shirts

7. Four chairs are placed in a row on the stage. The three candidates for class president, Adrian, Toni, and Miwa, are seated on the stage. How many different ways can the candidates seat themselves?24 ways

8. Leslie wants to take a picture of her four dogs. She has a beagle, a terrier, a collie, and a poodle. How many ways canshe arrange her dogs in a row if the beagle and terrier mustbe next to each other?12 ways

9. Using only dimes and nickels, how many different ways can aclerk make change for a dollar?11 ways

10. Earl attends a convention every three years. The year 1992 wasa leap year, and Earl attended a convention. What is the nextleap year that Earl will be attending a convention?2004

APPL ICAT IONS

SKILL

52


Name ______________________________________ Date ___________

Probability of Independent EventsThe probability of an event is the ratio of the number of ways an event canoccur to the number of possible outcomes.

Probability of an event �

Suppose you spin the two spinners. What is the probability that the sum of the numbers showing on the two spinners will be 4?

Make a tree diagram to show all possible outcomes of these events.

There are 3 outcomes that have a sum of 4 and there are 9 possible outcomes.

Probability of sum of 4 � or

The probability that the sum will be 4 is .

Use the spinners in the Example above to answerExercises 1–4.

1. What is the probability that the sum of the numbers showing on the two spinners is 3?

2�9

EXERCISES

1�3

1�3

3�9

EXAMPLE

number of ways the event can occur��

number of possible outcomes

1

3 2

1

3 2

First Spinner Second Spinner1

2

3

1

Sum2

3

4

1

2

3

2

3

4

5

1

2

3

3

4

5

6


2. What is the probability that the sum of the numbers showingon the two spinners is greater than 3?

3. What is the probability that the sum of the numbers showingon the two spinners is an even number?

4. What is the probability that the sum of the numbers showingon the two spinners is not a 5?

5. Make a tree diagram showing the possible outcomes of tossing a penny and a dime.

6. What is the probability that a tossed penny and a tossed dimewill both show heads?

7. What is the probability that a tossed penny and a tossed dimewill both show one head and one tail?

8. What is the probability that a tossed penny and a tossed dimewill show at least one tail?

Beau, Jiang, and Marci are playing a gamethat requires each player to toss two number cubes. Use this information toanswer Exercises 9–12.

9. Beau needs a sum of 4 on the number cubes to win. What isthe probability that Beau will toss a 4?

10. Jiang needs a sum of 9 on the number cubes to win. What isthe probability that Jiang will toss a 9?

11. Marci needs a sum of 7 on the number cubes to win. What isthe probability that Marci will toss a 7?

12. Who is most likely to win the game?

APPL ICAT IONS

3�4

1�2

1�4

7�9

5�9

2�3

Marci

1�6

1�9

1�12

SKILL

53


Name ______________________________________ Date ___________

Expected Value of an OutcomeMr. Eugene has four different colored markers in a cup on his desk.Each day he pulls a marker out of the cup at random. How often couldhe expect to use a given marker in 8 days? in 16 days? in 40 days?

The probability of choosing any one of the four different

colored markers is .

In 8 days, he could expect to use the given marker twice.

In 16 days, he could expect to use the given marker 4 times.

In 40 days, he could expect to use the given marker 10 times.

A number cube is rolled 12 times. How oftenwould you expect to get each of the followingoutcomes?

1. a 6 2. a 7twice never

3. a prime number 4. an even number6 times 6 times

5. a number greater than 2 6. a number less than 18 times never

7. a multiple of 1 8. a multiple of 412 times twice

EXERCISES

1�4

EXAMPLE


A coin is tossed 20 times. How often would you expect to get eachof the following outcomes?

9. a head 10. a tail10 times 10 times

11. a head or a tail 12. neither a head nor a tail20 times never

LeRoy has 15 different ties. He chooses a tieat random every day.

13. How many times could he expect to wear a given tie in 45 days?3 times

14. How many times could he expect to wear a given tie in 180 days?12 times

15. How many times could he expect to wear a given tie in a yearthat is not a leap year?about 24 times

16. Suppose LeRoy buys 5 more ties to add to his collection. How many times could he now expect to wear a given tie in 45 days? in 180 days? in a year that is not a leap year?about 2 times; 9 times; about 18 times

17. How many ties would LeRoy need to own in order to expectto wear each tie just 5 times in a year that is not a leap year?73 ties

APPL ICAT IONS

SKILL

54


Name ______________________________________ Date ___________

Theoretical and ExperimentalProbability

The theoretical probability of an event is the ratio of the number of ways theevent can occur to the number of possible outcomes.

The experimental probability of an event is the ratio of the number of successful trials to the number of trials.

Louis wants to determine the probability of getting a sum of 7 whenrolling a number cube. The sample space, or all the possible outcomes,for a roll of two number cubes is shown below.

1, 1 1, 2 1, 3 1, 4 1, 5 1, 62, 1 2, 2 2, 3 2, 4 2, 5 2, 63, 1 3, 2 3, 3 3, 4 3, 5 3, 64, 1 4, 2 4, 3 4, 4 4, 5 4, 65, 1 5, 2 5, 3 5, 4 5, 5 5, 66, 1 6, 2 6, 3 6, 4 6, 5 6, 6

What is the theoretical probability of rolling a sum of 7? What is theexperimental probability of rolling a sum of 7 if Louis rolls the numbercube 20 times and records 4 sums of 7?

There are 6 sums of 7 shown in the sample space above. So, the

theoretical probability of rolling a sum of 7 is or .

Since Louis rolled 4 sums of 7 on 20 rolls, the experimental probability

is or .

Find the theoretical probability of each of the following.

1. getting tails if you toss a coin

2. getting a 6 if you roll one number cube

3. getting a sum of 2 if you roll two number cubes1

�36

1�6

1�2

EXERCISES

1�5

4�20

1�6

6�36

EXAMPLE


4. getting a sum less than 6 if you roll two number cubes

or

5. Melissa rolled one number cube 30 times and got 8 sixes.

a. What is her experimental probability of getting a six?

or

b. What is her experimental probability of not getting a six?

or

6. Melissa rolled two number cubes 36 times and got 3 sums of 11.

a. What is her experimental probability of getting a sum of 11?

or

b. What is her experimental probability of not getting a sumof 11?

or

While playing a board game, Jerod rolled apair of number cubes 48 times and got doubles 10 times.

7. What was his experimental probability of rolling doubles?

or

8. How does his experimental probability compare to the theoretical probability of rolling doubles?The experimental probability is slightly greater.

