Skill Intervention Workbook (grade 8)

Skills Intervention for Algebra Diagnosis and Remediation

Student Workbook

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Algebra Intervention Student Workbook

Table of Contents Skill

Number and Operation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Multiplication Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Adding and Subtracting Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Multiplying and Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Adding and Subtracting Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Multiplying and Dividing Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 Greatest Common Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Ratios as Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 Adding and Subtracting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 Adding and Subtracting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 Multiplying and Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 Multiplying and Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 Percents as Fractions and Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 Percent of a Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 Percent Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 Percent of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Skill

Algebra

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Solve Equations Involving Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . .37 Solve Equations Involving Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . .39 Solve Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 Use an Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 Proportional Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47 Scale Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53 Function Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 Solve Equations With Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 Graphing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 Slope of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 Graphing Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65 Graphing Linear and Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67

Skill

Geometry

35 36 37 38 39 40 41

Sums of Angles of Polygons Similar Figures . . . . . . . . . . . Similar Triangles . . . . . . . . . . Congruent Figures . . . . . . . . Reflections . . . . . . . . . . . . . . Dilations and Rotations . . . . Translations . . . . . . . . . . . . .

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Algebra Intervention

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. . . . . . .69 . . . . . . .71 . . . . . . .73 . . . . . . .75 . . . . . . .77 . . . . . . .79 . . . . . . .81

Skill

Measurement

42 43 44 45 46 47 48

Perimeter and Area . . . . . . . . . . . . Area of Circles . . . . . . . . . . . . . . . . Area of Rectangles . . . . . . . . . . . . . Area of Triangles and Trapezoids . Area of Irregular Shapes . . . . . . . . Surface Area of Rectangular Prisms Volume of Rectangular Prisms . . . .

Skill

Data Analysis and Probability

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

Using Samples to Predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97 Mean, Median, and Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 Make a List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 Probability of Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103 Expected Value of an Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105 Theoretical and Experimental Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107 Probability Using Area Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109 Line Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113 Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 Bar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117 Circle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119 Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121 Constructing and Interpreting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123 Make a Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125 Interpreting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129 Predicting Distribution of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131 Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135

Skill

Problem Solving

69 70 71 72 73 74 75

Classify Information . . . . . . . . . . Problem-Solving Strategies . . . . Determine Reasonable Answers Work Backward . . . . . . . . . . . . . Solve a Simpler Problem . . . . . . Make a Model . . . . . . . . . . . . . . Make Tables . . . . . . . . . . . . . . . .

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Algebra Intervention

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.83 .85 .87 .89 .91 .93 .95

.137 .139 .141 .143 .145 .147 .149

SKILL

1

Name ______________________________________ Date ___________

Order of Operations

When you evaluate an expression in mathematics, you must do the

operations in a certain order. This order is called the order of operations.

EXAMPLE

Evaluate 56
(17  9)  7  3. 56
(17  9)  7  3  56
8  73

7 

21

Do all the operations within the grouping symbols.



Do multiplication and division from left to right.

28

Do addition and subtraction from left to right.

Therefore, 56
(17  9)  7  3  28.

EXERCISES

Evaluate each expression.

1. 2  9  5  3 33

2.

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

(9  4)
5 1

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

Glencoe/McGraw-Hill

1

Algebra Intervention

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

25. 150
10  3  5 0

26.

5(35 18)  1 86

A P P L I C AT I O N S

Use the price list at the right to answer 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

8

Sam’s Sporting Supplies Price List Ping Pong Balls 5 for $2 Ping Pong Paddles $8 Softballs $5 Soccer Balls $20

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.

Glencoe/McGraw-Hill

2

Algebra Intervention

Name ______________________________________ Date ___________

SKILL

2

Multiplication Properties

The table shows the properties for multiplication. Property

Examples

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

21  2  2  21 42  42

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

5  (3  6)  (5  3)  6 5  18  15  6 90  90

Identity The product of a number and 1 is the number.

81  1  81

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

7 8     1 8 7

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

EXERCISES 1.

6  11

11  6

2  (9  3)  (2  9)  (2  3) 2  12  18  6 24  24

Name the multiplicative inverse, or reciprocal, of each number. 2.

19  3

Glencoe/McGraw-Hill

3  19

3.

1  8

8

3

4. 9

1  9

Algebra Intervention

Name the property shown by each statement. 5. 67  89  89  67

6.

11 11  1  12 12

8.

commutative

7.

identity

identity 9.

3 5 5 3        4 6 6 4

10.

2

5

1

2

5

1  41 4

3 1 5 3 1 3 5             5 3 7 5 3 5 7












distributive 12.

inverse 13.

1


5  3   9  5 
3  9  associative

commutative 11.

1  45  45

45(23  3)  (45  23)  (45  3)

distributive

9 4    1 4 9

14.

inverse

4 3 3 4        5 4 4 5

commutative

A P P L I C AT I O N S 3

15. Jill runs for 1  4 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 2

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

22 inches

Glencoe/McGraw-Hill

4

Algebra Intervention

Name ______________________________________ Date ___________

SKILL

3

Adding and Subtracting Decimals

To add decimals, line up the decimal points. Then add the same way you add whole numbers.

EXAMPLES

4.76  3.62

12.8  3.467  8.56

4.76  3.62 8.38

12.800 3.467 Annex zeros.  8.560 24.827 The sum is 24.827.

The sum is 8.38.

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

EXAMPLES

15.05  4.86

35  13.631

15.05  4.86 10.19 The difference is 10.19.

EXERCISES 1.

45.9  12.7

Add or subtract. 2.

58.6 4.

6.83  3.77

3.

3.06

205.7  98.8

5.

106.9 7.

35.000 Annex zeros.  13.631 21.369 The difference is 21.369.

100.21

6.7  3.56

6.

10.26

17.93  33.5

8.

51.43 Glencoe/McGraw-Hill

43.89  56.32

18.75  7.2

11.55

77  12.66

9.

64.34

6.5  7.547

14.047 5

Algebra Intervention

10. 4.7  0.89

11.

12. 25  4.76

13.

14. 9.857  4.5

15.

16. 408.7  56.78

17.

18. 73.56  29

19.

3.81

23.49

20.24

6.43  7.8  13

27.23

5.357

65.8  15.75  7.854

89.404

351.92

7.9  1.22  6.1  11

26.22

44.56

A P P L I C AT I O N S

15.6  7.89

11.444  5.9  13.93

31.274

The results of the 1948 presidential election is given at the right. Use this information to answer Exercises 20–22.

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

Candidate

Percent of Popular Vote

Truman

49.5

Dewey

45.12

Thurmond

2.4

Wallace

2.38

Other

0.6

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 checks for $55.86 and $25.00. He deposited a check for $113.76. What is his new balance in his checking account? $317.63 24. For lunch, Connie buys a sandwich for $2.35 and a small lemonade 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

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Algebra Intervention

SKILL

4

Name ______________________________________ Date ___________

Multiplying and Dividing Decimals

EXAMPLE

Multiply 1.45 by 0.68. 1.45  0.68 1160 870 0.9860

2 decimal places 2 decimal places

The sum of the decimal places in the factors is 4, so the product has 4 decimal places.

4 decimal places

The product is 0.9860.

EXAMPLE

Divide 38.22 by 2.6. 1 4.7 2.6.3 8.2 .2 
26
12 2 10 4 182 182 0

Change 2.6 to 26 by moving the decimal point one place to the right. Move the decimal point in the dividend one place to the right. Divide as with whole numbers, placing the decimal point above the new point in the dividend.

The quotient is 14.7.

EXERCISES 1.

Multiply.

4.9  35

2.

18.9  3.7

5.

171.5 4.

69.93

53  3.7

3.

0.014  0.65

6.

196.1

9.80

0.0091

Glencoe/McGraw-Hill

2.8  3.5

53.98  71.2

3,843.376

7

Algebra Intervention

7.

4.55  41.8

8.

190.19

0.133  4.2

9.

3.91  8.5

0.5586

33.235

1.42 10. 68 .5 2

2.8 11. 236 4.4 

0.37 12. 531 9.6 1

84 13. 1.61 34.4 

73 14. 0.523 7.9 6

160 15. 0.233 6.8 

0.022 16. 1.70 .0 374

650 17. 0.1127 2.8 

5.13 18. 7.43 7.9 62

Divide.

A P P L I C AT I O N S The prices at Martha’s Meat Market are given at the right. Use this information to answer Exercises 19–21. 19. What is the cost of a chicken that weighs 3.4 pounds?

Martha’s Meat Market Specials of the Week Ground Beef Chicken Turkey Breast

$1.90/lb $1.15/lb $1.75/lb

$3.91 20. Willy buys a package of ground beef for $6.84. How many pounds of ground beef did he buy?

3.6 lb 21. A turkey breast costs $8.05. How much does the turkey breast weigh?

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

Glencoe/McGraw-Hill

8

Algebra Intervention

SKILL

5

Name ______________________________________ Date ___________

Adding and Subtracting Integers

When 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.

EXAMPLES

Solve j  –71  35.

Solve w  –41  (–73).

j  –71  35

w  –41  (–73)

–71
35 71  35

w  –41  73 w  32

The sum is negative.

71  35  36

The solution is 32.

j  –36 The solution is –36.

EXERCISES

Solve each equation.

1. –9  (–5)  a

2.

4. d  –16  9

5.

7. –10  8  g

8.

–4

–7

3.

–11  (–16)  e

6.

14  (–12)  h

9.

26 11.

–42

Glencoe/McGraw-Hill

c  11  (–14)

–3

5

–18

10. k  –56  (–14)

b  –3  9

6

f  –11  (–12)  (–9)

–32

j  –7  (–9)  (–9)

–25

m  –11  28

12. –37  11  n

–39

–26

9

Algebra Intervention

13. p  –15  (–36)

14. –12  (–23)  q

16. 20  (–13)  s

17. t  –5  23  (–10) 18. w  –16  (–35)

19. 19  (–15)  x

20. –23  23  y

21 7

34

11

15. –9  (–23)  r

32

8

–51

21.

0

(–10)  36  z

26

Evaluate each expression if c  4, x  –5, and h  6. 22. x  5  9  (–7)

23. –12  c  (–3)

24. –6  x  h

25.

2

–5

–11

–12  h  x  h

–5

A P P L I C AT I O N S 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 the night?

9°F 27. Trackmaster Bike shop reported these profits and losses for the last 5 years: 1988: profit of $25,000 1989: loss of $47,000 1990: loss of $30,000 1991: profit of $13,000 1992: 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 two years 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 bike shop to break even (have total losses and profit be $0) for the six years?

$5,000

Glencoe/McGraw-Hill

10

Algebra Intervention

Name ______________________________________ Date ___________

SKILL

6

Multiplying and Dividing Integers

If 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.

EXAMPLES

Solve each equation. d  (–5)  7 d  –35

One factor is negative and the other is positive. The product is negative.

The solution is –35. t  (–120)
(–6) t  20

Both integers are negative. The quotient is positive.

The solution is 20. k  (–6)  7  (–2) k  84

There is an even number of negative integers. The product is positive.

The solution is 84.

EXERCISES

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

1. 9  (–6)

2.

3. (–7)  (–7)

4.

5. (–1)  (–2)  3

6.

negative; –54

(–3)  (–6)

positive; 18

positive; 49

(–1)  2  (–1)  (–1)

negative; –2

positive; 6

(–5)  (–2)  (–1)  (–3)

positive; 30

Solve each equation. 7. (–56)
(–4)  a

14

9. c  72
(–12)

8. b  26  (–2)

–52

10.

–6

11. e  (–4)  21

–84

Glencoe/McGraw-Hill

(–22)  (–3)  d

66

12.

(–100)
5  f

–20 11

Algebra Intervention

13. 98
(–7)  g

14.

h  13  (–12)

15. (–125)
(–25)  j

16.

k  (–15)  (–7)

17. 240
(–16)  m

18.

q  (–88)
4

19. y  (–6)  (–9)

20.

21. 12  (–2)  3  n

22.

23. (–12)  (–2)  (–2)  a

24.

25. f  (–9)  2  (–2)

26.

–14

–156

5

105

–15

–22

t  (–90)
(–10)

54

9

(–2)  5  (–10)  p

–72

100

–48

e  10  (–5)  3

–150

36

(–2)  (–2)  (–2)  2  n

–16

A P P L I C AT I O N S 27. Find the value of each expression. a. (–1)2 1 e

(–1)6 1

b. (–1)3 –1 f.

(–1)10

1

c. (–1)4 1

d. (–1)5 –1

g. (–1)25

h. (–1)100

–1

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 an

even power will produce a positive number, and a negative number raised to an odd power will produce a negative number. 29. On Monday, Tuesday, and Wednesday, the low temperatures in Wisconsin were –20°F, –15°F, and –28°F. What is the average low temperature for the three days? –21°F

30. California had a record hot day in 1913 in Death Valley. It reached 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 last game. Each penalty was for 15 yards. What was the total lost yards due to penalties? 60 lost yards

Glencoe/McGraw-Hill

12

Algebra Intervention

Name ______________________________________ Date ___________

SKILL

7

Prime Factorization

Evelyn has 105 books. She is trying to decide how to put them on the shelves of 3 separate bookcases.

EXAMPLE

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  35 357 She can put 5 books on each of 7 shelves or 7 books on each of 5 shelves.

EXERCISES 1.

75

Find the prime factorization of each number. 2.

355

5. 90

2335

5  5  11

7.

385

66

2  3  11

Glencoe/McGraw-Hill

100

22233

8.

2255

11.

5  7  11

14.

3557

42

4. 72

77

237

10.

13. 525

3. 49

2233

6.

9. 275

36

210

11  11

12.

2357

15.

196

2277

13

121

147

377

16.

500

22555

Algebra Intervention

17. 136

18.

2  2  2  17

21. 234

22.

2  3  3  13

495

19. 231

3  3  5  11

84

3  7  11

23. 255

2237

A P P L I C AT I O N S

20.

1,001

7  11  13

24.

3  5  17

252

22337

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

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

19 ft and 5 ft

26. Monty changes his mind and decides that the garden should have an area of 100 square feet. What are the whole number dimensions 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 would be in Monty’s yard and one-half would be in his neighbor’s yard. 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 are possible for this garden?

50 ft by 5 ft, 25 ft by 10 ft

Glencoe/McGraw-Hill

14

Algebra Intervention

SKILL

8

Name ______________________________________ Date ___________

Greatest Common Factor (GCF)

Carlos is 20 years old and his brother Thomas is 24 years old. The greatest

common factor (GCF) of their ages is the same as their niece Cristina’s age.

EXAMPLE

How old is Cristina? To find Cristina’s age, find the GCF of 20 and 24. One way to find the GCF is to find the prime factorization of each number. 20  2  10 225

24  2  12 243 2223

Then find the common prime factors. 20  2  2  5 24  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.