9. How do you think the experimental probability compares tothe theoretical probability in most experiments? Explain.The experimental probability should be close to the theoreticalprobability.

10. Do you think the experimental probability is ever equal to thetheoretical probability? Explain.Sample answer: Yes, especially if many trials are used.

5�24

10�48

APPL ICAT IONS

11�12

33�36

1�12

3�36

11�15

22�30

4�15

8�30

5�18

10�36

SKILL

55


Name ______________________________________ Date ___________

Probability Using Area ModelsDetermine the probability that a randomly-dropped counter will fall in the shaded area.Each small square has an area of 1 squarefoot and the area of the shaded region isabout 30 square feet.

probability �

P � or

The probability that a counter will fall in the shaded area is .

Find the probability that a randomly-droppedcounter will fall in the shaded region.

1. 2. 3.

4. 5. 6.

4�13

1�5

5�27

7�22

32�81

25�64

EXERCISES

3�10

3�10

30�100

number of ways an event can occur��

number of possible outcomes

EXAMPLE


7. Draw a square, 4 units on a side, on a piece of grid paper.Shade in 12 squares. What is the probability that a randomly-dropped counter will fall in the shaded area?

8. Draw a square, 6 units on a side, on a piece of grid paper.Shade in 14 squares. What is the probability that a randomly-dropped counter will fall in the shaded area?

9. Find the probability that a golf ball will land on the green.

10. Suppose you throw 100 darts at each dart boardbelow. Use the formula below to determine how many darts you would expect to land in the shaded regions.

�

11. Zoe lost her contact lens while playing basketball. Ifit is equally likely that she lost the lens on any partof the court, estimate the probability that Zoe losther lens inside the circle.

about 1�47

area of shape��area of region

number landing in shape��number landing in region

1�6

APPL ICAT IONS

7�18

3�4

80 yd

1,600square yards

120 yd

1

94 ft

6 ft

50 ft

a. b. c.

4 cm

SKILL

56


Name ______________________________________ Date ___________

Line PlotsThe table at the right shows the final results of theGreat Frog Competition in Dickerson County. Thefrog jumps were measured in inches.

Organize this information using a line plot.

The shortest jump was 33 inches, and the longest jump was 65 inches.Draw a number line that includes the numbers 33 to 65. Place an Xabove the number line to represent the distance of each jump.

Make a line plot for each set of data.

1. 71, 74, 73, 71, 72, 74, 71, 75, 77, 79, 74, 72, 74, 75

2. 81, 81, 83, 84, 83, 85, 86, 77, 70, 65, 65, 80, 85

EXERCISES

×

30 65

×

35 40 45 50 55 60

× ××× × × ×

×× ×

EXAMPLE

Jumps

Frogs 1 2 3

Slippery 61 51 60

Spots 46 38 39

Inky 56 33 61

Popper 65 51 52


3. 560, 790, 800, 850, 760, 810, 650, 850, 790, 690, 600

4. 1,750, 2,000, 1,900, 1,950, 1,900, 1,900, 1,900, 1,800, 2,100, 2,000, 1,800

5. 7.1, 7.7, 7.8, 8.2, 8.4, 7.5, 7.8, 8.0, 8.3, 8.2, 8.4, 7.6, 8.0

6. The breathing rates (breaths/minute) of twelve friends are listed below.13, 11, 13, 14, 10, 16, 12, 13, 15, 13, 11, 13

7. The lengths in centimeters of nine ladybugs are listed below.0.77, 0.72, 0.87, 0.82, 0.77, 0.79, 0.87, 0.87, 0.77

8. Ask your classmates to open up one of their hands as wide as possible and measure the span from the tip of the smallest finger to the tip of the thumb in cen-timeters. Record the distances and organize this information into a line plot.See students’ work.

APPL ICAT IONS

SKILL

57


Name ______________________________________ Date ___________

Stem-and-Leaf PlotsA stem-and-leaf plot is one way to organize a list of numbers. The stemsrepresent the greatest place value in the numbers. The leaves represent thenext place value.

Make a stem-and-leaf plot for the following numbers.

$0.89, $1.12, $0.92, $1.28, $1.25, $1.02,$1.13, $1.02, $1.01, $1.10, $1.14, $1.23

Since a stem-and-leaf plot is represented with just two digits, round the values to the nearest ten cents.

$0.90, $1.10, $0.90, $1.30, $1.30, $1.00,$1.10, $1.00, $1.00, $1.10, $1.10, $1.20

Use the dollar values for the stems and the ten-cent values for the leaves.

0 9 91 0 0 0 1 1 1 1 2 3 3 0 9 means $0.90.

Make a stem-and-leaf plot for each set of data.

1. 18, 67, 35, 20, 45, 2. 500, 610, 720, 830, 870,55, 69, 23, 34, 58, 880, 750, 630, 520, 500,61, 43, 56, 63, 29 540, 580, 890, 780, 880

1 8 5 0 0 2 4 82 0 3 9 6 1 33 4 5 7 2 5 84 3 5 8 3 7 8 8 95 5 6 8 5 0 means 500.6 1 3 7 9

1 8 means 18.

EXERCISES

EXAMPLE


3. 6, 10, 23, 35, 30, 13, 4. 61.6, 51.9, 60.0, 46.019, 33, 1, 22, 28, 35 38.7, 39.1, 56.2, 33.024, 23, 14, 4, 19, 23 61.5, 65.4, 51.0, 52.3

0 1 4 6 3 3 9 91 0 3 4 9 9 4 62 2 3 3 3 4 8 5 1 2 2 63 0 3 5 5 6 0 2 2 5

0 1 means 1. 3 3 means 33.

Make a stem-and-leaf plot for each set of data.

5. The high temperatures (°F) on a December day for sixteensouthwestern cities are 54°, 67°, 64°, 61°, 70°, 65°, 72°, 63°, 49°,58°, 60°, 48°, 68°, 77°, 69°, and 65°.

4 8 95 4 86 0 1 3 4 5 5 7 8 97 0 2 7

4 8 means 48°.

6. The commuting times for fifteen workers are 33, 23, 18, 30, 44,28, 34, 17, 38, 28, 30, 18, 29, 23, and 21 minutes.

1 7 8 82 1 3 3 8 8 93 0 0 3 4 84 4

1 7 means 17 min.

7. The number of stories in the fifteen tallest buildings of a cityare 28, 52, 39, 27, 40, 32, 32, 54, 41, 37, 33, 43, 48, 41, and 45.

2 7 83 2 2 3 7 94 0 1 1 3 5 85 2 4

2 7 means 27 stories.

8. Lori sells tickets at a movie theater. Her sales for week 1 were$173, $194, $160, $182, $183, and $247. Her sales for week 2 were $137, $182, $151, $193, $199, and $194.

1 4 5 6 7 8 8 8 9 9 92 0 5

1 4 means $140.

APPL ICAT IONS

SKILL

58


Name ______________________________________ Date ___________

Line GraphsA line graph is usually used to show the change and direction of changeover time. All line graphs should have a graph title, a vertical-axis label, and ahorizontal-axis label.

Make a line graph for the data on the number of space flights carrying people during the 1960’s.

Make a line graph for each set of data.

1.

EXERCISES

EXAMPLE

Space Flights Carrying People

Year Number

1961 41962 51963 31964 11965 61966 51967 11968 31969 9

Sid’s Daily Jogging Timefor Three Miles

Time in Day Minutes

1 322 293 284 265 286 337 27

Space Flights Carrying People

Num

ber

of F

light

s

2

4

6

8

Year19

6119

6219

6319

6419

6519

6619

6719

6819

69

Sid’s Daily Jogging Timefor Three Miles

Tim

e in

Min

utes

Day1 2 3 4 5 6 7

0

5

10

15

20

25

30

35


Make a line graph for each set of data.

2.

3.

4.

5.

APPL ICAT IONS

Traffic on Maple Drive

Number of Day Vehicles

Monday 7,200Tuesday 8,050Wednesday 10,500Thursday 5,900Friday 9,990Saturday 3,400Sunday 900

Recorded Number of Hurricanes

Month Number

June 23July 36August 149September 188October 95November 21

Evans Family Electric Bill

Month Amount

March $129.90April $112.20May $105.00June $88.50

Home Runs by Hank Aaron1967 to 1976

Year Number

1967 391968 291969 441970 381971 471972 341973 401974 201975 121976 10

SKILL

59


Name ______________________________________ Date ___________

Bar Graphs

Bar graphs are used to compare numbers. All bar graphs should have agraph title, a vertical-axis label, and a horizontal-axis label.

Make a bar graph for the data on women’s NCAA gymnastics championships between 1982 and 1993.

Make a tally.

UtahGeorgiaAlabama

Make a bar graph.

Make a bar graph for each set of data.

1.

EXERCISES

EXAMPLE

NCAA Women’s Gymnastics

Year Champion

1982 Utah1983 Utah1984 Utah1985 Utah1986 Utah1987 Georgia1988 Alabama1989 Georgia1990 Utah1991 Alabama1992 Utah1993 Georgia

Preference for Brands

Brand Number of Students

A 15B 35C 30D 25

Utah Georgia Alabama

Colleges

Num

ber

of C

ham

pion

ship

s

NCAA Women’s Gymnastics1982–1993

0

1

2

3

4

5

6

7

8

9


2.

3.

4. Survey the students in your math class to find out their favoritemovie. Use this data to make a bar graph.See students’ work.

5. Survey your friends to find out their favorite television show.Use this data to make a bar graph.See students’ work.

APPL ICAT IONS

NCAA Women’s Cross Country

Year Champion

1981 Virginia1982 Virginia1983 Oregon1984 Wisconsin1985 Wisconsin1986 Texas1987 Oregon1988 Kentucky1989 Villanova1990 Villanova1991 Villanova1992 Villanova

NCAA Women’s Volleyball

Year Champion

1981 Southern California

1982 Hawaii1983 Hawaii1984 UCLA1985 Pacific1986 Pacific1987 Hawaii1988 Texas1989 California State,

Long Beach1990 UCLA1991 UCLA1992 Stanford

SKILL

60


Name ______________________________________ Date ___________

Circle GraphsThe air surrounding Earth is referred to as the atmosphere. Without airthere would be no life on Earth. Air is a mixture of gases. By volume, dry airis composed of 78% nitrogen, 21% oxygen, and 1% other gases.

Make a circle graph to show the composition of the Earth’s atmosphere.

To make a circle graph, first find the number of degrees that correspond to each percent. Use a calculator and round to the nearest degree.

Nitrogen: 78% of 360° � 281°Oxygen: 21% of 360° � 76°Other: 1% of 360° � 4°

Use a compass and a protractor to draw the circle graph.

Note that the sum of the degrees is not 360°because of rounding.

Make a circle graph to show the data in eachchart.

1.

EXERCISES

EXAMPLE

Earth’s Atmosphere

Other1%

Nitrogen 78%

Oxygen21%

Favorite TV Shows

Movies 12%Sports 20%News 4%Drama 16%Comedy 20%Music 28%


2.

Make a circle graph to show the data ineach chart.

3.

4.

5.

6. Make a circle graph showing how you spent your time lastSaturday.See students’ work.

APPL ICAT IONS

Daily Activities

Sleeping 8 hoursEating 1 hourSchool 6 hoursHomework 3 hoursTeam practice 2 hoursMiscellaneous 4 hours

Area of Continents

Area in MillionsContinent of Square Miles

Europe 3.8Asia 17.4North America 9.4South America 6.9Africa 11.7Oceania 3.3Antarctica 5.4

World Cup Winners

Country Number of Wins

Argentina 2Brazil 4England 1Italy 3Uruguay 2West Germany 3

Area of New England States

Area inState Square Miles

Maine 33,215New Hampshire 9,304Vermont 9,609Massachusetts 8,257Connecticut 5,009Rhode Island 1,214

SKILL

61


Name ______________________________________ Date ___________

Scatter PlotsA scatter plot shows the relationship, if any, between two values.

Positive Relationship Negative Relationship No Relationship

The pattern of dots The pattern of dots The dots are spreadslants upward to slants downward to out. There is nothe right. the right. pattern.

The Marysville Garden Club sells stationery each year. The scatter plot at the right relates the length a personhas belonged to the club to the numberof boxes sold. What does the point with the box around it represent?

It represents a person who has belonged to the club for 3 years and sold 20 boxes of stationery.

What type of relationship is shown in the Stationery Sales scatter plot?

The pattern of dots slants upward to the right, so the relationship ispositive.

20 4 6 8 10Years in Club

20

40

Box

es S

old

Stationery SalesEXAMPLES

0 6 7 8 9 10Shoe Size

5

12345

Num

ber

of P

ets

0 10 20 50Age

1

5432

New

Cav

ities

1 2 3 4 5

252015105

0Time

Dis

tanc

e


Determine whether a scatter plot of the datawould show a positive relationship, a negativerelationship, or no relationship.