EXERCISES 1.

16, 24

Find the GCF for each set of numbers. 2.

8

5.

28, 70

40, 56

8

3.

3

6.

14

9.

15, 18

72, 96

14, 28

14

Glencoe/McGraw-Hill

4. 32, 48

18

7.

24

10.

18, 36

16

81, 48

8.

3

11.

48, 36

12

30, 18

6

12.

84, 154

14

15

Algebra Intervention

13. 24, 64

14.

8

35, 25

5

17. 2, 4, 8

18.

2

22.

6

8, 12, 16

24, 36, 48

75, 120

15

19. 18, 30, 36

20.

6

12

A P P L I C AT I O N S

16.

20

4

21. 18, 12, 24

15. 100, 80

15, 25, 30

5

23. 8, 16, 40

8

24.

12, 18, 72

6

Nita is making a baby quilt. She is using strips of material that are cut from pieces of material that are 36 inches wide and 48 inches wide. Use this information to answer 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 use for the quilt. The piece of material is 54 inches wide. If all of the strips from the three pieces of material 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 of the three 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

Glencoe/McGraw-Hill

16

Algebra Intervention

SKILL

9

Name ______________________________________ Date ___________

Ratios as Fractions

A ratio is a comparison of two numbers by division. EXAMPLE

A store sold 360 newspapers last week and 440 this week. Write a ratio 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

360 to 440, 360 : 440, or  440 . 360 last week’s sales    this week’s sales 440 9

 11 9

The ratio written as a fraction in simplest form is  11 .

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

Are 12 : 18 and 14 : 21 equivalent ratios? Express each ratio as a fraction in simplest form. 12
6 12 2      18
6 18 3 2

14
7 14 2      21
7 21 3

2

 Since  3  3 , the ratios are equivalent.

EXERCISES

Express each ratio as a fraction in simplest form.

1. 8 cheese pizzas out of 14 pizzas 4  7

2. 27 feet to 24 feet

3. 35 sopranos in an 84-member chorus 5  12

4. 14 hours to 3 days

Glencoe/McGraw-Hill

9  8

7  36 17

Algebra Intervention

5. read 75 pages out of 90 5  6

6.

68 : 18 34  9

7. 6 pounds : 12 ounces 8  1

8.

exercise 45 minutes out of 63 5  7

9. 15 : 50 3  10

10.

40 minutes per hour 2  3

Tell whether the ratios in each pair are equivalent. Show your answer by simplifying. 11. 42 to 49 and 54 to 63 6 6    ; yes 7 7

12.

18 : 42 and 20 : 44 3 5   ; no 7 11

13. 144 : 36 and 72 : 32 4 9   ; no 1 4

14.

16 to 96 and 1 to 6 1 1    ; yes 6 6

15. 3 lbs. : 12 ozs. and 6 lbs : 24 ozs. 4 4    ; yes 1 1

16.

6 hours to 4 days and 12 hours to 10 days 1 1   ; no 16 20

A P P L I C AT I O N 17. Sound waves travel at about 740 miles per hour. In 1947, Chuck Yeager became the first person to fly a plane at a speed greater than the speed of sound. If he had flown at the speed of sound, he would have flown at mach 1. The mach number is the ratio of an object’s speed to the speed of sound. Find the mach number of each of the following and express it as a decimal rounded to the nearest hundredth. a. a Bell X-15A2 rocket plane that flew at 4,520 miles per hour in 1964 mach 6.11 b. a Boeing 747 passenger plane that can fly at a speed of 625 miles per hour mach 0.84 c. the space shuttle Columbia that has traveled through space at over 16,600 miles per hour mach 22.43 d. Carl Lewis who ran 100 meters in 9.86 seconds in 1991 (that is about 20 miles per hour) mach 0.03 e. a Cessna passenger plane that can fly 176 miles per hour

mach 0.24 Glencoe/McGraw-Hill

18

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

10

Adding and Subtracting Fractions

To add fractions, you must have a common denominator. EXAMPLE

Find each sum. 2

3

5

1

 a.  7  7

 b.  4  6

2  7 3

 7

1   4

3  12

5

10

  6   12 1 13   1 12 12

5  7 5

1

The sum  7.

The sum is 1  12 .

To subtract fractions, you must have a common denominator. EXAMPLE

Find each difference. 11

2

5

 a.  12  12

1

 b.  8  2

11  12 2

 12

5   8

5  8

1

4

  2  8

9 3   12 4

1  8 3

1

The difference is  4.

EXERCISES 1.

7  9

The difference is  8.

Add or subtract. Write each answer in simplest form. 2.

3  8

3.

5  6 1

 9

4

 8

1

 6

1  3

1  2

2  3

Glencoe/McGraw-Hill

19

Algebra Intervention

4.

7.

4  5

5.

6.

1

3  4

 2

1

 3

 6

3  10

1 21

4

7  12

3  4

8.

5

10.

6  7

1

1  7

9.

13  15

 12

 5

4

 3

1  3

33  35

1  5

1  2

11.

2

1  8

12.

3  5

3  5

3  4

7  10

 5

2

 2

1

 4

1

1 8

3

1 20

1 2

1

11

A P P L I C AT I O N S 2

1

 13. Reginald planted  5 of his garden with tomatoes and 4 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? 7 13  of the garden;  of the garden 20 20 2

1

 14. Tina rode her bicycle  3 mile to the park and then 2 mile to 3 the library. Finally she rode her bicycle  5 mile to her home.

How far did Tina ride her bike? 23 53  or 1  mi 30 30 2

15. In a survey,  7 of the people said they preferred Brand A, and 1  of the people said they preferred Brand B. What is the 5 difference between the fraction of people who prefer Brand A and the fraction of people who prefer Brand B? 3  35 Glencoe/McGraw-Hill

20

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

11

Adding and Subtracting Fractions

Lina is making trail mix for a hiking trip. She has 2 1

2

1  cups of peanuts, 2

 3 4 cups of raisins, and 2 3 cups of carob chips.

EXAMPLES

How many cups of trail mix will Lina have? 1

2 12

1

3 12

2 2  3 4  2

6 3 8

  2 3   2 12 17

12

5

  7 12 7  12  12 5

71  12 5

8  12 5

 8 12 5

Lina will have 8  12 cups of trail mix. If Lina wants 15 cups of trail mix, how many more cups of trail mix does she have to make? 12

12

 15  14  1  14   12  14 12 12

 14  12

15 5

5

  8 12   8 12 7

6 12 7

She needs to make another 6  12 cups of trail mix.

EXERCISES 1.

7 2    12 12

Add or subtract. Write each answer in simplest form. 3  4

2.

Glencoe/McGraw-Hill

9 3    10 10

3  5

3. 21

7 5    9 9

1

1 3 Algebra Intervention

4.

7 3    16 16

7.

1 7    4 8

10. 13.

1  4

5.

1 1    6 2

1 8

8.

9 3    10 5

11 1    15 3

2  5

11.

1 1    9 6

3 4    10 5

1 10

14.

4 1    5 6

1

1

1

1

1

4 3

 16. 9  2 56

3

2  3

2 1    3 2

9.

4 1    5 12

53  60

5  18

12.

1 7    2 16

1  16

19  30

 15. 7  10  2 5

3  10

5

 17. 5  4 28

1  6

6.

1

3

8 8

18.

3

3

1

9 10 7

1

7 12

 9 4 26

A P P L I C AT I O N S 19. The route from Ramon’s house to city hall and then to the 3

9

 school is  10 mile. It is 10 mile from city hall to the school. What is the distance from Ramon’s house to city hall?

3  mile 5

3

20. To make a salad, Henry used  4 pound of Boston lettuce and 2  pound of red lettuce. How much lettuce did he use? 3

5

1 lb 12

1

3

 21. Donna has 10  4 yards of ribbon. She needs 3 2 yards of ribbon to make a bow. How much ribbon will she have after she 1 7 yd makes the bow? 4

22. Part of the daily diet of polar bears at the Bronx Zoo is 1

1

 1 4 pounds of apples and a 1 2 -pound mixture of oats and barley. What is the combined weight of these items?

3

2 lb 4

23. Ani has two chores to do on Saturday. She has to wash the car 3

which will take her  4 hour and rake the leaves which will 1 take her 1  2 hours. How much time should she plan to spend

1

2 hr 4

on these chores? 24. Mr. Vazquez wants to put a fence around his rectangular 3

1

 vegetable garden. If the garden is 18  4 feet long and 10 2 feet 1 58  feet wide, how much fence will he need? 2 Glencoe/McGraw-Hill

22

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

12

Multiplying and Dividing Fractions

To multiply fractions, multiply the numerators and multiply the denominators.

EXAMPLE

3

2

 What is the product of  8 and 3 ? Multiply the numerators. 3 2 32      8 3 83 Multiply the denominators. 6

1

  24 or 4

Simplify.

1

The product is  4.

To divide by a fraction, multiply by its reciprocal. EXAMPLE

3

1

 What is the quotient of  5 and 2 ? 3 1 3 2 
     5 2 5 1

1

Multiply by the reciprocal of  2. Multiply the numerators. Multiply the denominators.

32

 51 6

1

  5 or 1 5 6

Simplify.

1

 The quotient is  5 or 1 5 .

EXERCISES

Multiply or divide. Write each answer in simplest form.

1.

1 2    2 3

1  3

2.

1 2 
 2 3

3  4

3.

4 1    5 6

2  15

4.

5 5 
 7 6

6  7

5.

4 3    5 4

3  5

6.

3 1 
 5 3

9 4  or 1  5 5

7.

4 2    7 3

8  21

8.

5 2 
 6 3

5 1  or 1  9. 4 4

3 5    4 6

5  8

Glencoe/McGraw-Hill

23

Algebra Intervention

10.

1 2 
 7 3

5 1    6 3

11.

3  14

13.

5  18

2 3    5 4

4 1    5 2

15.

3 1  or 1  2 2

7 2 
 9 3

6 2    7 3

2  5

3 4    8 5

17.

7 1  or 1  6 6

19.

21 1  or 5  4 4

1 1 
 6 9

14.

3  10

16.

7 1 
 8 6

12.

8 2 
 9 3

18.

3  10

4 1  or 1  3 3

3 2 
 7 3

20.

4  7

1 6    8 7

21.

9  14

3  28

A P P L I C AT I O N S 22. Of the 48 NBA World Championship Series from 1947 to 1994, 5

the Boston Celtics won  16 of the championships. Two thirds of the Celtics’ championships occurred before 1970. What fraction represents the championships that were won by the Celtics before 1970?

5  24

1

23. About  11 of the land in the continental United States is in 5 Texas. About  9 of the land in Texas is used as rural pastureland. What fraction of the land in the continental United States is Texas pastureland?

5  99

24. Helen planted vegetables and flowers in her garden. Three 1

fourths of her garden is planted in flowers. If  10 of the total garden is planted in roses, what fraction of the flower garden is planted in roses?

2  15

25. One third of the videos at Vinnie’s Video Store are appropriate 2

for young children. If  5 of the children’s videos are cartoons, what fraction of the videos in the store are children’s cartoons?

Glencoe/McGraw-Hill

24

2  15

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

13

Multiplying and Dividing Fractions

A new industrial park is being developed. The ABC Manufacturing 2

Company owns a rectangular piece of property that is  5 mile long and 1  mile wide. 4

EXAMPLES

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.

2 1 21      5 4 54 2

1

  20 or 10

Simplify. 1

The ABC Manufacturing Company owns  10 square mile of land. 1

The A to Z Distribution Company owns  8 square mile of land in the industrial park. If the land is in the shape of a rectangle and the 1

length of the land is  3 mile, what is the width of their land? To find the width, divide the area of the rectangle by the length. 1 1 1 3 
 =    8 3 8 1

1

Multiply by the reciprocal of
3. Multiply the numerators. Multiply the denominators.

13

 81 3

 8 3

The width of the land owned by A to Z Distributing Company is  8 mile.

EXERCISES 1.

2 1    3 4

1  6

Multiply or divide. Write each answer in simplest form. 2.

Glencoe/McGraw-Hill

1 2 
 4 5

5  8

3.

25

3 1    7 2

3  14

Algebra Intervention

4.

5 4 
 8 5

25  32

5.

1 3    3 5

1  5

6.

2 3 
 9 5

10  27

7.

1 6    2 7

3  7

8.

2 2 
 5 3

3  5

9.

3 1    8 6

1  16

10.

1 2 
 3 5

5  6

11.

7 5    10 7

1  2

12.

2 1 
 3 2

4 1  or 1  3 3

13.

2 5    3 6

5  9

14.

3 3 
 5 10

2

15.

3 1    4 3

1  4

16.

1 5 
 9 6

2  15

17.

2 5    3 7

10  21

18.

1 1 
 4 12

19.

4 5    7 9

20  63

20.

1 7 
 2 8

4  7

21.

2 2    3 3

3 4  9

A P P L I C AT I O N S 1

1

 22. About  8 of the world’s population lives in Africa. About 13 of the population of Africa lives in Ethiopia. About what 1  about fraction of the world’s population lives in Ethiopia? 104 1

5

 23. About  20 of the world’s water supply is fresh water. If about 7 of Earth’s surface is covered with water, about what fraction 4 about  28 of Earth is covered with fresh water? 1

24. Two thirds of Esma’s garden is planted in flowers. If  4 of the 1 flowers are gladiolas, what fraction of the garden is planted  6 in gladiolas? 3

25. One eighth of Jonas’ garden is planted in green beans. If  4 of his garden is planted in vegetables, what fraction of the vegetable garden is planted in green beans?

1  6

26. Three fourths of the books sold at Bernie’s Book Store are 1

paperbacks. If  3 of the paperbacks sold are adventure stories, what fraction of the books sold are paperback adventure books?

1  4

1

27. A honeybee can produce  10 pound of honey1 in its lifetime. How many honeybees does it take to make  2 pound 5 honeybees of honey? Glencoe/McGraw-Hill

26

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

14

Multiples

Bryan noticed that every time he spent $1 at the department store, he paid 8¢ in sales tax. He decided to make a table of the amount of sales tax charged on whole-dollar purchases.

EXAMPLE

Can you help him make the table? The amount of sales tax charged on whole-dollar purchases can be found using multiples of 8. A multiple of a number is the product of that number and any whole number.

EXERCISES

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¢

List the first four multiples of each number.

1. 10

2. 9

10, 20, 30, 40 4. 7

3. 15

9, 18, 27, 36

15, 30, 45, 60

5. 18

7, 14, 21, 28 7. 20

6. 12

18, 36, 54, 72 8. 25

20, 40, 60, 80 Glencoe/McGraw-Hill

12, 24, 36, 48 9. 16

25, 50, 75, 100 27

16, 32, 48, 64 Algebra Intervention

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

10. 56; 7

yes

A P P L I C AT I O N S

Kyle is planning a trip. He plans to drive 55 miles per hour. Use this information to answer 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 answers to 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 that are 4 inches long, and the next row has tiles that are 5 inches long. In how many inches will the ends of the two rows be even and the pattern start to repeat?

20 inches

Glencoe/McGraw-Hill

28

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

15

Percents as Fractions and Decimals

A stereo is on sale for 33 EXAMPLE

1  % off the original price. 3

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 r

in the form  100 and simplify. 3313 1   33 3 %  100

100

1

  3  100

1

100


Write 33
3 as 3 and multiply by the reciprocal of 100.

1

 3 r

To express a percent as a decimal, express the percent in the form  100 and then express the fraction as a decimal. 1

1

 33  3% 3

You found this in the example above.

 0.33

EXERCISES

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

1. 40% 2  ; 0.4 5

2. 8% 2  ; 0.08 25

3. 29% 29  ; 0.29 100

4. 55% 11  ; 0.55 20

5. 25% 1  ; 0.25 4

6. 81% 81  ; 0.81 100

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29

Algebra Intervention

7.

2

66  3%

8.

2  ; 0.67 3

10. 30% 3  ; 0.3 10

A P P L I C AT I O N S

98%

9. 16.5%

49  ; 0.98 50

33  ; 0.165 200

11. 240% 12. 0.05% 12 2 1  or 2  ; 2.4  ; 0.0005 5 5 2,000

Between 1980 and 1990, the population of New Hampshire increased by 20.5%. Use this information to answer Exercises 13–17.

13. Write this percent as a fraction in simplest form. 41  200 14. Write this percent as a decimal.

0.205 15. When is it best to use the percent instead of the fraction or the decimal?

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

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

Answers will vary. 18. Between 1975 and 1985, the disposable personal income in the United States more than doubled. Does this mean the income has increased by more than 200%? Explain.

yes; When something doubles, it increases by a 200 factor of 2 and 200%    2. 100

Glencoe/McGraw-Hill

30

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

16

Percent of a Number

To find the percent of a number, you can either change the percent to a fraction and then multiply, or change the percent to a decimal and then multiply.

EXAMPLE

In the Washington County championship basketball game, Lee made 55% of his 20 attempted field goals. How many field goals did he make? Find 55% of 20. Method 1 Change the percent to a fraction. 55

11

Method 2 Change the percent to a decimal. 55

 55%   100  20

55%   100 or 0.55

11   20  11 20

0.55  20  11

Lee made 11 field goals.

EXERCISES

Find the percent of each number.

1. 25% of 200 50

2.

30% of 55

16.5

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.

3. 3% of 610

140% of 125 175 9. 100% of 50

18.3

50

10. Which of the following does not belong? d a. 25% of 80 b. 80% of 25 c. 2.5% of 800 d. 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 they win? 21 games Glencoe/McGraw-Hill

31

Algebra Intervention

Write a percent to represent the shaded area. 12.

13.

30%

35%

14.

15.

57%

44%

A P P L I C AT I O N S 16. Kleema owns 40 music CD’s. Fifteen of her CD’s are recordings done by rap groups. What percent of her CD collection is rap music? 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? 0.15 b. What should their tip be? $5.25 c. What is their total food and tip bill? $40.25 18. Angie wants to put a winter coat in layaway at a store. To do so, she must pay the store 20% of the cost of the coat so they will hold it. If the coat costs $48.99, about how much of a deposit 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 Glencoe/McGraw-Hill

32

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

17

Percent Proportion

Use the percent proportion to solve problems dealing with percent. P r    B 100

EXAMPLES

P  percentage

B  base

37.2 is what percent of 186?

What number is 15% of 280?

P r    B 100

P r    B 100

37.2 r    186 100

P 15    280 100

(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.

EXERCISES

r   rate 100

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 3. What percent of 49 is 7?

base: 190, rate: 6.25% 4.

percentage: 7, base: 49

40% of what number is 82?

percentage: 82, rate: 40%

Write a proportion for each problem. Then solve. Round answers to 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

24 is 20% of what number?

120

9. 36 is 75% of what number?

48

10.

36% of what number is 18?

50 Glencoe/McGraw-Hill

33

Algebra Intervention

11. What percent of 224 is 28?

12.

12.5%

What number is 40% of 250?

100

13. 15% of 290 is what number?

43.5

14.

50% of what number is 74?

148 1

15. Use a proportion to find 55  2 % of 66. Round to the nearest tenth.

36.6 1

16. Use a proportion to find 19  4 % of 45. Round to the nearest tenth.

8.7

A P P L I C AT I O N S 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 at King 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 bus to school, about how many students do not ride the bus to school?

about 77 students 20. There are 20 students running for student council at Pine Bluff High School. If the school will elect a president, vice president, treasurer, and secretary, what percent of the students running will win in the election?

20% 21. There were 102,269 tickets available for a rock concert. If The Ticket Company sold 72.5% of the tickets available, about how many tickets did they sell for the concert?

about 74,145 tickets Glencoe/McGraw-Hill

34

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

18

Percent of Change

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

EXAMPLE

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 to the original amount. 137,053 r    2,913,808 100

137,053  100  2,913,808  r 13,705,300  2,913,808r 13,705,300 2,913,808r    2,913,808 2,913,808

5r The population of Iowa decreased by about 5%.

EXERCISES

Find the percent of change. Round to the nearest whole percent.

1. old: $5 new: $7

2. old: 45 students new: 50 students

40% increase 3. old: 32 dogs new: 30 dogs

6% decrease Glencoe/McGraw-Hill

11% increase 4. old: $56 new: $52

7% decrease 35

Algebra Intervention

5. old: 345 adults new: 450 adults

6.

30% increase

2% decrease

7. old: 150 pounds new: 138 pounds

8.

8% decrease

A P P L I C AT I O N S

old: $648 new: $635

old: 9.5 hours new: 8 hours

16% decrease

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

9. This year, the value of his car is $11,994. What was the percent change in the car’s value?

40% decrease

10. The year before last the value of his car was $24,500. What was the percent change in the car’s value? How does this change 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 blazers were put on sale at 65% off the listed price. Did the store break even, make a profit, or lose money? Explain.

The store lost money because 65% of the original list price is greater than 65% of the store’s cost.

Glencoe/McGraw-Hill

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Algebra Intervention

SKILL

19

Name ______________________________________ Date ___________

Solve Equations Involving Addition and Subtraction

Addition Property of Equality: If you add the same number to each side of an

equation, the two sides remain equal.

EXAMPLE

Solve t  57  46. t  57  46 t  57  57  46  57

Add 57 to each side.

t  103 Check:

t  57  46 ? 46 103  57 

Replace t with 103.

46  46 ✓ The solution is 103.

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

EXAMPLE

Solve t  24.4  25.1. t  24.4  25.1 t  24.4  24.4  25.1  24.4 t  0.7 Check:

t  24.4  25.1 ? 25.1 0.7  24.4  25.1  25.1

Subtract 24.4 from each side of the equation.

Replace t with 0.7. ✓

The solution is 0.7.

EXERCISES 1.

Complete each statement.

y  18  39 y  18  18  39  18 Glencoe/McGraw-Hill

2.

m  23  17 m  23  23  17  23 37

Algebra Intervention

Solve each equation. Check your solution. 3.

w  6  19

4. n  4.7  8.4

5.

18.42  t  63

7. e  0.9  17.4

8. b  43  18

13

6.

44.58

9.

3

h  32  5  44 3 76  5

12. g  6.3  9.5

13.1

61

947  p  43

11. 7.36  w  8.94

990 13.

15.8

A P P L I C AT I O N S

60

18.3

10.

m  18  78

1.58

r  18  36

14.

54

2.17  k  4.19

2.02

Each of Exercises 15–18 can be modeled by one of these equations: n  2  10 n  2  10 Choose the correct equation. Then solve the problem.

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 does she already have? n  2  10; $8 17. The width of the rectangle shown at the right is 2 inches less than the length. What is the length?

n  2  10; 12 inches 10 in.

18. In the figure at the right, the length of  AC  is 10 centimeters. The length of  BC  is 2 centimeters. What is the length of  AB ? n  2  10; 8 cm

Glencoe/McGraw-Hill

38

A

Algebra Intervention

B

C

SKILL

20

Name ______________________________________ Date ___________

Solve Equations Involving Multiplication and Division

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

EXAMPLE

Solve 156  4r. 156  4r 156 4r    4 4

Divide each side by 4.

39  r 156  4r

Check:

? 4  39 156 

Replace r with 39.

156  156

The solution is 39.

✓

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

EXAMPLE

w

Solve  21 = 4.2. w   4.2 21 w   21  4.2  21 21

Multiply each side by 21.

w  88.2 Check:

w   4.2 21 88.2 ?   4.2 21

4.2  4.2

Glencoe/McGraw-Hill

39

Replace w with 88.2. ✓

The solution is 88.2.

Algebra Intervention

EXERCISES 1.

Complete the solution of each equation.

12h  48

r

34   3

2.

48 12h    12 12

h 

r

34  3   3  3

102  r

4

Solve each equation. Check your solution. 3. 3.6t  11.52 3.2 6.

1.4j  0.7 0.5

9.

1.3z  3.9 3

4.

n   15 4

7.

4.1m  13.12 3.2 8.

10.

7 1   f 8 2

60

5.

7  4

11.

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

A P P L I C AT I O N S

3  4

1 3 w   2 8 c   16 5

80

d   0.6 3.5 1

2.1

1

 c
 4  2

1  8

Each of Exercises 15–17 can be modeled by one of these equations: 2n  10

n   10 2

Choose the correct equation. Then solve the problem. 15. Chum earned $10 for working two hours. How much did he earn per hour?

2n  10; $5

16. Kathy and her brother won a contest and shared the prize equally. Each received $10. What was the amount of the prize? n   10; $20 2 17. In the triangle at the right, the length of  PQ  is twice the length of Q QR R . What is the length of  ?

Q 60° 10 cm

2n  10; 5 cm

P

Glencoe/McGraw-Hill

40

30°

Algebra Intervention

R

SKILL

Name ______________________________________ Date ___________

21

Solve Two-Step Equations

To solve two-step equations, you need to add or subtract first. You also need to multiply or divide.

EXAMPLES

Solve each equation. 7v  3  25 7v  3  3  25  3

Add 3 to each side.

7v  28 7v 28    7 7

Divide each side by 7.

v  4 The solution is 4. 1  (r  3)  –5 6 1

6  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 The solution is –27.

EXERCISES

Name the first step in solving each equation. Then solve each equation.

1. 6n  2  22

Add 2 to each side; 4

Glencoe/McGraw-Hill

2.

1  (y  3)  12 2

Multiply each side by 2; 27

41

Algebra Intervention

Solve each equation. 3.

–5t  5  –5

4.

–5h  6  24

7.

0

6.

k

7  4 9

5.

6  3b  –9

8.

5

–6

9.

4x  5  15

–3.5

5

10.

8

24  17  2c

12  4n  4

2

5  (d  20)  –10 7

–34

11.

2  (a  18)  –6 3

9

Translate each sentence into an equation. Then solve the equation. 12. Six less than a number divided by 3 is 12. n   6  12; 54 3 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

A P P L I C AT I O N S 15. On a July day in Detroit, Michigan, the temperature rose to 9

80°F. Find this temperature in degrees Celsius. (F =  5 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 the cost be for a 2-mile trip?

$3 17. Three pens cost $1.55 including $0.08 sales tax. How much did each pen cost?

$0.49

Glencoe/McGraw-Hill

42

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

22

Use an Equation

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

EXAMPLE

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  6.00 0.25s  3.50  3.50  6.00  3.50 0.25s  2.50 0.25s
0.25  2.50
0.25 s  10

Subtract 3.50 from each side. Divide each side by 0.25.

Lucy bought 10 stickers.

EXERCISES

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. The result is 26. What is the number?

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

15

Glencoe/McGraw-Hill

43

Algebra Intervention

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

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

10

A P P L I C AT I O N S 7. Ruiz earned $117. If his pay is $6.50 per hour, how many hours did he work?

18 hr 8. There are 425 students at Dayville Elementary School. If 198 of the students are girls, how many students are boys?

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

111 miles 10. Pachee bought some baseballs for $4 each and a batting glove for $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. How much more money must he save?

$260 12. Cindy went to the hobby shop and bought 2 model sports cars at $8.95 each and some paints. If she spent $23.65, what was the cost of the paints?

$5.75 13. Arlen drove for 3 hours at 52 miles per hour. How fast must he drive during the next 2 hours in order to have traveled a total of 254 miles?

49 mph 14. Postage costs $0.29 for the first ounce and $0.23 for each additional ounce. Peter spent $1.44 to send a package. How much did it weigh?

6 oz Glencoe/McGraw-Hill

44

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

23

Proportions

A proportion is an equation that shows that two ratios are equivalent. The cross products of a proportion are equal.

EXAMPLE

2

12

 Determine if the ratios  3 and 18 form a proportion. 2

12

 Find the cross products of  3  18 .

2  18  36 3  12  36 12

2

 So,  3  18 is a proportion.

If one term of a proportion is not known, you can use cross products to find

the term. This is called solving the proportion.

EXAMPLE

r

7

 Solve  24  8 . r 7    24 8

r  8  24  7

Find the cross products.

8r  168 8r 168    8 8

Divide each side by 8.

r  21 Therefore, r equals 21.

EXERCISES 1.

2 5    n 10

Solve each proportion. 2.

4

5 m    8 24

3.

15

Glencoe/McGraw-Hill

12 k    20 15

9

45

Algebra Intervention

4.

3.5 16    m 32

5.

7 7.

6 n    30 50

6.

10

f 2    0.8 8

8.

0.2

75 6    r 2

25

15 t    120 16

9.

2

7 c    9 36

28

A P P L I C AT I O N S 10. Holly was absent from school 8 out of 36 days. Juan was absent 9 out of 45 days. Do these ratios form a proportion?

no 11. Denise needed 4 hours to paint 1,280 square feet of wall space. How much time would she need to paint 1,600 square feet of space?

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

distance if the map distance is 4  2 inches?

562.5 miles 13. If you spend 1.5 hours per day doing homework, how many hours would you spend doing homework in 8 days?

12 hours 14. Jenny got 3 hits in her first 8 at-bats this season. How many hits 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 the rest in a savings account. If Josh earns $32.00 per week cutting lawns, how much does he save per week?

$14.72

Glencoe/McGraw-Hill

46

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

24

Proportional Reasoning

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

EXAMPLE

How many perch should the ranger stock? 296 ← sunfish sunfish → 4    3 p perch → ← perch

4p  296  3 Cross multiply. 4p  888 4p 888    4 4

Divide each side by 4.

p  222 The ranger should stock 222 perch.