1. number of minutes a candle burns and the candle’s heightnegative relationship

2. length of a taxi ride and the amount of the farepositive relationship

3. number of letters in a person’s first name and height of a person in centimetersno relationship

The scatter plot at the right shows the relationship between age and physical activity. Use this information to answer Exercises 4–6.

4. How many people are represented on the plot? 14 people

5. What happens to the number of hours of physical activity as people grow older? It decreases.

6. What relationship (positive, negative, or none) does this data show beween physical activity and age? negative

The scatter plot at the right shows the relationshipbetween a preschool child’s English vocabulary and their age. Use this information to answer Exercises 7–8.

7. Make a general statement about the scatter plot.As the age increases, the vocabularyincreases.

8. What are some probable causes of the data that do not seem to fit? Possible answers: speakers of another language, developmentally disabled

9. Measure the heights of some of your friends. Then measure the circumference of their heads. Make a scatter plot that relates height with circumference of the head. See students’ work.

0 3 6 9 12 15 18 21 24 27 30

12345

Words Used (100’s)

Age

Vocabulary ofPreschool Children

0 2 4 6 8 10 12 14 16 18 20 22Age in Years

2468

101214

Hou

rs o

f Phy

sica

l Act

ivity

Relationship of PhysicalActivity and Age

APPL ICAT IONS

EXERCISES

SKILL

62


Name ______________________________________ Date ___________

Constructing and Interpreting GraphsThe chart at the right shows scores in theSpringboard Diving event in the SummerOlympic Games.

Construct and interpret a graph of the data.

To graph the data, first label the axes and graph the points named by the data. Thenconnect the points as shown in the graph at the right.

The graph shows that the scores generallytended to increase with each successiveOlympic game.

Construct and interpret a graph of each set of data.

1.

The speed generally tends to increase over time.

EXERCISES

Year

Summer OlympicsSpringboard Diving Scores

Scor

e

0

100

200

300

400

500

600

700

800

1964

1968

1972

1976

1980

1984

1988

1992

EXAMPLE

Summer Olympics Springboard DivingYear Score1964 145.001968 150.771972 450.031976 506.191980 725.911984 530.701988 580.231992 572.40

Time (seconds) Speed (mph)5 510 815 2020 1825 3030 4035 55


2.

The speed fluctuates withthe distance.

The chart at the rightlists the winning timesfor the men’s 110-meterhurdles at the SummerOlympic Games. Use the data to answerExercises 3–6.

3. Construct a graph of the data.See students’ work.

4. Interpret the graph of the data. The time generally tends to decrease with each successiveOlympic game.

5. Why do you think the times do not always show a consistentpattern? Many factors can affect the time, such as weather conditions,health of the competitor, and so on.

6. What would you predict the time for this event to be in thenext Olympic games? Explain why you chose this time. Answers may vary.

7. Suppose you are driving down a street that has many trafficlights. What do you think a graph of your time versus yourspeed would look like? Why? Sketch your graph. Answers may vary.

APPL ICAT IONS

Distance (feet) Speed (mph)40 1580 28120 42160 60200 46240 37280 55

Year Time (seconds)

1964 13.61968 13.31972 13.241976 13.301980 13.201984 13.201988 12.981992 13.12

SKILL

63


Name ______________________________________ Date ___________

Make a GraphAll graphs should have a graph title. A bar graph should also have a vertical-axis label and a horizontal-axis label.

Make a bar graph to show theduration of the five spaceflights listed in the table below.


1.

2.

EXERCISES

EXAMPLE

Spaceflights Carrying People July, 1992 to December, 1992

Spacecraft Days in Space

Soyuz TM-15 15Atlantis 8Endeavour 8Columbia 10Discovery 7

Soyuz

TM-1

5

Day

s in

Spa

ce

Spacecraft

Spaceflights Carrying People,July, 1992 to December, 1992

4

8

12

16

Atlant

is

Endea

vour

Colum

bia

Discov

ery

Class President Election Results

Name Number of Votes

Joan 18Ron 11Ramona 15Chi Wan 9

Cases of Shampoo Sold

Brand Number of Cases

X 35Y 20Z 15


3.


4.

5.

6.

APPL ICAT IONS

Wins of Basketball Teams

Team Number of Wins

Bears 7Hawks 6Tigers 3Wildcats 9

Days Exceeding Carbon Dioxide Standards in 1990

Metropolitan Area Number of Days

Los Angeles 47Spokane 13Las Vegas 17Sacramento 11Anchorage 12

Silver Medals Won at 1992 Summer Olympics

Number of Country Silver Medals

China 22Germany 21Hungary 12Unified Team 38United States 34

Average Rating for TV Shows 1990–1993

Show Rating

Roseanne 19.7%Murder, She Wrote 17.0%Full House 16.3%America’s Funniest

Home Videos 14.6%Fresh Prince of Bel Air 14.1%

SKILL

64


Name ______________________________________ Date ___________

Interpreting GraphsThe graph below shows the height of a baby girl at certain ages.

Describe the shape of the graph. Then use the graph to predict whatwill happen to the baby’s height over the next year. What do you predict her height will be after 2 years?

The graph increases rapidly at first and then slows down. Over the next year, the baby’s height will probably continue to increase, but at a slower rate of growth. After 2 years, the baby will probably be around 32 inches tall.

Describe the shape of each graph. Then use thegraph to answer each question.

1. What do you think will happen tothe height of the plant over the next2 weeks? What do you predict theheight will be after 48 days? The graph rises rapidly at first and thenslows down. Over the next two weeks, theplant will continue to grow but the rate ofgrowth will be slower. Predictions for theheight after 48 days will vary.

Plant Growth

Heig

ht (i

nche

s)

0 4 8 12 16 20 24 28

2468

10121416

Day

EXERCISES

EXAMPLE

Age (months)

Female Growth Rate

Heig

ht (i

nche

s)

2 4 6 8 10 120

20

22

24

26

28

30


2. What do you think will happen tothe income over the next 5 years?What do you predict the income willbe in 1999? The graph rises at a steady rate of about$1,200 a year. The income will continue torise over the next 5 years at about the samerate. The income in 1999 will probably beabout $37,000.

3. What do you think will happen tothe weight of the baby over the nextyear? What do you predict his weightwill be after 2 years? The graph rises rapidly at first and thenslows down. Over the next year his weightwill probably continue to rise but at a slower rate. Predictions of his weight after2 years will vary.

4. What do you think will happen to the attendance over the next 5 years? What do you predict theattendance will be in 1999? The graph rises at a steady rate of about200 a year and then rises rapidly. The attendance will continue to rise over thenext 5 years at about the same rate.Predictions of the attendance in 1999 willvary.