EXERCISES

Write a proportion to solve each problem. Then solve.

1. 40 nails hold 5 rafters. 96 nails hold r rafters. 2. 2 quarts fill 8 cups. 5 quarts fill c cups.

40 96    ; 12 rafters 5 r

2 5    ; 20 cups 8 c

3. 81 rivets on 3 panels. r rivets on 13 panels.

81 r    ; 351 rivets 3 13

4. 32 addresses are on 2 pages of the address book. a addresses are on 9 pages. 5. 60 sliced mushrooms on 4 pizzas. m sliced mushrooms on 15 pizzas. 6. 98 beats a minute. y beats per hour.

Glencoe/McGraw-Hill

32 a    ; 144 addresses 2 9

60 m    ; 225 mushrooms 4 15 98 y    ; 5,880 beats 1 60

47

Algebra Intervention

A P P L I C AT I O N S

Solve by using proportional reasoning.

7. Naturalists can determine the number of fish in a pond by using 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 and mark 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. Record the number of recaptured pieces and the number recaptured with an X. Return the pieces and repeat the recapture 9 times. Find the sum of the recaptured pieces and the sum of the recaptured ones with an X. Use the following proportion to determine the total number of pieces in the bag. original number captured total recaptured with an X    number in the bag total recaptured

Answers will vary.

8. A shop produces 47 surf boards in 6 days. How long will it take 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

Glencoe/McGraw-Hill

48

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

25

Scale Drawings

Chuck has a scale drawing of Detroit’s Tiger Stadium. The scale of the 1

drawing is  4 inch equals 25 feet. On the drawing, the home-run distance 1

from home plate to right field is 3  4 inches.

EXAMPLE

What is the actual home-run distance from home plate to right field? 1

1

 Think of  4 inch as 0.25 inch and 3 4 inches as 3.25 inches. Use the scale 0.25 inch equals 25 feet and write a proportion to find the actual distance. drawing 3.25 ← drawing → 0.25

 

actual distance → 25  x ← actual distance

0.25x  25  3.25

Cross multiply.

0.25x  81.25 0.25x 81.25    0.25 0.25

Divide each side by 0.25.

x  325 The actual distance is 325 feet.

EXERCISES 1. 3 inches

On a map, the scale is 1 inch equals 150 miles. For each map distance, find the actual distance. 2. 8 inches

450 miles

4. 5 inches

3.

1,200 miles

75 miles

1

1

5. 1  2 inches

750 miles

Glencoe/McGraw-Hill

1  inch 2

6. 4  2 inches

225 miles

675 miles

49

Algebra Intervention

1

On a scale drawing of a floor plan for a new building, the scale is  4 inch equals 1 foot. Find the actual dimensions of the rooms if the measurements from the drawing are given. 7. 5 inches by 3 inches

8.

20 ft by 12 ft

8 ft by 16 ft

1

9. 2 inches by 3  2 inches

1

18 ft by 18 ft

1

 11. 3  4 inches by 2 2 inches

13 ft by 10 ft

A P P L I C AT I O N S

1

 10. 4  2 inches by 4 2 inches

8 ft by 14 ft 1

2 inches by 4 inches

12.

3

1

 3 4 inches by 4 4 inches

15 ft by 17 ft

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

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

Scale _____ 1 inch = 8 feet

Storage 11–4 in.

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

Living 1 in.

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

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

Living 1 in.

Recreation Living 1 in.

11–2 in.

Entry 3 – in. 4

17. Estimate the actual distance from the entry chamber to the back of the storage chamber.

about 28 ft Glencoe/McGraw-Hill

50

Algebra Intervention

Living 1 in.

SKILL

Name ______________________________________ Date ___________

26

Square Roots

If a  b, then a is the square root of b. 2

EXAMPLE

Joanna wants to buy a house. The realtor told her that the family room in a certain house has a floor area of 144 square feet. What is the length of a side of the room if all four sides of the room are the same length? If all four sides of the room are the same length, then the room is shaped 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 s2 144   12  s

To solve this equation, find the square root of each side. The square root of 144 is 12.

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

EXERCISES

Find each square root.

1.

9 3

2.

25 5

3.

81 9

4.

169 13

5.

36 6

6.

16 4

7.

64 8

8.

121 11

9.

100 10

10.

400 20

11.

900 30

12.

10,0 00 100

13.

196 14

14.

0.0 9 0.3

15.

0.8 1 0.9

16.

1.4 4 1.2

Glencoe/McGraw-Hill

51

Algebra Intervention

17.

21.

0.4 9 0.7

 4  9

2  3

18.

22.

A P P L I C AT I O N S

.04 0  0.2

 16  25

19.

4  5

23.

2.2 5 1.5

 49  100

20.

7  24. 10

0.1 6 0.4 5  6

 25  36

The area of a square picture is 64 square inches. Use this information to answer Exercises 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 square inches?

11 inches 27. Will a square mat with an area of 81 square inches be large enough 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 in on 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 254 square 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 the sides of the box at their midpoints, what is the diameter of the pizza?

10 inches

Glencoe/McGraw-Hill

52

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

27

Ordered Pairs

A horizontal number line and a vertical number line meet at their zero

points 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

EXAMPLES

y-coordinate

Name the ordered pair for point P. Start at O. Move along the x-axis until you are above point P. Then move down until you reach point P. Since you moved 4 units to the right and 3 units down, the ordered 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.

EXERCISES

y 5 (-2, 4) 4 3 2 4 -2 1

O

-5 -4 -3 -2 -1 -1 -2 -3 -4 -5

4 x 1 2 3 4 5 -3 P

Name the ordered pair for each point.

1. G (–1, 4)

3. J (–3, –4)

5. M (5, –2)

Glencoe/McGraw-Hill

2. H (5, 3)

4. K (3, –3)

6. N (2, 0)

53

y 5 G 4 3 2 1

H

O

-5 -4 -3 -2 -1 -1 -2 -3 -4 J -5

Algebra Intervention

x N 1 2 3 4 5 M K

Graph and label each point. 7. A(–5, 5)

y

8.

B(2, 4)

9. C(0, 5)

10.

D(–4, 0)

11. E(2, 2)

12.

F(4, –3)

A P P L I C AT I O N S

5 4 3 2 1 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5

x

O 1 2 3 4 5

A botanist is interested in what part of a certain leaf is being infested by an insect that leaves black spots. She places a clear coordinate plane over several leaves that are about the same size and shape. Complete each of the following.

13. Find the coordinates of the black spots on the leaf at the right.

(0, 2), (2, –2) (7, 3), (8, –3) 1

(–5  , 1), (–3, –5) 2

14. Draw and label the spots having the following coordinates on the leaf at the right. A(2, –3) E(–5, 3)

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

Glencoe/McGraw-Hill

54

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

28

Function Tables

The data at the right shows the shipping and

Maximum Purchase (dollars) 50 100 150 200 250 300 350

handling charged by a catalog company.

EXAMPLE

Complete the table.

First look for a pattern in the data that is already given. Each entry in the shipping and handling column is $3 greater than the previous entry. So, to complete the table, add $3 to each entry in the second column to get the next entry. The entries for the last 3 rows of the table are given below.

EXERCISES 1.

Maximum Purchase

Shipping and Handling

250 300 350

18.95 21.95 24.95

Complete each table. 2.

Distance (feet)

Time (seconds)

10

5

7.5

1,500

15

10

15

2,000

20

15

22.5

2,500

25

20

30

3,000

30 35 40 45

25

37.5

30

45 52.5 60

Principal (dollars)

Interest (dollars)

1,000

3,500 4,000 4,500

Shipping and Handling (dollars) 6.95 9.95 12.95 15.95

Glencoe/McGraw-Hill

35 40 55

Algebra Intervention

3.

Purchase (dollars)

Tax (dollars)

4.

Length of call (minutes)

Cost (dollars)

10

0.60

1

1.00

20

1.20

2

1.35

30

1.80

3

1.70

40

2.40

4

2.05

50

3.00 3.60 4.20 4.80

5

2.40 2.75 3.10 3.45

60 70 80

A P P L I C AT I O N S

6 7 8

The table at the right shows the amount of Federal individual income tax for 1993 for different amounts of adjusted gross income between $22,100 and $53,500 for single taxpayers. Use the data to answer Exercises 5–7.

Adjusted Gross Income (dollars) 25,000 30,000 35,000 40,000 45,000 50,000

Income Tax (dollars) 7,000 8,400 9,800

11,200 12,600 14,000

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 contains more data? Why or why not? How can you add more data to the 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 be

17,050, 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.

Glencoe/McGraw-Hill

56

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

29

Graphing Functions

gallons of gasoline. The function table at the right shows this relationship.

EXAMPLE

Graph the function.

Gallons of Gasoline 2 4 6 8 10 12 14 16

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

Miles 25 50 75 100 125 150 175 200

200 Miles

Carin’s motor home averages about 25 miles on two

150 100 50 0

EXERCISES 1.

4 8 12 16 Gallons of Gasoline

Graph each function.

Length of Side (cm) 1 2 3 4 5 6 7

Glencoe/McGraw-Hill

Area (sq cm) 1 4 9 16 25 36 49

57

Algebra Intervention

2.

Time (years) 1 2 3 4 5 6 7

A P P L I C AT I O N S

Savings (dollars) 100 250 150 300 600 550 650

The function table at the right shows the apparent temperature for the given room temperatures for a relative humidity of 80%. Use the data to answer Exercises 3–5.

Room Temperature (in °F) 69 70 71 72 73 74 75

Apparent Temperature (in °F) 70 71 73 74 75 76 77

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 temperatures of 70°F and 71°F; Answers will vary.

Glencoe/McGraw-Hill

58

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

30

Solve Equations With Two Variables

An ordered pair that makes an equation true is a solution for the equation. Find four solutions for the equation y  5x  1.

EXAMPLE

Choose values for x. Let Let Let Let

x  –4. x  –2. x  0. x  2.

Calculate y values.

Write ordered pairs.

y  5(–4)  1  –21 y  5(–2)  1  –11 y  5(0)  1  –1 y  5(2) 1  9

(–4, –21) (–2, –11) (0, –1) (2, 9)

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

EXERCISES

Complete the table for each equation. Then use the 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

x

3x  2

y

x

4x

y

1 2 3 4

3(1)  2 3(2)  2 3(3)  2 3(4)  2

5 8 11 14

–1 0 1 2

4(–1) 4(0) 4(1) 4(2)

–4 0 4 8

(1, 5), (2, 8), (3, 11), (4, 14)

(–1, –4), (0, 0), (1, 4), (2, 8)

x

–3x  4

y

–1 –3(–1)  4 0 –3(0)  4 1 –3(1)  4 2 –3(2)  4

–1 –4 –7 –10

(–1, –1), (0, –4), (1, –7), (2, –10)

Find four solutions for each equation. Write your solutions as ordered pairs. Answers will vary. Samples are given. 4. y  x  4

(0, –4), (1, –3), (2, –2), (3, –1)

5. y  3x  1

Glencoe/McGraw-Hill

(–1, –2), (0, 1), (1, 4), (2, 7)

59

6. y  –3

(0, –3), (1, –3), (2, –3), (3, –3)

Algebra Intervention

7.

y  –2x 2

8. y  2.5x

(–2, 2), (–1, 0), (0, –2), (2, –6)

1

10. y  –  2x4

9. y  –2x  4

(–2, –5), (–1, –2.5), (0, 4), (1, 2), (0, 0), (1, 2.5) (2, 0), (3, –2)

1

11. y   3x1

(–2, –3), (0, –4), (2, –5), (4, –6)

12.

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

1

y  2x3

(–2, 2), (0, 3), (2, 4), (4, 5)

A P P L I C AT I O N S 13. One number is three more than half another number. Determine which ordered pairs in the set {(0, 3), (–2, 2), (4, –1), 1

(1, 3  2 )} are solutions for the two numbers. 1 (0, 3), (–2, 2), (1, 3  ) 2

14. An organization donates one third of all the money it raises for housing the homeless. How much will it donate if it raises $6,000? 1 y  x; $2,000 3

15. You can show the distance in feet it takes a car to stop when traveling at a certain speed on a dry, concrete surface by using the formula d  0.042 s2  1.1s. Complete the table to find the distance for each speed. Round the distances to the nearest foot. speed in mph (s)

30

35

distance in feet (d)

71

90

Glencoe/McGraw-Hill

40

45

50

55

60

65

70

75

111 135 160 188 217 249 283 319

60

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

31

Graphing Equations

EXAMPLE

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

y 8 6 4

y  2x  2

EXERCISES

x

2x  2

y

(x, y)

0 1 2 3

2(0)  2 2(1)  2 2(2)  2 2(3)  2

2 4 6 8

(0, (1, (2, (3,

2 x

2) 4) 6) 8)

-2 O

2

4

-2

Complete each function table. Then graph the equation.

1. y  x  1

y 6

x

x1

y

1 2 3 4 5

11 21 31 41 51

0 1 2 3 4

(x, y)

(1, (2, (3, (4, (5,

4

0) 1) 2) 3) 4)

2 x -2 O

2

4

6

2

4

6

-2

2. y  5  x

y 6

x

5x

y

0 1 2 3 4

50 51 52 53 54

5 4 3 2 1

Glencoe/McGraw-Hill

(x, y)

(0, (1, (2, (3, (4,

4

5) 4) 3) 2) 1)

2 x -2 O -2

61

Algebra Intervention

6

8

Graph each equation. 3. y  x  2

1

4. y  3x

y

5. y   2 x1 y

y

6

6

6

4

4

4

2

2

2

x -2 O

2

4

6

-2

x -2 O

2

4

6

x -2 O

-2

2

4

6

-2

A P P L I C AT I O N S y

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

120 80

y  50x  40

40 x O

1

7. A blizzard at the Slippery Ski Area deposited  2 foot of snow per hour atop a 3-foot snow base. Let x represent the number of hours and y represent the total amount of snow. Write an equation to represent the total amount of snow. Graph the equation. 1 y  x3 2 8. Yukari averages 40 miles per hour when she drives from Los Angles to San Francisco. Let x represent the number of hours and y represent the distance traveled. Write an equation to represent the distance traveled. Graph the equation.

y  40x

1

2

3

2

4

6

1

2

3

y 6 4 2 x O y 120 80 40 x O

Glencoe/McGraw-Hill

62

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

32

Slope of a Line

The graph of a line is shown below. y

x O

EXAMPLE

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

y run = –5

(–2,8) 1. Choose any two points on the line. The points chosen at the right have coordinates (3, 4) and (–2, 8). 2. Draw a vertical line and then a horizontal line to connect the two points. O 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.

rise

rise = 4 (3,4)

4

 5. slope   run  –5

Glencoe/McGraw-Hill

63

Algebra Intervention

x

EXERCISES 1.

Find the slope of each line shown.

–1

y

2.

x

x

O

3.

1  3

y

O

–3  4

y

4.

x

x

O

O

Paula works as a sales representative Paula’s Monthly Earnings for a computer manufacturer. She 6,000 earns a base pay of $1,000 each 5,000 month. She also earns a commission 4,000 based on her sales. The graph at the 3,000 right shows her possible monthly 2,000 earnings. Use the graph to answer 1,000 Exercises 5–8. 0 Pay

A P P L I C AT I O N S

4

y

5,000

10,000 15,000 Sales

5. What is the slope of the line? 1  5 6. What information is given by the slope of the line? 1 The rate of commission Paula earns is  or 20% of her sales. 5 7. If Paula’s base pay changed to $1,100, would it change a. 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 1 the graph? Why or why not? Yes, the slope would be  . 4 Glencoe/McGraw-Hill

64

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

33

Graphing Exponential Equations

Jamie conducted an experiment that began with 400 bacteria. He found that the number of bacteria, y, after x hours was given by the equation y  400(2x).

EXAMPLE

Use a graphing calculator to graph this equation. Follow the steps below to graph the equation. key. Then enter the equation by pressing 400  2

1. Press the X,T,


.

2. Press WINDOW 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 GRAPH to draw the graph shown below.

EXERCISES

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

Glencoe/McGraw-Hill

65

Algebra Intervention

1 x

3. y 
 8

4.

y  22x

5. y  30(0.5x)

6.

y  500(0.25x)

A P P L I C AT I O N S

Carbon-14 has a half-life of 5,730 years. Manford has a sample that contains 200 g of 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 x hours was given by the equation y  200(3x). Use a graphing calculator to graph this equation. How would you use the information shown on this graph?

See students’ work. Answers will vary. Glencoe/McGraw-Hill

66

Algebra Intervention

SKILL

34

Name ______________________________________ Date ___________

Graphing Linear and Exponential Equations

The value of manufacturing equipment with an initial value of $25,000

depreciates at a rate of 10% a year. The value, V, of the equipment after n years is given by the equation V
25,000(1  0.10)n.

EXAMPLE

Use a graphing calculator to graph this equation. Before you graph the equation on a graphing calculator, you must rewrite the equation using Y for V and X for n. So the equation you will use is Y
25,000(1  0.10)X. Then follow the steps below to graph the equation. 1. Press the key. Use the key to delete any equations from the Y
list. Enter the equation by pressing X,T,
. 25000  ( 1  . 1 ) 2. Press WINDOW 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 GRAPH to draw the graph shown below.

Glencoe/McGraw-Hill

67

Algebra Intervention

EXERCISES

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 P P L I C AT I O N S

A $500 deposit is made into an account that earns 3.5% interest and is compounded monthly. If no deposits or withdrawals are made, the amount of money, A, in this account after t years is given by the equation 0.035

12t A
500
1   12  .

7. Change A and t to y and x and use a graphing calculator to graph 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 give you a better graph than your original boundaries? Why or why 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 an average speed of 157.2 mph. Write an equation that can be used to compute the distance, d, traveled after t hours. Use a graphing calculator to graph your equation.

d
157.2t; Graphs will vary. Glencoe/McGraw-Hill

68

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

35

Sums of Angles of Polygons

A 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 sum of the measures of the angles of any polygon.

EXAMPLE

Find the sum of the measures of the angles 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 of the figure is 900°.

EXERCISES

Pick one vertex and draw all the diagonals possible from that vertex. How many diagonals can be drawn from that one vertex in each figure below?

1.

2.

2

3

3.

4.

1

0 Glencoe/McGraw-Hill

69

Algebra Intervention

Find the sum of the measures of the angles of each of the following polygons. 5. quadrilateral

6.

360

triangle

180

7.

8.

STOP 720

1,080

A P P L I C AT I O N S 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 is the 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°

Glencoe/McGraw-Hill

70

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

36

Similar Figures

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

EXAMPLE

Determine if each pair of figures is similar.

The figures are the same shape. Therefore, the figures are similar.

EXERCISES

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

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

1.

yes

2.

yes

3.

no

4.

yes

5.

no

6.

Glencoe/McGraw-Hill

yes

71

Algebra Intervention

7. List the pairs or groups of similar figures. a.

b.

c.

d.

e.

f.

a, d, and e; c and f

A P P L I C AT I O N S 8. List three real-world examples of similar figures.

Sample answers: T-shirt in different sizes, a photograph and an enlargement 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 is not the same size as the original figure.

See students’ work. 10. Use toothpicks to make a figure. Then make two 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.

Glencoe/McGraw-Hill

72

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

37

Similar Triangles

The triangles below are similar. 8 cm 4 cm

4 cm

2 cm

3 cm

6 cm

EXAMPLE

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. side of the first triangle  side of the second triangle

3 1 
 6 2

2 1 
 4 2

4 1 
 8 2 1

The ratios of the corresponding sides all equal  2.

EXERCISES

Use the similar triangles below to answer Exercises 1–3.

1. Measure each side of each triangle to the nearest centimeter. Glencoe/McGraw-Hill

73

Algebra Intervention

2. Find the ratios of the corresponding sides.

9 3 9 3 6 3 
, 
, 
 6 2 6 2 4 2

3. What do you notice about the ratios of the corresponding sides? 3 All the ratios equal  . 2

Determine if each pair of triangles is similar. 4.

5. 12 in. 6 in.

18 in. 9 in.

8 in.

12 cm

11 in.

15 cm

20 cm

16 cm

9 cm

no

12 cm

yes

Find the value of x in each pair of similar triangles. 6. 10 ft

7.

26 ft 5 ft 24 ft

13 ft

12 m

9m x

12 ft

4m

3m

x

5m

15 m

A P P L I C AT I O N S 8. A lamppost casts a shadow 16 feet. A girl standing nearby casts a shadow of 4 feet. The two triangles formed are similar. If the girl is 5 feet tall, how tall is the lamppost? ?

20 ft

5 ft 16 ft

4 ft

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

32 m 10

40

m

m

8m

? m 10

40 Glencoe/McGraw-Hill

74

m

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

38

Congruent Figures

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

EXAMPLE

Determine if each pair of figures is congruent.

The figures are the same shape, but not the same size. The figures are not congruent.

EXERCISES 1.

The figures are the same size, but not the same shape. The figures are not congruent.

The figures are the same shape and the same size. The figures are congruent.

In Exercises 1–3, determine whether each pair of figures is congruent. 2.

no

3.

yes

Glencoe/McGraw-Hill

yes

75

Algebra Intervention

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

b.

c.

d.

e.

f.

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

6.

7.

8.

A P P L I C AT I O N S 9. An architect wants to use ceramic tiles on the floor of a building she is designing. She wants to create a design using four-by-four squares of tiles. She plans to divide each four-byfour square into two congruent halves. There are six ways to divide the squares in this manner. One way is shown at the right. Show the five other ways below.

10. Draw a six-by-six tile pattern. Show at least three ways it can be divided into congruent halves. See students’ work. Glencoe/McGraw-Hill

76

Algebra Intervention

SKILL

39

Name ______________________________________ Date ___________

Reflections

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

EXAMPLE

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. Figure

EXERCISES

Reflection

Reflect each figure over the given line.

1.

2.

Glencoe/McGraw-Hill

77

Algebra Intervention

3.

4. x x

Use your reflections to answer Exercises 5–8. 5. Are the reflections in Exercises 1–4 smaller, larger, or the same size 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 its reflection? no; no 7. In Exercise 3, the x’s and the dot are in a straight line. In the reflection, 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

A P P L I C AT I O N S

M. C. Escher used transformations such as reflections to create interesting art. A simple example of his type of art starts with a square. A simple change is made and this change is reflected over the dashed line. Other reflections are made over other dashed 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. Glencoe/McGraw-Hill

78

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

40

Dilations and Rotations

In mathematics, there are several ways that a figure may be moved or changed. Two of these ways are dilations and rotations.

EXAMPLES

Draw the image of the triangle ABC for a dilation with a scale factor of 2. Draw a dashed line from the origin of the coordinate plane to point A. Extend the dashed line so that its length is twice as long as the distance from the origin to point A. This is one vertex of the dilated triangle. Repeat the procedure for the other two vertices and draw the dilated triangle.

y

B

A

x

C O

Draw three rotated images of triangle DEF. Rotate the image around the 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 the image point must be the same distance from the origin as the original point. In this case the image of (0, 3) is (3, 0). Find the image points for the other two vertices and draw the rotated triangle. Rotate the image two more times.

Glencoe/McGraw-Hill

79

y E

D

F O

Algebra Intervention

x

EXERCISES

Draw a dilation for the given scale drawing.

1. Scale factor: 3

2.

1

Scale factor:  2

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

A P P L I C AT I O N S 7. Does the movement of a Ferris wheel represent a dilation or a rotation? rotation 8. Does an enlargement of a photograph represent a dilation or a rotation? dilation 9. Make a design using rotations. See students’ work. 10. Make a design using dilations. See students’ work. Glencoe/McGraw-Hill

80

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

41

Translations

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

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

B

1 2 3 4 5 A

1 C 2 3

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.

EXERCISES

Translate each figure as indicated.

1. 7 units to the left

Glencoe/McGraw-Hill

2. 8 units to the right and 2 units down

81

Algebra Intervention

3. 5 units to the right and 2 units up

4.

2 units to the left and 1 unit down

5. 5 units to the left and 2 units down

6.

6 units to the right and 4 units up

Answer each question. 7. Are the translated figures congruent or similar to the original figures? congruent 8. In Exercise 5, are the arrows pointing in the same direction? Is direction 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 the translation, 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

A P P L I C AT I O N S 11. Describe the dive from A to B in terms of a translation.

4 units to the right and 6 units down

A

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

Glencoe/McGraw-Hill

82

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

42

Perimeter and Area

EXAMPLE

Tova Albert wants to make a garden with a perimeter of 54 feet because that is the amount of fence that she has. She wants the least area possible because she doesn’t have that much space in her yard. What should be the dimensions of her garden? Dimensions

Perimeter

Area

1  26 2  25 3  24 4  23

54 54 54 54

26 50 72 92

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

EXERCISES

Find the perimeter and area of each figure.

1.

2.

P
18 units A
18 units2

4.

3.

P
16 units A
16 units2

5.

P
20 units A
24 units2

Glencoe/McGraw-Hill

P
16 units A
12 units2

6.

P
18 units A
12 units2

83

P
18 units A
14 units2

Algebra Intervention

A P P L I C AT I O N S 7. A cardboard tube has a circumference of 7 inches and a length of 15 inches. When it is cut straight down its length, it becomes a rectangle. How much cardboard is used to make this 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 is space for the length to be up to 20 feet, but the width cannot be 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 that the bottom has an area of 96 square inches but has the least perimeter possible. What would be the whole number dimensions of the bottom of the box?

8 in.  12 in.

Glencoe/McGraw-Hill

84

Algebra Intervention

SKILL

43

Name ______________________________________ Date ___________

Area of Circles

The parts of a circle are illustrated ra di us

at the right. Notice that the radius is one-half of the diameter.

er et am di

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

center

A

r 2 The value of
is approximately 3.14.

EXAMPLE

Find the area of the circle.

14

The diameter of the circle is 14 meters. The radius of the 1

s

er

et

m

circle is  2 (14) or 7 meters. A

r2 A

(7)2 A

(49) A  3.14(49) A  153.86

Use 3.14 for
.

The area is about 153.86 square meters.

Find the area of each circle. Use 3.14 for
.

EXERCISES 1.

2. 6m

3. 22

ete

rs

113.04 m2 Glencoe/McGraw-Hill

8

inc

in

ch

he

s

es

379.94 in2

200.96 in2 85

Algebra Intervention

4.

5. 14 meters

6. 16 feet 3 centimeters

153.86 m2 7.

803.84 ft2

28.26 cm2

8.

16 kilometers

200.96 km2

9. 9 kilo

mete

24 fe

rs

63.585 km2

et

1,808.64 ft2

A P P L I C AT I O N S 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 its area? about 346 ft2 15. A California earthquake in 1989 sent horizontal shock waves about 60 miles from its epicenter. Find the area affected by the earthquake. about 11,304 mi2 16. The stage of a theater is a semicircle. If the radius of the stage is 28 feet, what is the area of the stage? about 1231 ft2

Glencoe/McGraw-Hill

86

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

44

Area of Rectangles

Area is the number of square units needed to cover a surface. The area of

a rectangle is the product of its length (
) and its width (w). A

w

EXAMPLE

Find the area of the rectangle at the right.

4m

A

w A
94 A
36

9m

The area of the rectangle is 36 square meters.

EXERCISES

Find the area of each rectangle.

1.

2.

3.

3 ft 3 cm

12 ft

10 yd

5 cm

36 ft2

15 yd

15 cm2

4.

150 yd2

5.

6. 3 in.

6m

18 cm 7 in.

9m

54

20 cm

m2

21

7.

in2

360 cm2

8.

2m

5 in.

9.

9m

3 ft

13 in.

3 ft

65 in2 Glencoe/McGraw-Hill

18 m2

9 ft2 87

Algebra Intervention

A P P L I C AT I O N S

Find the area of each playing field.

10. volleyball court

11.

polo field

5 YD

' 30

30 YD MARK 40 YD MARK

60 YD MARK

CENTER OF FIELD

60 YD MARK

GOAL LINE

40 YD MARK 30 YD MARK

7'

60'

GOAL POSTS

7'

MARKS ON GUARD BOARDS

160 YD

15 YD

MARKS ON GUARD BOARDS

TURF

8'

300 YD

1,800 ft2

48,000 yd2

12. four-wall handball court

13.

squash court

20' 12'

40'

20'

18' 6" 32'

800 ft2

592 ft2

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

Soccer Field Size Maximum Minimum

225 ft by 360 ft 195 ft by 330 ft

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 soccer field 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 much will it cost to carpet the room?

$885.00 Glencoe/McGraw-Hill

88

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

45

Area of Triangles and Trapezoids

A triangle is a polygon that has three sides. The area of a triangle is equal to one-half the product of its base and height.

EXAMPLE

Find the area of the triangle shown at the right.

5m

1

A
 2 bh

12 m

1

A
 2  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 of

a trapezoid is equal to the product of half the height and the sum of the bases.

EXAMPLE

Find the area of the trapezoid shown at the right.

12 cm

1

8 cm

A
 2 h(a  b) 1

A
 2 (8)(15  12)

15 cm

1

A
 2 (8)(27) A
108 The area of the trapezoid is 108 square centimeters.

EXERCISES 1.

Find the area of each triangle or trapezoid. 2.