The graph below shows the public debt from the years 1970 to 1992.

5. Describe the shape of the graph. It increases gradu-ally at first and then starts to increase rapidly.

6. Use the graph to predict what will happen to the publicdebt over the next 10 years. It will probably continue to increase rapidly.

7. According to this graph, what do you think the public debtwill be in 2002? Answers will vary.

8. What do you think could happen to cause your prediction tonot hold true? Answers will vary.

Year

Public Debt, 1970-1992

Amou

nt o

f Deb

t (bi

llion

s)

1970

1975

1980

1985

1990

1995

0500

1,0001,5002,0002,5003,0003,5004,0004,500

APPL ICAT IONS

Year

Conference Attendance

Num

ber o

f Peo

ple

1988

1989

1990

1991

1992

1993

1994

0

200

600

1,000

1,400

1,800

2,200

Age (months)

Male Growth Rate

Wei

ght (

poun

ds)

0 2 4 6 8 10 12

4

8

12

16

20

24

Year

Kirk’s Family Income

Inco

me

(thou

sand

s of

dol

lars

)

1988

1989

1990

1991

1992

1993

1994

022242628303234

SKILL

65


Name ______________________________________ Date ___________

Standard DeviationThe heights of a group of young pine trees in a reforestation plot are58 cm, 56 cm, 51 cm, 54 cm, 49 cm, 61 cm, 54 cm, and 49 cm. Find thestandard deviation.

Follow the steps below to find the standard deviation. You can use acalculator to help you.

1. Find the mean. The meanof this data is 54.

2. Subtract each measurement from themean.

3. Square each difference.

4. Find the mean of thesquares. The mean of the

squares is � 16.

5. Find the positive squareroot of this mean. The square root of 16 is 4.

The standard deviation is 4.

Find the standard deviation for each set of data.

1. 1, 4, 11, 7, 2 3.6

2. 39, 47, 51, 38, 45, 29, 37, 40, 36, 48 6.3

3. 250, 275, 325, 300, 200, 225, 175 50

4. 10, 20, 30, 40, 50 14.1

EXERCISES

128�

8

EXAMPLE

Height Difference Square

58 4 16

56 2 4

51 –3 9

54 0 0

49 –5 25

61 7 49

54 0 0

49 –5 25


The number of millions of persons viewingprime time television each day of the weekis given below.

101.5, 96.7, 98.8, 96.6, 88.0, 88.4, 109.6

Use this data to answer Exercises 5–8.

5. Find the mean of the data. about 97.1

6. Find the standard deviation of the data. about 6.9

7. If each piece of data increased by 1, would it change a. the mean?

Yes, the mean increases by 1.

b. the standard deviation? No, the standard deviation will remain the same.

8. If just one piece of data decreased by 5, would it change a. the mean?

Yes, the sum of the numbers will be less and the mean willbe less.

b. the standard deviation? Yes, the differences will all be different.

9. Find the length of the arms of ten people from their elbow totheir wrist. Measure and record each length to the nearest centimeter. Find the mean and the standard deviation for thedata you collected. Answers will vary. See students’ work.

APPL ICAT IONS

SKILL

66


Name ______________________________________ Date ___________

Predicting Distribution of DataThe lifetimes of 10,000 light bulbs are normally distributed. The meanlifetime is 300 days, and the standard deviation is 40 days. How manylight bulbs will last more than 380 days?

Since the lifetimes of the light bulbs are normally distributed, the distribution of the data is shown by the graph below.

The graph shows that 2.35% of the lightbulbs will last more than 380 days.

2.35% of 10,000 � 0.0235 � 10,000� 235

So, 235 lightbulbs will last more than 380 days.

The frequencies of 50,000 scores on a collegeentrance examination are normally distributed.The mean score is 510, and the standard deviationis 80.

1. How many scores are between 510 and 590? 17,000


3. How many scores are less than 350? 1,175

EXERCISES

220 260 300 380340

2.35%2.35%13.5% 13.5%34%34%

EXAMPLE


4. The number of scores above 590 is the same as the number ofscores below what score? 430


6. How many scores are greater than 670 or less than 350? 2,350

7. Between what two scores do 68% of the scores fall? 430 and 590

The weights of 150 oranges picked in a citrus grove are normally distributed. The mean weight is 7.5 ounces, and thestandard deviation is 2.1 ounces. Use thisinformation to answer Exercises 8–10.

8. How many oranges weigh between 5.4 ounces and 9.7 ounces? 102

9. What percent of the oranges would you expect to weigh lessthan 3.3 ounces? 2.35%

10. Would you expect any of the oranges to weigh a. less than 1 ounce? Why or why not?

No, 99.7% of the oranges will have a weight greater than orequal to 1.2 ounces and 0.997 � 150 � 149.55.

b. more than 14 ounces? Why or why not? No, 99.7% of the oranges will have a weight less than orequal to 13.8 ounces and 0.997 � 150 � 149.55.

11. Ask all of the students in your class for their height in inches.Record each height. Find the mean and the standard deviationof the data. Draw a graph of the data. Does it appear to benormally distributed? Why or why not? Answers will vary. See students’ work.

APPL ICAT IONS

SKILL

67


Name ______________________________________ Date ___________

Arithmetic Sequences

� ��

An arithmetic sequence is given below.

6, 8.5, 11, 13.5, 16, ...

Find the next three terms in the sequence given above.

To extend the arithmetic sequence, find the common differencebetween any two consecutive terms.

6 8.5 11 13.5 16

2.5 2.5 2.5 2.5

The common difference is 2.5. Therefore, the sixth term is 16 � 2.5 � 18.5, the seventh term is 18.5 � 2.5 � 21, and the eighth term is 21 � 2.5 � 23.5.

Find the next three terms in each arithmeticsequence.

1. 3, 5, 7, 9, 11, ... 2. 15.1, 15.2, 15.3, 15.4, 15.5, ... 13, 15, 17 15.6, 15.7, 15.8

3. 98, 93, 88, 83, 78, ... 4. 6.7, 7.5, 8.3, 9.1, 9.9, ...73, 68, 63 10.7, 11.5, 12.3

5. 0.3, 0.6, 0.9, 1.2, 1.5, ... 6. 76, 75.5, 75, 74.5, 74, ...1.8, 2.1, 2.4 73.5, 73, 72.5

7. 3, 15, 27, 39, 51, ... 8. 15, 18.6, 22.2, 25.8, 29.4, ...63, 75, 87 33, 36.6, 40.2

EXERCISES

EXAMPLE


9. Find the sixth term in the arithmetic sequence 8, 12, 16, 20, ... 28

10. Find the seventh term in the arithmetic sequence 90, 86, 82,78, ... 66

11. Find the tenth term in the sequence 84, 79, 74, 69, ... 39

12. Find the twelfth term in the sequence 65, 69.5, 74, 78.5, ... 114.5

The State General Sales Tax in 1993 inMaryland was 5%. The amount of sales taxon certain purchases is given in the chartbelow. Use this information to answerExercises 13–15.

13. What is the tax on a $7 purchase? $0.35

14. What is the tax on a $10 purchase? $0.50

15. The State General Sales Tax in 1993 in Mississippi was 7%. Make a chart like the one above to find the tax on a $15 purchase. See students’ work.; $1.05

16. A house rents for $800 a month. The owner expects themonthly rent to increase $25 each year. What will the monthlyrent be at the end of five years? $925

17. The cost for the first three minutes of a long-distance phonecall is $0.59. Each minute after that costs $0.13. What is thecost of a 10-minute long-distance phone call? $1.50

APPL ICAT IONS

Amount of Purchase(dollars) 1 2 3 4 5 6 7

Amount of Tax(dollars) 0.05 0.10 0.15 0.20

SKILL

68


Name ______________________________________ Date ___________

Geometric Sequences

��

The value of a car over several years is given in the chart below.

When consecutive terms of a sequence are formed by multiplying by a constant factor, the sequence is called a geometric sequence. The factor iscalled the common ratio.

What will the value of the car be after years 4 and 5?

To find the ratio between each pair of successive terms, divide the second term by the first.

18,000 � 20,000 � 0.9 Use a calculator.So the common ratio is 0.9.

20,000 18,000 16,200 14,580

�0.9 �0.9 �0.9

To find the next two terms, multiply the last term in the sequence by0.9. So, the value after year 4 is 14,580 � 0.9, or 13,122, and the valueafter year 5 is 13,122 � 0.9 or 11,809.80.

EXAMPLE

Year Value

0 $20,000

1 $18,000

2 $16,200

3 $14,580

4

5

Year Value

4 $13,122.00

5 $11,809.80


State whether each sequence is a geometricsequence. Write yes or no. If so, state the common ratio and write the next three terms of the sequence.

1. 3, 9, 27, 81, ... 2. 100, 50, 25, 12.5, ...

yes; ratio: 3; yes; ratio: ;

243, 729, 2,187 6.25, 3.125, 1.56253. 16, 18, 20, 22, ... 4. 5, –10, 20, –40, ...

no yes;ratio: –2; 80, –160, 320

5. –243, 81, –27, 9, ... 6. 90, 85, 75, 60, ...yes; noratio: – ; –3, 1, –

7. 1.2, 4.8, 19.2, 76.8, ... 8. 100, 80, 64, 51.2, ...yes; ratio: 4; yes; 307.2, 1,228.8, 4,915.2 ratio: 0.8; 40.96, 32.768, 26.2144

9. , , , , ... 10. , , , 1, ...

yes; noratio: 2; 1, 2, 4

Suppose you take a job for 31 days helpingyour cousin mow lawns. Your cousin offersyou a choice of two payment plans. He willeither pay you $100 a day or 1¢ the firstday, 2¢ the second day, 4¢ the third day, 8¢ the fourth day, and so on, continuing todouble the amount each day.

11. Without doing any calculations, which way would you chooseto be paid? Why? Answers will vary.

12. Determine the total amount you will make if you choose to bepaid $100 a day for 31 days. $3,100

13. Determine how much you will be paid on the 31st day if youchoose to be paid 1¢ the first day and double the amounteach day. $10,737,418.24

14. Now look back at your answer to Exercise 11. Would you change your choice? Why or why not? Answers will vary.

APPL ICAT IONS

3�4

1�2

1�4

1�2

1�4

1�8

1�16

1�3

1�3

1�2

EXERCISES

SKILL

69


Name ______________________________________ Date ___________

Classify InformationIn 1980, the United States film industry took in $2,748,500,000 in box officereceipts. The average admission charge was $2.69. In 1990, the box officereceipts were $5,021,800,000, and the average admission charge was $4.75.

How much more were the box office receipts in 1990 than in 1980?

What is the question?How much more were the receipts in 1990 than 1980?

What information is needed?The total receipts in 1980 and 1990 are needed.

What information is not needed?The average admission charges in 1980 and 1990 are not needed.

Solve the problem.

5,021,800,000� 2,748,500,000

2,273,300,000

In 1990, the receipts were $2,273,300,000 more than in 1980.

Classify information in each problem by writing“not enough information” or “too much information.” Then solve, if possible.

1. The sum of three numbers is 78. If one of the numbers is 14,what are the other two numbers?not enough information

2. If the product of 56 and 77 is 4,312, what is the sum of thenumbers?too much information; 133

EXERCISES

EXAMPLE


3. If the sum of 18 and a number is 54 and their product is 648,what is their difference?too much information; 36

4. If the product of two numbers is 100, what is the difference ofthe numbers?not enough information

Classify information in each problem bywriting “not enough information” or “toomuch information.” Then solve, if possible.

5. Phien bought 3 address books that cost $4.98 each. She gavethe cashier a $20 bill. What was the total cost of the books?too much information; $14.94

6. Jimmy grew 3 inches last year and 2 inches so far this year.How tall is Jimmy now?not enough information

7. Carla, a carpenter, has two tape measures. The steel tape is 8feet long. The cloth tape is marked in metric measure at one-centimeter intervals. How much longer is the steel tape thanthe cloth tape?not enough information

8. Jonathan bought 10 computer disks for $1.39 each. The disksusually sell for $1.99 each, or ten for $18. How much did hepay for the disks?too much information; $13.90

9. The Sheng family drove 1,287 miles on their vacation. Abouthow many miles did they drive per day?not enough information

10. Gerda pays a delivery service $18 for priority delivery, $15 forstandard delivery, and $21 for Saturday delivery. How muchwill she save by sending a package by standard deliveryinstead of Saturday delivery?too much information; $6

11. Alan ran the same number of miles for 6 days. How far did herun?not enough information

APPL ICAT IONS

SKILL

70


Name ______________________________________ Date ___________

Problem-Solving StrategiesThere are many strategies that can be used to solve a problem. A few ofthese strategies are listed below.

• Draw a diagram • Use logical reasoning• Use a matrix or chart • Draw a picture• Make a list • Guess and check

For each problem you solve, you must decide which strategy would work bestfor you.

A 1-inch spool holds 100 inches of line, a 2-inchspool holds 400 inches of line, and a 3-inch spoolholds 900 inches of line. How many inches of lineare on a 5-inch spool?