3.

5 cm 3 cm

4m

20 in. 7 cm 3m

6 m2

24 in.

18 cm2 Glencoe/McGraw-Hill

240 in2 89

Algebra Intervention

4.

5.

20 ft

6.

4m

8 mm 40 mm 4m

15 ft

6m

12 ft

240 ft2

160 mm2

7.

8.

18 yd

3.7 cm

3.2 cm

20 m2 9.

8 in.

7 yd

5 in.

8 yd

5.92 cm2 10.

91 yd2

20 in2

11.

8 ft

12.

7 ft

4.7 m 3.1 m

9 cm

16 ft

8.6 m 15 cm

84 ft2

67.5 cm2

20.615 m2

A P P L I C AT I O N S 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 Glencoe/McGraw-Hill

90

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

46

Area of Irregular Shapes

One way to estimate the area of an irregular figure is to find the mean of the inner measure and the outer measure of the figure.

EXAMPLE

Shaun had his friend draw the outline of his body on a piece of paper. The diagram at the right shows this outline on a piece of grid paper in which 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 200  1,225

mean: 
712.5 square inches 2 An estimate of the area of Shaun’s body is 712.5 square inches.

EXERCISES 1.

4.

Estimate the area of each figure. 2.

3.

11  30 
20.5 units2 2

17  38 
27.5 units2 2

5.

6.

7  24 
15.5 units2 2

Glencoe/McGraw-Hill

14  39 
26.5 units2 2

91

5  19 
12 units2 2

5  19 
12 units2 2

Algebra Intervention

7. Estimate the area of the two leaves below. a.

b.

19.5 units2

20 units2

A P P L I C AT I O N S 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 B 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. 39  60 A C 
49.5 units2 2 b. Draw a line from A to C on the grid and find the actual area of the figure using the E D 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? 50  66 yes: 
58 ft2 2 Glencoe/McGraw-Hill

92

Algebra Intervention

Name ______________________________________ Date ___________

SKILL

47

Surface Area of Rectangular Prisms

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 the formula A
2(
h 
w  wh).

EXAMPLE

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

9.25 in.



11 in., w
9.25 in., h
1.5 in.

11 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.

EXERCISES

Find the surface area of each prism shown or described below. Round answers to the nearest tenth.

1.

2.

3.

7 in.

10 cm

9 in. 15 in. 6 1–2 cm

606 in2 4. length, 15 m width, 12 m height, 9 m

846 m2

8.5 cm

8 1–2 cm

4 cm 3 cm

410.5 cm2

143 cm2

5. length, 9.6 cm width, 7.5 cm height, 7.7 cm

Glencoe/McGraw-Hill

6. length, 100 in. width, 100 in. height, 50 in.

407.3 cm2

40,000 in2

93

Algebra Intervention

1.5 in.

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

72 in2

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

18 ft2

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

9 in.

A P P L I C AT I O N S 10. Jesse is making enclosed storage cubes for his room. The sides 3

of the cubes will each be 1  4 feet. He has three 32-square-foot sheets of plywood. How many storage containers can he make?

5 storage cubes 11. Twenty-seven cubes are used to make a large cube that is three cubes long by three cubes wide by three cubes high. The outside of the large cube is painted red. How many of the small 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 a senior citizens’ activity room. This room is 12 feet long, 16 feet wide, and 9 feet high. There are two windows that are 3 feet by 5 feet each and a door that is 2 feet by 6.5 feet in this room. a. How much area will they have to paint?

653 ft2

b. How many gallons of paint will be needed if a gallon of paint 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 Glencoe/McGraw-Hill

94

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

48

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

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

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

EXERCISES

Find the volume of each rectangular prism shown or described below. Round decimal answers to the nearest tenth.

1.

2.

3.

9 ft 3.8 in. 6.5 in.

8 ft 13 ft

4. length, 14 meters width, 23 meters height, 18 meters

3.4 mm 2.1 mm

4.7 in.

936 ft3

116.1 in3

69.3 mm3

1

5. length, 4  3 feet

5,796 m3

Glencoe/McGraw-Hill

9.7 mm

6. cube: side, 9.2 cm

3

width, 3  4 feet

778.7 cm3

height, 5 feet 1 81  ft3 4 95

Algebra Intervention

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 the volume of the cube?

729 in3

10. A cube has a volume of 1,000 cubic inches. What is the surface area of the cube?

600 in2

11. What is the height of a rectangular prism if the volume is 2,112 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 dimensions of the prism.

1136; 1218; 1312; 149; 166; 229; 236; 334

A P P L I C AT I O N S 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 1

14. An aquarium is 3 feet long and 1  2 feet wide. It is filled with water to a height of 1 foot. How many gallons of water are in the aquarium? (Hint: 1 cubic foot  7.5 gallons.)

about 33.75 gal

Glencoe/McGraw-Hill

96

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

49

Using Samples to Predict

EXAMPLE

In a community of 35,000 people, 45 out of 100 randomly selected people responded that they prefer vanilla ice cream to chocolate. How many 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.

EXERCISES

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 bicycle paths. Seventy-eight of those surveyed said yes. How many people in the town can be expected to think that the town needs more bicycle paths?

about 10,920 people

2. Mr. Tata surveyed his class about their favorite foods. Ten of the 30 students surveyed said their favorite food was pizza. How many students out of the 250 students in the school would you expect to like pizza best?

about 83 students

Glencoe/McGraw-Hill

97

Algebra Intervention

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

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 video camera?

about 45 families

A P P L I C AT I O N S

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

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

5. How many households in a town with 40,000 households would you expect to have watched 60 Minutes?

about 8,760 households 6. How many households in a town with 80,000 households would you expect to have watched the show in fourth place?

about 14,320 households 7. How many households in a town with 100,000 households would you expect did not watch Home Improvement?

about 80,800 households 8. Suppose that 2,124 of the households responding to this survey watched Murder, She Wrote. How many households were surveyed?

about 12,000 households

Glencoe/McGraw-Hill

98

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

50

Mean, Median, Mode

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

EXAMPLE

Hakeem Olajuwon, 1994’s Most Valuable Player in the National Basketball Association, helped the Houston Rockets win the NBA championship. In winning the 7-game series, Olajuwon scored 28, 25, 21, 32, 27, 30, and 25 points. Find the mean, median, and mode of his scores. 28  25 21  32  27  30  25   26.857 7

Mean:

The mean is about 27 points. Median:

21, 25, 25, 27, 28, 30, 32 ↑ median The median is 27.

Mode:

EXERCISES

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 Glencoe/McGraw-Hill

99

Algebra Intervention

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

A P P L I C AT I O N S

11. 12. 13. 14.

The data at the right shows the record high temperatures for several states in the U.S. Use the data to answer Exercises 11–15.

State

Record High Temperature (°F)

Alabama

112

Alaska

100

Michigan

112

Oklahoma

120

What is the mode? 112°F Vermont 105 What is the median? 112°F Wyoming 114 What is the mean? 110.5°F 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 be 113°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, would it change a. the mode? Why or why not? No, 112°F would still be the number

occurring 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 be

greater but will still be divided by the same number. 16. Find the hand spans of ten people. Ask each person to spread apart the little finger and thumb of his or her right hand as far as possible. Then measure and record the distance from tip to tip to the nearest centimeter. Find the mean, median, and mode for the data you collected.

Answers will vary. See students’ work. Glencoe/McGraw-Hill

100

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

51

Make a List

Pat’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.

EXAMPLE

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 PBO SBG SOM PBG

PSG PBM SBO BGO POM

PSO PGO SBM BGM SGM

PSM PGM SGO BOM GOM

There are 20 different pizzas the Davis family can order.

EXERCISES

Solve by making a list.

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

6 ways

2. How many different four-digit numbers can be formed from the numbers 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 Glencoe/McGraw-Hill

101

Algebra Intervention

4. How many different two-digit numbers can be formed from the numbers 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

A P P L I C AT I O N S 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 can she arrange her dogs in a row if the beagle and terrier must be next to each other?

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

11 ways 10. Earl attends a convention every three years. The year 1992 was a leap year, and Earl attended a convention. What is the next leap year that Earl will be attending a convention?

2004

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SKILL

Name ______________________________________ Date ___________

52

Probability of Independent Events

The probability of an event is the ratio of the number of ways an event can

occur to the number of possible outcomes.

number of ways the event can occur

Probability of an event
 number of possible outcomes

EXAMPLE

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.

1

First Spinner

Second Spinner

Sum

1

2

2

3

3

4

1

3

2

4

3

5

1

4

2

5

3

6

2

3

1

1 3

2

2 3

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

1

 Probability of sum of 4
 9 or 3 1

The probability that the sum will be 4 is  3.

EXERCISES

Use the spinners in the Example above to answer Exercises 1–4.

1. What is the probability that the sum of the numbers showing on the two spinners is 3? Glencoe/McGraw-Hill

103

2  9

Algebra Intervention

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

2  3

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

5  9

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

7  9

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 dime will both show heads?

1  4

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

1  2

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

3  4

A P P L I C AT I O N S

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

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

1  12

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

1  9

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

1  6

12. Who is most likely to win the game?

Marci

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SKILL

Name ______________________________________ Date ___________

53

Expected Value of an Outcome

EXAMPLE

Mr. 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 could he 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 1

colored markers is  4. 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.

EXERCISES

A number cube is rolled 12 times. How often would you expect to get each of the following outcomes?

1. a 6

2. a 7

twice

never

3. a prime number

6 times

5. a number greater than 2

8 times

7. a multiple of 1

12 times

Glencoe/McGraw-Hill

4. an even number

6 times

6. a number less than 1

never

8. a multiple of 4

twice

105

Algebra Intervention

A coin is tossed 20 times. How often would you expect to get each of the following outcomes? 9. a head

10.

10 times

10 times

11. a head or a tail

12.

20 times

A P P L I C AT I O N S

a tail

neither a head nor a tail

never

LeRoy has 15 different ties. He chooses a tie at 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 year that 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 expect to wear each tie just 5 times in a year that is not a leap year?

73 ties

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54

Name ______________________________________ Date ___________

Theoretical and Experimental Probability

The theoretical probability of an event is the ratio of the number of ways the event 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.

EXAMPLE

Louis wants to determine the probability of getting a sum of 7 when rolling a number cube. The sample space, or all the possible outcomes, for a roll of two number cubes is shown below. 1, 2, 3, 4, 5, 6,

1 1 1 1 1 1

1, 2, 3, 4, 5, 6,

2 2 2 2 2 2

1, 2, 3, 4, 5, 6,

3 3 3 3 3 3

1, 2, 3, 4, 5, 6,

4 4 4 4 4 4

1, 2, 3, 4, 5, 6,

5 5 5 5 5 5

1, 2, 3, 4, 5, 6,

6 6 6 6 6 6

What is the theoretical probability of rolling a sum of 7? What is the experimental probability of rolling a sum of 7 if Louis rolls the number cube 20 times and records 4 sums of 7? There are 6 sums of 7 shown in the sample space above. So, the 6

1

 theoretical probability of rolling a sum of 7 is  36 or 6 .

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

1

 is  20 or 5 .

EXERCISES

Find the theoretical probability of each of the following.

1. getting tails if you toss a coin

1  2

2. getting a 6 if you roll one number cube

1  6

3. getting a sum of 2 if you roll two number cubes Glencoe/McGraw-Hill

107

1  36 Algebra Intervention

4. getting a sum less than 6 if you roll two number cubes 5 10  or  36 18 5. Melissa rolled one number cube 30 times and got 8 sixes. a. What is her experimental probability of getting a six? 4 8  or  30 15 b. What is her experimental probability of not getting a six? 11 22  or  30 15 6. Melissa rolled two number cubes 36 times and got 3 sums of 11. a. What is of 11? 3  or 36 b. What is of 11? 33  or 36

her experimental probability of getting a sum 1  12

her experimental probability of not getting a sum 11  12

A P P L I C AT I O N S

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

7. What was his experimental probability of rolling doubles? 5 10  or  48 24 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 to the theoretical probability in most experiments? Explain.

The experimental probability should be close to the theoretical probability. 10. Do you think the experimental probability is ever equal to the theoretical probability? Explain.

Sample answer: Yes, especially if many trials are used.

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Name ______________________________________ Date ___________

55

Probability Using Area Models

EXAMPLE

Determine the probability that a randomlydropped counter will fall in the shaded area. Each small square has an area of 1 square foot and the area of the shaded region is about 30 square feet.

number of ways an event can occur

probability
 number of possible outcomes 30

3

 P
 100 or 10 3

The probability that a counter will fall in the shaded area is  10 .

EXERCISES 1.

Find the probability that a randomly-dropped counter will fall in the shaded region. 2.

25  64

4.

3.

32  81

7  22

5.

5  27

6.

1  5

Glencoe/McGraw-Hill

4  13

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Algebra Intervention

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 randomlydropped counter will fall in the shaded area? 3  4 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 randomlydropped counter will fall in the shaded area? 7  18

A P P L I C AT I O N S 9. Find the probability that a golf ball will land on the green. 1  6 10. Suppose you throw 100 darts at each dart board below. Use the formula below to determine how many darts you would expect to land in the shaded regions.

1,600 square yards 1

80 yd

120 yd

number landing in shape area of shape 
 number landing in region area of region a.

b.

c.

4 cm

11. Zoe lost her contact lens while playing basketball. If it is equally likely that she lost the lens on any part of the court, estimate the probability that Zoe lost her lens inside the circle. 1 about  47

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110

50 ft 6 ft 94 ft

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

56

Line Plots

The table at the right shows the final results of the Great Frog Competition in Dickerson County. The frog jumps were measured in inches.

EXAMPLE

Jumps Frogs

1

2

3

Slippery

61

51

60

Spots

46

38

39

Inky

56

33

61

Popper

65

51

52

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 X above the number line to represent the distance of each jump. × 30

EXERCISES

×× 35

× ××

× 40

45

50

× 55

× ××

×

60

65

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

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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

A P P L I C AT I O N S 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 centimeters. Record the distances and organize this information into a line plot.

See students’ work.

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SKILL

Name ______________________________________ Date ___________

57

Stem-and-Leaf Plots

A stem-and-leaf plot is one way to organize a list of numbers. The stems

represent the greatest place value in the numbers. The leaves represent the next place value.

EXAMPLE

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 1

EXERCISES

9 9 0 0 0 1 1 1 1 2 3 3

8 0 4 3 5 1 1

9 means $0.90.

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

1. 18, 67, 35, 20, 45, 55, 69, 23, 34, 58, 61, 43, 56, 63, 29

1 2 3 4 5 6

0

3 9 5 5 6 8 3 7 9 8 means 18.