First make a chart.

Study the chart. You know that 12 � 1, 22 � 4, and 32 � 9. Using logical reasoning, a 5-inch spool holds 52 � 100 or 2,500 inches.

Solve using any strategy.

1. Juan has a mixture of pennies and dimes worth $2.28. He has between 39 and 56 pennies. How many dimes does Juan have? 18 dimes

EXERCISES

EXAMPLE

Spool Size Inches of Line

1 in. 100

2 in. 400

3 in. 900


2. Arrange the digits 1 through 7 in the squares so that the sumalong any line is 10.

3. There are three cubes each measuring a different whole number of inches on an edge. When the cubes are stacked, the stack is six inches high. What is the length of the edge of each cube? 1 in., 2 in., and 3 in.

4. Mr. Patel asked five of his students to line up by height. Juan is not the shortest and is not standing next to Pamela. Tad isthe tallest and is not standing next to Juan. Marco is taller than Pamela and Caroline is next to Tad. Who is standing in the middle? Juan

5. If it takes 20 seconds to inflate a balloon with helium from a tank, how many balloons can be inflated in 6 minutes? 18 balloons

6. A vending machine dispenses products that each cost 60¢. Itaccepts quarters, dimes, and nickels only. If it only accepts exactchange, how many different combinations of coins must themachine be programmed to accept? 13 combinations

7. The bus leaves the downtown for the mall at 7:35 A.M., 8:10 A.M., 8:45 A.M., and 9:20 A.M.. If the bus continues to run on this schedule, what time does the bus leave between10:00 A.M. and 11:00 A.M.? 10:30 A.M.

8. Bob needs to go to the bank, the post office, and the bicycle shop. In how many different orders can he do his errands? 6 different orders

9. Ronda spent 22 minutes on the telephone talking long-distanceto her cousin. If the rate is $0.20 for each of the first 3 minutesand $0.15 for each minute after that, how much did the call cost? $3.45

APPL ICAT IONS

SKILL

71


Name ______________________________________ Date ___________

Determine Reasonable AnswersThe product manager at Taylor Dairy reported that about 2 billion poundsof cheese were sold in the United States in 1992. The average American eats28 pounds of cheese a year, and the population of the United States wasabout 255,200,000 in 1992.

Is the product manager’s statement reasonable?

The average American eats about 30 pounds of cheese a year. Therewere between 200 million and 300 million people in the United Statesin 1992. Multiply to estimate the amount of cheese sold in 1992.

200,000,000 � 30 � 6,000,000,000300,000,000 � 30 � 9,000,000,000

There were between 6 billion and 9 billion pounds of cheese sold inthe United States that year. The product manager’s statement is notreasonable.

Determine whether the answers shown are reasonable. Write yes or no.

1. 27 � 38 � 65 2. 604 � 225 � 729 3. 168 � 35 � 153yes no no

4. 535 � 5 � 107 5. 112 � 4 � 484 6. 5,962 � 11 � 542yes no yes

7. 56 � 35 � 91 8. 168 � 17 � 111 9. 205 � 5 � 31yes no no

10. 6,657 � 7 � 95 11. 3,137 � 4 � 1,258 12. 36 � 22 � 792no no yes

EXERCISES

EXAMPLE


13. 812 � 5 � 16.24 14. 510 � 490 � 1,000 15. 5,988 � 635 � 6,623no yes yes

16. Carrie thinks she can buy five CD’s at the Compact Disc Depot sale for less than $55.00 including tax. Does this seem reasonable?yes

17. Jake bought three CD’s on sale at the Compact Disc Depot. Hegave the clerk two $20.00 bills. The clerk gave him $5.06 inchange. Does this seem reasonable?no

18. The Ortiz family is planning a 1,880-mile trip. They want todrive between 200 and 250 miles per day. Is 6 days a reason-able time for the trip?no

19. There are 478 people who are planning to take buses to arally. Each bus carries 37 people. Thirteen buses have beenordered. Is that a reasonable number?yes

20. The Kowalski family spent $1,500 on a one-week vacation.Their calculator showed an average cost of $150 a day. Is thisanswer reasonable?no

21. Out-of-town newspapers cost 60¢ each, and local papers cost35¢ each. Bill buys two out-of-town papers and three localpapers. He hands the cashier $3.00. Should he expect morethan 50¢ change?yes

22. A bottler is putting 120 gallons of juice into one-quart bottles.The empty bottles are packed 50 in a crate. What is a reasonable number of crates for the bottler to order?10 crates

Compact Disc Depot Sale!!!

Selected CD’s $9.98

APPL ICAT IONS

SKILL

72


Name ______________________________________ Date ___________

Work BackwardRupesh earned some money mowing lawns one month. He put half of hismoney into savings. With the rest, he spent $15 on a new CD, $6 to see amovie, and $3 on food. He still had $24 left in his pocket.

How much money did Rupesh earn mowing lawns?

Work backward to answer this question. Undo each step.

Start with $24. $24

Add the $3 spent on food. $24 � $3 � $27

Add the $6 spent to see the movie. $27 � $6 � $33

Add the $15 spent on the CD. $33 � $15 � $48

Since Rupesh saved half of the money, multiply by 2. $48 � 2 � $96

Rupesh made $96 mowing lawns.

Solve by working backward.

1. A number is added to 8, and the result is multiplied by 10. Thefinal answer is 140. Find the number.6

2. A number is divided by 8, and the result is added to 12. Thefinal answer is 75. Find the number.504

3. A number is decreased by 12. The result is multiplied by 5, and30 is added to the new result. The final result is 200. What isthe number?46

EXERCISES

EXAMPLE


4. Twenty five is added to a number. The sum is multiplied by 4, and 35 is subtracted from the product. The result is 121.What is the number?14

5. Take a number, divide it by 3, add 14, multiply by 7, and double the answer. The result is 252. What is the number?12

6. Dwayne’s weight is twice Beth’s weight minus 24 pounds.Dwayne weighs 120 pounds. How much does Beth weigh?72 lb

7. Kara wants to buy a certain leather jacket, but she did nothave enough money. The leather jacket went on sale and wasreduced by $15.00, then by $13.50 more, and finally by anadditional $12.15. Kara bought the jacket at the final saleprice of $109.35. What was the original price?$150.00

8. James arrived for piano practice at 4:45 P.M. On the way fromschool, he stopped at the video store for 15 minutes and alsomade a call from the phone booth for 10 minutes. It usuallytakes 25 minutes to get from the school to the piano teacher’shouse. What time did James leave school?3:55 P.M.

9. Dave has 12 baseball cards left after trading cards. This is onethird as many as he had yesterday, which is 8 less than the daybefore. How many cards did Dave have on the day before yesterday?44 cards

10. A fence is put around a dog run 10 feet wide and 20 feetlong. Enough fencing is left over to also fence a square gardenwith an area of 25 square feet. If there is 3 feet left after thefencing is completed, how much fencing was available at thebeginning?83 ft

APPL ICAT IONS

SKILL

73


Name ______________________________________ Date ___________

Solve a Simpler ProblemFind the sum of the whole numbers from 1 to 300.

This would be a tedious problem to solve using a calculator or addingthe numbers yourself. The problem is easier to solve if you solve simpler problems. First consider the partial sums indicated below.

1, 2, 3, 4, 5, . . . , 150, 151, . . . , 296, 297, 298, 299, 300150 � 151 � 301

.

.

.5 � 296 � 3014 � 297 � 3013 � 298 � 3012 � 299 � 3011 � 300 � 301

Notice that each sum is 301. There are 150 of these partial sums.

301 � 150 � 45,150

The sum of the whole numbers from 1 to 300 is 45,150.

Solve by solving a simpler problem.

1. Find the sum of the whole numbers from 1 to 150.11,325

2. Find the sum of the whole numbers from 101 to 300.40,100

3. Find the sum of the even numbers from 2 to 200.10,100

EXERCISES

EXAMPLE


4. What is the total number of triangles of any sizein the figure at the right?26 triangles

5. What is the total number of squares of any sizein the figure at the right?30 squares

6. Shea is planning to carpet a large areain her basement as shown at the right. How much carpet will she need to carpet this area?1,072 ft2

7. Cliff heard a funny joke on the radio onSunday. On Monday (day 1), he told the joke to Sarah, Rich, and Claire. These people each told the joke to 3 more people on Tuesday (day 2), who told the joke to 3 more people on Wednesday (day 3). This patterncontinued. How many people heard the joke on the sixth day?729 people

8. How many days passed before at least 100 people had heardthe joke in Exercise 7?4 days

9. By the end of the day 6, how many people altogether hadheard the joke in Exercise 7? (Remember to count Cliff!)1,093 people

10. A summer camp has 7 buildings arranged in a circle. Pathsmust be constructed joining every building to every otherbuilding. How many paths are needed?21 paths

40 ft

8 ft

16 ft 16 ft

16 ft16 ft

30 ft 30 ft

APPL ICAT IONS

SKILL

74


Name ______________________________________ Date ___________

Make a ModelA box like the one at the right is a rectangular prism. It has six sidesand each one is the shape of a rectangle.

How many different shapes of rectangularprisms can be formed using exactly 20 cubes?

Use 20 cubes to model this problem. Make as many different shapes of rectangular prisms as you can.

There are four different shapes of rectangular prisms that can be made.

Solve by making a model.

1. How many different shapes of rectangular prisms can beformed using exactly 12 cubes?4 shapes

2. How many different shapes of rectangular prisms can beformed using exactly 24 cubes?6 shapes

EXERCISES

EXAMPLE

1 × 1 × 20

1 × 4 × 5 2 × 2 × 5

1 × 2 × 10


3. How many cubes are needed to make the displayshown at the right?30 cubes

4. How many cubes are needed to make the displayshown at the right?35 cubes

5. Ronnie used blocks to build a “fort”. The blocks were cubesand were stacked five high. The top, front, and side viewswere all squares. How many blocks did Ronnie need to buildhis fort?80 blocks

6. Twelve one-inch-tall square snack cakes are packed in a box.No two cakes are stacked on top of one another. What are thepossible dimensions of the box if the top view of each cake isa two-inch by two-inch square?24 in. by 2 in. by 1 in., 12 in. by 4 in. by 1 in., 8 in. by 6 in. by 1 in.

7. The town playground is to have a hedge around it. The playground is in the shape of a pentagon with two sides of 40 feet, two sides of 60 feet, and one side of 70 feet. The bushes will be planted every 5 feet. How many bushes will be needed?54 bushes

8. Rita collects miniature lamps. She is building a shelf aroundthe rectangular family room to display them. If the familyroom is 15 feet wide and 18 feet long, how many feet ofshelving will she need?66 feet

9. A carton is 8 inches by 4 inches by 12 inches. How many four-inch cubes can Brian pack in the carton?6 cubes

APPL ICAT IONS

SKILL

75


Name ______________________________________ Date ___________

Make TablesTables can help you organize information so it can be understood easier.

Shauna needed to give a customer $1.40 in change. The customerrequested that she not give him any bills. He also did not want to beable to make change for a dime or a nickel. She gave the customer 10 United States coins. What ten coins did Shauna give the customer?

This problem can be solved by making a table. Try to find differentcombinations of ten coins that make $1.40 and do not include changefor 10¢ or 5¢.

The combination in the last row satisfies the requirements. There are 10 coins in the group, the coins have a value of $1.40, and you cannot make change for 10¢ or 5¢. Shauna gave the customer 1 nickel,6 dimes, and 3 quarters.

Solve. Make a table.

1. How many ways can you make change for a $50-bill using only$5-, $10-, and $20-bills?12 ways

2. Gregg has a penny, a nickel, a dime, and a quarter in his pocket.Without looking, Gregg picks two coins out of his pocket. Howmany different amounts of money could he choose?6 amounts

EXERCISES

EXAMPLE

pennies nickels dimes quarters total

5 1 3 1 $0.65

0 3 6 1 $1.00

0 1 7 2 $1.25

0 1 6 3 $1.40


3. Norma’s Repair Shop charges $35 for a service call and $25 anhour for each hour of labor. How much does she charge for an8-hour service call?$235

Jake and June Washington started a collegefund for their daughter. They started thefund by depositing $800 at the beginningof the first month. They plan to add $75 tothe fund at the end of every month. Usethis information to answer Exercises 4–6.

4. How much will be in the account after

a. 1 month? $875

b. 6 months? $1,250

c. 1 year? $1,700

d. 2 years? $2,600

5. How can you extend your table from Exercise 4 to find out howmuch will be in the account after every year?Add $900 to the previous year’s amount.

6. Suppose the Washingtons deposited $800 at the end of the firstmonth and then $75 at the end of every month after that. Howwould this change your table?The amounts would all be reduced by $75.

7. Find out how much your long distance phone company chargesfor calls. How much would it cost you to make a 15-minute longdistance phone call?Answers may vary.

APPL ICAT IONS

Documents

Skill Intervention Workbook (Grade 8)