Glencoe/McGraw-Hill

2. 500, 610, 720, 830, 870, 880, 750, 630, 520, 500, 540, 580, 890, 780, 880

5 6 7 8

0 1 2 3 5

113

0 2 4 8 3 5 8 7 8 8 9 0 means 500.

Algebra Intervention

3. 6, 10, 23, 35, 30, 13, 19, 33, 1, 22, 28, 35 24, 23, 14, 4, 19, 23

0 1 2 3

1 0 2 0 0

4. 61.6, 51.9, 60.0, 46.0 38.7, 39.1, 56.2, 33.0 61.5, 65.4, 51.0, 52.3

4 6 3 4 9 9 3 3 3 4 8 3 5 5 1 means 1.

A P P L I C AT I O N S

3 4 5 6

3 9 9 6 1 2 2 6 0 2 2 5 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 sixteen southwestern cities are 54°, 67°, 64°, 61°, 70°, 65°, 72°, 63°, 49°, 58°, 60°, 48°, 68°, 77°, 69°, and 65°.

4 5 6 7

8 4 0 0 4

9 8 13455789 27 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 2 3 4

7 8 8 1 3 3 8 8 9 0 0 3 4 8 4 1 7 means 17 min.

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

2 3 4 5

7 2 0 2 2

8 2 3 7 9 1 1 3 5 8 4 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 2

4 5 6 7 8 8 8 9 9 9 0 5 1 4 means $140. Glencoe/McGraw-Hill

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SKILL

Name ______________________________________ Date ___________

58

Line Graphs

A line graph is usually used to show the change and direction of change

over time. All line graphs should have a graph title, a vertical-axis label, and a horizontal-axis label.

EXAMPLE

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

EXERCISES

1961 1962 1963 1964 1965 1966 1967 1968 1969

4 5 3 1 6 5 1 3 9

8 6 4 2

19

61 19 62 19 63 19 64 19 65 19 66 19 67 19 68 19 69

Number

Year

Make a line graph for each set of data. Sid’s Daily Jogging Time for Three Miles

Sid’s Daily Jogging Time for Three Miles Day

Time in Minutes

1 2 3 4 5 6 7

32 29 28 26 28 33 27

Glencoe/McGraw-Hill

Time in Minutes

1.

Year

Number of Flights

Space Flights Carrying People

35 30 25 20 15 10 5 0

1

115

2

3

4 5 Day

6

7

Algebra Intervention

A P P L I C AT I O N S 2.

Traffic on Maple Drive Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

3.

Make a line graph for each set of data.

Number of Vehicles 7,200 8,050 10,500 5,900 9,990 3,400 900

Recorded Number of Hurricanes Month

Number

June July August September October November 4.

Evans Family Electric Bill Month March April May June

5.

23 36 149 188 95 21

Amount $129.90 $112.20 $105.00 $88.50

Home Runs by Hank Aaron 1967 to 1976 Year

Number

1967 1968 1969 1970 1971 1972 1973 1974 1975 1976

39 29 44 38 47 34 40 20 12 10

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SKILL

Name ______________________________________ Date ___________

59

Bar Graphs

Bar graphs are used to compare numbers. All bar graphs should have a

graph title, a vertical-axis label, and a horizontal-axis label.

EXAMPLE

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

NCAA Women’s Gymnastics

1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993

Utah Georgia Alabama

Champion Utah Utah Utah Utah Utah Georgia Alabama Georgia Utah Alabama Utah Georgia

Make a bar graph. NCAA Women’s Gymnastics 1982–1993 9 Number of Championships

Year

8 7 6 5 4 3 2 1 0

Utah

Georgia

Alabama

Colleges

EXERCISES 1.

Make a bar graph for each set of data.

Preference for Brands Brand

Number of Students

A B C D

15 35 30 25

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Algebra Intervention

A P P L I C AT I O N S 2.

NCAA Women’s Volleyball Year

Champion

1981

Southern California Hawaii Hawaii UCLA Pacific Pacific Hawaii Texas California State, Long Beach UCLA UCLA Stanford

1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992

3.

NCAA Women’s Cross Country Year 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992

Champion Virginia Virginia Oregon Wisconsin Wisconsin Texas Oregon Kentucky Villanova Villanova Villanova Villanova

4. Survey the students in your math class to find out their favorite movie. 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. Glencoe/McGraw-Hill

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Algebra Intervention

SKILL

Name ______________________________________ Date ___________

60

Circle Graphs

The air surrounding Earth is referred to as the atmosphere. Without air

there would be no life on Earth. Air is a mixture of gases. By volume, dry air is composed of 78% nitrogen, 21% oxygen, and 1% other gases.

EXAMPLE

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: Oxygen: Other:

78% of 360°  281° 21% of 360°  76° 1% of 360°  4°

Earth’s Atmosphere

Nitrogen 78%

Use a compass and a protractor to draw the circle graph. Note that the sum of the degrees is not 360° because of rounding.

EXERCISES 1.

Oxygen 21%

Make a circle graph to show the data in each chart.

Favorite TV Shows Movies Sports News Drama Comedy Music

12% 20% 4% 16% 20% 28%

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Algebra Intervention

Other 1%

2.

Daily Activities Sleeping Eating School Homework Team practice Miscellaneous

A P P L I C AT I O N S 3.

8 1 6 3 2 4

hours hour hours hours hours hours

Make a circle graph to show the data in each chart.

Area of Continents Continent

Area in Millions of Square Miles

Europe Asia North America South America Africa Oceania Antarctica 4.

3.8 17.4 9.4 6.9 11.7 3.3 5.4

World Cup Winners Country

Number of Wins

Argentina Brazil England Italy Uruguay West Germany 5.

2 4 1 3 2 3

Area of New England States State

Area in Square Miles

Maine New Hampshire Vermont Massachusetts Connecticut Rhode Island

33,215 9,304 9,609 8,257 5,009 1,214

6. Make a circle graph showing how you spent your time last Saturday.

See students’ work. Glencoe/McGraw-Hill

120

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

61

Scatter Plots

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

0

1

2

3 4 Time

5

The pattern of dots slants upward to the right.

EXAMPLES

5 4 3 2 1 0

10

5 4 3 2 1

20 50 Age

0

The pattern of dots slants downward to the right.

The Marysville Garden Club sells stationery each year. The scatter plot at the right relates the length a person has belonged to the club to the number of boxes sold. What does the point with the box around it represent?

5

6

7 8 9 10 Shoe Size

The dots are spread out. There is no pattern.

Stationery Sales Boxes Sold

25 20 15 10 5

No Relationship Number of Pets

Negative Relationship New Cavities

Distance

Positive Relationship

40 20 0

2

4 6 8 10 Years in Club

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 is positive.

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EXERCISES

Determine whether a scatter plot of the data would show a positive relationship, a negative relationship, or no relationship.

1. number of minutes a candle burns and the candle’s height

negative relationship 2. length of a taxi ride and the amount of the fare

positive relationship 3. number of letters in a person’s first name and height of a person in centimeters

no 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.

Relationship of Physical Activity and Age Hours of Physical Activity

A P P L I C AT I O N S

14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 Age in Years

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 relationship between a preschool child’s English vocabulary and their age. Use this information to answer Exercises 7–8.

As the age increases, the vocabulary increases.

Age

7. Make a general statement about the scatter plot.

Vocabulary of Preschool Children 5 4 3 2 1 0 3 6 9 12 15 18 21 24 27 30 Words Used (100’s)

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. Glencoe/McGraw-Hill

122

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SKILL

Name ______________________________________ Date ___________

62

Constructing and Interpreting Graphs

The chart at the right shows scores in the

Summer Olympics Springboard Diving Year Score 1964 145.00 1968 150.77 1972 450.03 1976 506.19 1980 725.91 1984 530.70 1988 580.23 1992 572.40

Springboard Diving event in the Summer Olympic Games.

EXAMPLE

Construct and interpret a graph of the data.

Summer Olympics Springboard Diving Scores 800 700 600 Score

500 400 300 200

The graph shows that the scores generally tended to increase with each successive Olympic game.

100

EXERCISES 1.

Construct and interpret a graph of each set of data.

Time (seconds) 5 10 15 20 25 30 35

Speed (mph) 5 8 20 18 30 40 55

The speed generally tends to increase over time. Glencoe/McGraw-Hill

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Algebra Intervention

92

88

19

84

19

80

Year

19

76

19

72

19

68

19

19

64

0 19

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

2.

Distance (feet) 40 80 120 160 200 240 280

Speed (mph) 15 28 42 60 46 37 55

The speed fluctuates with the distance.

A P P L I C AT I O N S

The chart at the right lists the winning times for the men’s 110-meter hurdles at the Summer Olympic Games. Use the data to answer Exercises 3–6.

3. Construct a graph of the data.

See students’ work.

Year

Time (seconds)

1964 1968 1972 1976 1980 1984 1988 1992

13.6 13.3 13.24 13.30 13.20 13.20 12.98 13.12

4. Interpret the graph of the data.

The time generally tends to decrease with each successive Olympic game. 5. Why do you think the times do not always show a consistent pattern?

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 the next Olympic games? Explain why you chose this time.

Answers may vary. 7. Suppose you are driving down a street that has many traffic lights. What do you think a graph of your time versus your speed would look like? Why? Sketch your graph.

Answers may vary.

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SKILL

Name ______________________________________ Date ___________

63

Make a Graph

All graphs should have a graph title. A bar graph should also have a vertical-axis label and a horizontal-axis label.

EXERCISES 1.

Number of Votes 18 11 15 9

Cases of Shampoo Sold Brand X Y Z

Number of Cases 35 20 15

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ve

bi

Algebra Intervention

D

is

co

um

C

ol

av

Make a bar graph for each set of data.

Joan Ron Ramona Chi Wan

ry

a

r ou

is de

Spacecraft

Class President Election Results Name

2.

15 8 8 10 7

4

nt

Soyuz TM-15 Atlantis Endeavour Columbia Discovery

8

la

Days in Space

12

S TM oy u -1 z 5

Spacecraft

16

En

Spaceflights Carrying People July, 1992 to December, 1992

Spaceflights Carrying People, July, 1992 to December, 1992

At

Make a bar graph to show the duration of the five spaceflights listed in the table below.

Days in Space

EXAMPLE

3.

Wins of Basketball Teams Team

Number of Wins

Bears Hawks Tigers Wildcats

A P P L I C AT I O N S 4.

7 6 3 9

Make a bar graph for each set of data.

Days Exceeding Carbon Dioxide Standards in 1990 Metropolitan Area

Number of Days

Los Angeles Spokane Las Vegas Sacramento Anchorage 5.

47 13 17 11 12

Silver Medals Won at 1992 Summer Olympics Country

Number of Silver Medals

China Germany Hungary Unified Team United States 6.

22 21 12 38 34

Average Rating for TV Shows 1990–1993 Show

Rating

Roseanne Murder, She Wrote Full House America’s Funniest Home Videos Fresh Prince of Bel Air

19.7% 17.0% 16.3%

Glencoe/McGraw-Hill

14.6% 14.1% 126

Algebra Intervention

SKILL

Name ______________________________________ Date ___________

64

Interpreting Graphs

The graph below shows the height of a baby girl at certain ages. Female Growth Rate

Height (inches)

30 28 26 24 22 20 0

EXAMPLE

2

4 6 8 10 12 Age (months)

Describe the shape of the graph. Then use the graph to predict what will 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.

EXERCISES 1.

Describe the shape of each graph. Then use the graph to answer each question.

Height (inches)

Plant Growth 16 14 12 10 8 6 4 2

What do you think will happen to the height of the plant over the next 2 weeks? What do you predict the height will be after 48 days?

The graph rises rapidly at first and then slows down. Over the next two weeks, the plant will continue to grow but the rate of growth will be slower. Predictions for the height after 48 days will vary. 0 4 8 12 16 20 24 28 Day Glencoe/McGraw-Hill

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Algebra Intervention

2. 34 32 30 28 26 24 22 0

19 8 19 8 8 19 9 9 19 0 9 19 1 9 19 2 9 19 3 94

Income (thousands of dollars)

Kirk’s Family Income

What do you think will happen to the income over the next 5 years? What do you predict the income will be in 1999?

The graph rises at a steady rate of about $1,200 a year. The income will continue to rise over the next 5 years at about the same rate. The income in 1999 will probably be about $37,000.

Year

Male Growth Rate

Weight (pounds)

24 20 16 12 8 4 0

4.

2

4 6 8 10 12 Age (months)

Number of People

Conference Attendance 2,200 1,800 1,400 1,000 600 200 8 19 9 9 19 0 9 19 1 9 19 2 9 19 3 94

19

19

88

0 Year

A P P L I C AT I O N S

What do you think will happen to the weight of the baby over the next year? What do you predict his weight will be after 2 years?

The graph rises rapidly at first and then slows down. Over the next year his weight will probably continue to rise but at a slower rate. Predictions of his weight after 2 years will vary. What do you think will happen to the attendance over the next 5 years? What do you predict the attendance will be in 1999?

The graph rises at a steady rate of about 200 a year and then rises rapidly. The attendance will continue to rise over the next 5 years at about the same rate. Predictions of the attendance in 1999 will vary.

The graph below shows the public debt from the years 1970 to 1992.

Public Debt, 1970-1992 Amount of Debt (billions)

3.

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 public debt over the next 10 years. It will probably

continue to increase rapidly.

19 7 19 0 7 19 5 8 19 0 8 19 5 9 19 0 95

7. According to this graph, what do you think the public debt will be in 2002? Answers will vary.

4,500 4,000 3,500 3,000 2,500 2,000 1,500 1,000 500 0 Year

8. What do you think could happen to cause your prediction to not hold true? Answers will vary. Glencoe/McGraw-Hill

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SKILL

Name ______________________________________ Date ___________

65

Standard Deviation

EXAMPLE

The heights of a group of young pine trees in a reforestation plot are 58 cm, 56 cm, 51 cm, 54 cm, 49 cm, 61 cm, 54 cm, and 49 cm. Find the standard deviation. Follow the steps below to find the standard deviation. You can use a calculator to help you. 1.

Find the mean. The mean of this data is 54.

2.

Subtract each measurement from the mean.

Height

Difference

Square

58

4

16

56

2

4

51

–3

9

3.

Square each difference.

54

0

0

4.

Find the mean of the squares. The mean of the

49

–5

25

61

7

49

54

0

0

49

–5

25

128 squares is  8
16.

5.

Find the positive square root of this mean. The square root of 16 is 4.

The standard deviation is 4.

EXERCISES

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

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A P P L I C AT I O N S

The number of millions of persons viewing prime time television each day of the week is 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 will be 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 to their wrist. Measure and record each length to the nearest centimeter. Find the mean and the standard deviation for the data you collected.

Answers will vary. See students’ work.

Glencoe/McGraw-Hill

130

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SKILL

Name ______________________________________ Date ___________

66

Predicting Distribution of Data

EXAMPLE

The lifetimes of 10,000 light bulbs are normally distributed. The mean lifetime is 300 days, and the standard deviation is 40 days. How many light 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.

2.35%

2.35% 13.5%

220

34% 260

34% 300

13.5% 340

380

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.

EXERCISES

The frequencies of 50,000 scores on a college entrance examination are normally distributed. The mean score is 510, and the standard deviation is 80.

1. How many scores are between 510 and 590?

17,000 2. How many scores are between 430 and 510?

17,000 3. How many scores are less than 350?

1,175 Glencoe/McGraw-Hill

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Algebra Intervention

4. The number of scores above 590 is the same as the number of scores below what score?

430 5. How many scores are between 430 and 670?

40,750 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

A P P L I C AT I O N S

The weights of 150 oranges picked in a citrus grove are normally distributed. The mean weight is 7.5 ounces, and the standard deviation is 2.1 ounces. Use this information 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 less than 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 or equal 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 or equal 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 deviation of the data. Draw a graph of the data. Does it appear to be normally distributed? Why or why not?

Answers will vary. See students’ work.

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SKILL

Name ______________________________________ Date ___________

67

Arithmetic Sequences

An arithmetic sequence is given below. 6, 8.5, 11, 13.5, 16, ...

EXAMPLE

Find the next three terms in the sequence given above. To extend the arithmetic sequence, find the common difference between any two consecutive terms. 6

   8.5

2.5

11

2.5

13.5

2.5

16

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.

EXERCISES

Find the next three terms in each arithmetic sequence.

1. 3, 5, 7, 9, 11, ...

13, 15, 17

3. 98, 93, 88, 83, 78, ...

73, 68, 63

5. 0.3, 0.6, 0.9, 1.2, 1.5, ...

1.8, 2.1, 2.4

7. 3, 15, 27, 39, 51, ...

63, 75, 87

Glencoe/McGraw-Hill

2. 15.1, 15.2, 15.3, 15.4, 15.5, ...

15.6, 15.7, 15.8

4. 6.7, 7.5, 8.3, 9.1, 9.9, ...

10.7, 11.5, 12.3

6. 76, 75.5, 75, 74.5, 74, ...

73.5, 73, 72.5

8. 15, 18.6, 22.2, 25.8, 29.4, ...

33, 36.6, 40.2

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Algebra Intervention

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

A P P L I C AT I O N S

The State General Sales Tax in 1993 in Maryland was 5%. The amount of sales tax on certain purchases is given in the chart below. Use this information to answer Exercises 13–15.

Amount of Purchase (dollars) Amount of Tax (dollars)

1

2

3

4

0.05

0.10

0.15

0.20

5

6

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 the monthly rent to increase $25 each year. What will the monthly rent be at the end of five years?

$925 17. The cost for the first three minutes of a long-distance phone call is $0.59. Each minute after that costs $0.13. What is the cost of a 10-minute long-distance phone call?

$1.50 Glencoe/McGraw-Hill

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7

SKILL

Name ______________________________________ Date ___________

68

Geometric Sequences

The value of a car over several years is given in the chart below. Year

Value

0

$20,000

1

$18,000

2

$16,200

3

$14,580

4 5

When consecutive terms of a sequence are formed by multiplying by a constant factor, the sequence is called a geometric sequence. The factor is called the common ratio.

EXAMPLE

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 by 0.9. So, the value after year 4 is 14,580  0.9, or 13,122, and the value after year 5 is 13,122  0.9 or 11,809.80. Year

Value

4

$13,122.00

5

$11,809.80

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EXERCISES

State whether each sequence is a geometric sequence. 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.

yes; ratio: 3;

100, 50, 25, 12.5, ... 1 yes; ratio:  ; 2

243, 729, 2,187

6.25, 3.125, 1.5625

3. 16, 18, 20, 22, ...

4.

no

yes; ratio: –2; 80, –160, 320

5. –243, 81, –27, 9, ...

6.

yes; 1 1  ratio: –  ; –3, 1, – 3 3 7. 1.2, 4.8, 19.2, 76.8, ... 1 1 1 1  ,  ,  ,  , ... 16 8 4 2

8.

A P P L I C AT I O N S

100, 80, 64, 51.2, ...

yes; ratio: 0.8; 40.96, 32.768, 26.2144 10.

yes; ratio: 2; 1, 2, 4

90, 85, 75, 60, ...

no

yes; ratio: 4; 307.2, 1,228.8, 4,915.2 9.

5, –10, 20, –40, ...

1 1 3  ,  ,  , 1, ... 4 2 4

no

Suppose you take a job for 31 days helping your cousin mow lawns. Your cousin offers you a choice of two payment plans. He will either pay you $100 a day or 1¢ the first day, 2¢ the second day, 4¢ the third day, 8¢ the fourth day, and so on, continuing to double the amount each day.

11. Without doing any calculations, which way would you choose to be paid? Why? Answers will vary. 12. Determine the total amount you will make if you choose to be paid $100 a day for 31 days. $3,100 13. Determine how much you will be paid on the 31st day if you choose to be paid 1¢ the first day and double the amount each 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.

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136

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SKILL

Name ______________________________________ Date ___________

69

Classify Information

In 1980, the United States film industry took in $2,748,500,000 in box office receipts. The average admission charge was $2.69. In 1990, the box office receipts were $5,021,800,000, and the average admission charge was $4.75.

EXAMPLE

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.

EXERCISES

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 the numbers?

too much information; 133

Glencoe/McGraw-Hill

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Algebra Intervention

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 of the numbers?

not enough information

A P P L I C AT I O N S

Classify information in each problem by writing “not enough information” or “too much information.” Then solve, if possible.

5. Phien bought 3 address books that cost $4.98 each. She gave the 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 8 feet long. The cloth tape is marked in metric measure at onecentimeter intervals. How much longer is the steel tape than the cloth tape?

not enough information 8. Jonathan bought 10 computer disks for $1.39 each. The disks usually sell for $1.99 each, or ten for $18. How much did he pay for the disks?

too much information; $13.90 9. The Sheng family drove 1,287 miles on their vacation. About how many miles did they drive per day?

not enough information 10. Gerda pays a delivery service $18 for priority delivery, $15 for standard delivery, and $21 for Saturday delivery. How much will she save by sending a package by standard delivery instead of Saturday delivery?

too much information; $6 11. Alan ran the same number of miles for 6 days. How far did he run?

not enough information

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SKILL

Name ______________________________________ Date ___________

70

Problem-Solving Strategies

There are many strategies that can be used to solve a problem. A few of these strategies are listed below. • Draw a diagram • Use a matrix or chart • Make a list

• Use logical reasoning • Draw a picture • Guess and check

For each problem you solve, you must decide which strategy would work best for you.

EXAMPLE

A 1-inch spool holds 100 inches of line, a 2-inch spool holds 400 inches of line, and a 3-inch spool holds 900 inches of line. How many inches of line are on a 5-inch spool? First make a chart. Spool Size

Inches of Line

1 in.

100

2 in.

400

3 in.

900

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.

EXERCISES

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

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2. Arrange the digits 1 through 7 in the squares so that the sum along 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.

A P P L I C AT I O N S 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 is the 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¢. It accepts quarters, dimes, and nickels only. If it only accepts exact change, how many different combinations of coins must the machine 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 between 10:00 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-distance to her cousin. If the rate is $0.20 for each of the first 3 minutes and $0.15 for each minute after that, how much did the call cost? $3.45

Glencoe/McGraw-Hill

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Algebra Intervention

SKILL

Name ______________________________________ Date ___________

71

Determine Reasonable Answers

The product manager at Taylor Dairy reported that about 2 billion pounds

of cheese were sold in the United States in 1992. The average American eats 28 pounds of cheese a year, and the population of the United States was about 255,200,000 in 1992.

EXAMPLE

Is the product manager’s statement reasonable? The average American eats about 30 pounds of cheese a year. There were between 200 million and 300 million people in the United States in 1992. Multiply to estimate the amount of cheese sold in 1992. 200,000,000  30
6,000,000,000 300,000,000  30
9,000,000,000 There were between 6 billion and 9 billion pounds of cheese sold in the United States that year. The product manager’s statement is not reasonable.

EXERCISES

Determine whether the answers shown are reasonable. Write yes or no. 2. 604  225
729

3.

4. 535  5
107

5. 112  4
484

6. 5,962  11
542

7. 56  35
91

8. 168  17
111

9. 205  5
31

1.

27  38
65

yes

yes

yes

10. 6,657  7
95

no

no

no

11.

no

Glencoe/McGraw-Hill

168  35
153

no

yes

no

3,137  4
1,258 12. 36  22
792

no

yes

141

Algebra Intervention

13. 812  5
16.24

14.

no

510  490
1,000 15.

yes

5,988  635
6,623

yes

A P P L I C AT I O N S 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?

Compact Disc Depot Sale!!! Selected CD’s

yes 17. Jake bought three CD’s on sale at the Compact Disc Depot. He gave the clerk two $20.00 bills. The clerk gave him $5.06 in change. Does this seem reasonable?

no 18. The Ortiz family is planning a 1,880-mile trip. They want to drive between 200 and 250 miles per day. Is 6 days a reasonable time for the trip?

no 19. There are 478 people who are planning to take buses to a rally. Each bus carries 37 people. Thirteen buses have been ordered. 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 this answer reasonable?

no 21. Out-of-town newspapers cost 60¢ each, and local papers cost 35¢ each. Bill buys two out-of-town papers and three local papers. He hands the cashier $3.00. Should he expect more than 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

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Algebra Intervention

$9.98

SKILL

Name ______________________________________ Date ___________

72

Work Backward

Rupesh earned some money mowing lawns one month. He put half of his money into savings. With the rest, he spent $15 on a new CD, $6 to see a movie, and $3 on food. He still had $24 left in his pocket.

EXAMPLE

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.

EXERCISES

Solve by working backward.

1. A number is added to 8, and the result is multiplied by 10. The final answer is 140. Find the number.

6 2. A number is divided by 8, and the result is added to 12. The final answer is 75. Find the number.

504 3. A number is decreased by 12. The result is multiplied by 5, and 30 is added to the new result. The final result is 200. What is the number?

46 Glencoe/McGraw-Hill

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Algebra Intervention

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

A P P L I C AT I O N S 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 not have enough money. The leather jacket went on sale and was reduced by $15.00, then by $13.50 more, and finally by an additional $12.15. Kara bought the jacket at the final sale price of $109.35. What was the original price?

$150.00 8. James arrived for piano practice at 4:45 P.M. On the way from school, he stopped at the video store for 15 minutes and also made a call from the phone booth for 10 minutes. It usually takes 25 minutes to get from the school to the piano teacher’s house. What time did James leave school?

3:55 P.M. 9. Dave has 12 baseball cards left after trading cards. This is one third as many as he had yesterday, which is 8 less than the day before. 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 feet long. Enough fencing is left over to also fence a square garden with an area of 25 square feet. If there is 3 feet left after the fencing is completed, how much fencing was available at the beginning?

83 ft

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SKILL

Name ______________________________________ Date ___________

73

Solve a Simpler Problem

EXAMPLE

Find the sum of the whole numbers from 1 to 300. This would be a tedious problem to solve using a calculator or adding the 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, 300 150  151
301 . . . 5  296
301 4  297
301 3  298
301 2  299
301 1  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.

EXERCISES

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 Glencoe/McGraw-Hill

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Algebra Intervention

4. What is the total number of triangles of any size in the figure at the right?

26 triangles

5. What is the total number of squares of any size in the figure at the right?

30 squares

A P P L I C AT I O N S 6. Shea is planning to carpet a large area in her basement as shown at the right. How much carpet will she need to carpet this area?

1,072 ft2

16 ft

16 ft

16 ft

16 ft

30 ft

30 ft 8 ft

7. Cliff heard a funny joke on the radio on Sunday. On Monday (day 1), he told the joke to Sarah, Rich, and Claire. These 40 ft 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 pattern continued. How many people heard the joke on the sixth day?

729 people 8. How many days passed before at least 100 people had heard the joke in Exercise 7?

4 days 9. By the end of the day 6, how many people altogether had heard the joke in Exercise 7? (Remember to count Cliff!)

1,093 people 10. A summer camp has 7 buildings arranged in a circle. Paths must be constructed joining every building to every other building. How many paths are needed?

21 paths

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Algebra Intervention

SKILL

Name ______________________________________ Date ___________

74

Make a Model

A box like the one at the right is a rectangular prism. It has six sides and each one is the shape of a rectangle.

EXAMPLE

How many different shapes of rectangular prisms 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.

1 × 1 × 20

1 × 2 × 10

1×4×5

2×2×5

There are four different shapes of rectangular prisms that can be made.

EXERCISES

Solve by making a model.

1. How many different shapes of rectangular prisms can be formed using exactly 12 cubes?

4 shapes 2. How many different shapes of rectangular prisms can be formed using exactly 24 cubes?

6 shapes

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Algebra Intervention

3. How many cubes are needed to make the display shown at the right?

30 cubes

4. How many cubes are needed to make the display shown at the right?

35 cubes

A P P L I C AT I O N S 5. Ronnie used blocks to build a “fort”. The blocks were cubes and were stacked five high. The top, front, and side views were all squares. How many blocks did Ronnie need to build his 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 the possible dimensions of the box if the top view of each cake is a 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 around the rectangular family room to display them. If the family room is 15 feet wide and 18 feet long, how many feet of shelving will she need?

66 feet 9. A carton is 8 inches by 4 inches by 12 inches. How many fourinch cubes can Brian pack in the carton?

6 cubes

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SKILL

Name ______________________________________ Date ___________

75

Make Tables

Tables can help you organize information so it can be understood easier. EXAMPLE

Shauna needed to give a customer $1.40 in change. The customer requested that she not give him any bills. He also did not want to be able 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 different combinations of ten coins that make $1.40 and do not include change for 10¢ or 5¢. 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

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.

EXERCISES

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. How many different amounts of money could he choose?

6 amounts

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Algebra Intervention

3. Norma’s Repair Shop charges $35 for a service call and $25 an hour for each hour of labor. How much does she charge for an 8-hour service call?

$235

A P P L I C AT I O N S

Jake and June Washington started a college fund for their daughter. They started the fund by depositing $800 at the beginning of the first month. They plan to add $75 to the fund at the end of every month. Use this 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?

d. 2 years?

$1,700 $2,600

5. How can you extend your table from Exercise 4 to find out how much 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 first month and then $75 at the end of every month after that. How would this change your table?

The amounts would all be reduced by $75.

7. Find out how much your long distance phone company charges for calls. How much would it cost you to make a 15-minute long distance phone call?

Answers may vary.

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Algebra Intervention