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math sense® FOCUS ON OPERATIONS
Math for the High School Equivalency Tests
Cathy Fillmore Hoyt
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5
Unit 3: Fractions . . . . . . . . . . . . . . . . . . . 66 Relating Decimals and Fractions . . . . . . . . . . . . . . . 68
Skill Preview . . . . . . . . . . . . . . . . . . . . . . . . . 6
Different Forms of Fractions . . . . . . . . . . . . . . . . . . 70 Equivalent Fractions . . . . . . . . . . . . . . . . . . . . . . . . 72
Unit 1: Whole Numbers . . . . . . . . . . . . 14 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Comparing Fractions . . . . . . . . . . . . . . . . . . . . . . . . 74 Adding and Subtracting Like Fractions . . . . . . . . . . 76 Problem Solver: Multiples and Factors . . . . . . . . . . 78
Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Finding Common Denominators . . . . . . . . . . . . . . . 80
Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Adding and Subtracting Unlike Fractions . . . . . . . . 82
Division by Two or More Digits . . . . . . . . . . . . . . . . 24
Working with Whole and Mixed Numbers . . . . . . . . 84
Problem Solver: Mental Math and Estimation . . . . . 26
Mixed Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Problem Solver: The Five-Step Plan . . . . . . . . . . . . 28
Multiplying Fractions . . . . . . . . . . . . . . . . . . . . . . . . 88
Tools: Using Your Calculator . . . . . . . . . . . . . . . . . . 30
Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Test Taker: Choose an Operation . . . . . . . . . . . . . . 32
Multiplying and Dividing with Mixed Numbers . . . . . 92
Unit 1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Problem Solver: Does the Answer Make Sense? . . 94 Tools: Calculators and Fractions . . . . . . . . . . . . . . . 96
Unit 2: Decimals . . . . . . . . . . . . . . . . . . . . 36 Understanding Decimals . . . . . . . . . . . . . . . . . . . . . 38
Test Taker: Know When to Use Decimals . . . . . . . . 98 Unit 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Writing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Rounding Decimals . . . . . . . . . . . . . . . . . . . . . . . . . 42 Comparing Decimals . . . . . . . . . . . . . . . . . . . . . . . . 44
Unit 4: Ratio, Proportion, and Percent . . . . . . . . . . . . . . . . . . . . . . . . 102
Adding and Subtracting Decimals . . . . . . . . . . . . . . 46
Relating Fractions and Ratios . . . . . . . . . . . . . . . . 104
Problem Solver: Solving Multistep Problems . . . . . . 48
Writing Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Mixed Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Problem Solver: Unit Rates . . . . . . . . . . . . . . . . . . 108
Multiplying Decimals . . . . . . . . . . . . . . . . . . . . . . . . 52
Ratios in Word Problems . . . . . . . . . . . . . . . . . . . . 110
Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Writing Proportions . . . . . . . . . . . . . . . . . . . . . . . . 112
Problem Solver: Powers of Ten . . . . . . . . . . . . . . . . 56
Solving Problems with Proportions . . . . . . . . . . . . 114
Problem Solver: Figuring Unit Price and Total Cost 58
Tools: Using a Calculator with Proportion . . . . . . . 116
Tools: Calculators and Decimals . . . . . . . . . . . . . . . 60
Problem Solver: Scale Drawings . . . . . . . . . . . . . . 118
Test Taker: Use Estimation to Choose an Answer . . 62
Mixed Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Unit 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Understanding Percents . . . . . . . . . . . . . . . . . . . . 122
Decimals, Fractions, and Percents . . . . . . . . . . . . 124
Using Metric Units . . . . . . . . . . . . . . . . . . . . . . . . . 154
The Percent Equation . . . . . . . . . . . . . . . . . . . . . . 126
Problem Solver: Making Conversions . . . . . . . . . . 156
Solving Percent Equations . . . . . . . . . . . . . . . . . . 128
Working with Time and Temperature . . . . . . . . . . . 158
Problem Solver: Common Percent Applications . . 130
Problem Solver: Figuring Distance, Rate, and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Two-Step Percent Problems . . . . . . . . . . . . . . . . . 132 Tools: Using a Calculator with Percents . . . . . . . . 134 Problem Solver: Use Proportion to Solve Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . 136 Percent of Increase/Decrease . . . . . . . . . . . . . . . . 138
Problem Solver: Basic Perimeter Problems . . . . . 162 Test Taker: Make Your Own Drawing . . . . . . . . . . . 164 Unit 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Half-Length Simulated GED Test . . . . . . . . . . . . . . 168
Test Taker: Use Fractions to Solve Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . 140
Answer Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Unit 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Unit 5: Measurement . . . . . . . . . . . . . 144
Tool Kit
Customary Units of Length . . . . . . . . . . . . . . . . . . 146
Fraction Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Working with Length . . . . . . . . . . . . . . . . . . . . . . . 148
Decimal, Fraction, and Percent Equivalencies . . . 205
Measuring Capacity . . . . . . . . . . . . . . . . . . . . . . . . 150
Common Units of Measurement . . . . . . . . . . . . . . 206
Measuring Weight . . . . . . . . . . . . . . . . . . . . . . . . . 152
Calculator Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Introduction Math skills play an increasingly vital role in today’s world. Everyone needs to work confidently with numbers to solve problems on the job and in daily life. The increasing role of mathematics is reflected on the new GED® Mathematics Test and other high school equivalency tests, such as the HiSET® and TASC. This book is part of a three-book series designed to help you pass these tests and prepare for new opportunities in education and employment. In Math Sense 2, Book One: Focus on Operations, you will acquire the skills you need to solve problems with whole numbers, fractions, decimals, and percents. Each unit is organized around four key areas that will build your competence and confidence.
• Skill pages present instruction and practice with both computation and word problems.
• Tools pages provide insight on how to use calculators, key ideas, and mathematical properties to solve math problems.
• Problem Solver pages present strategies that will help you find the best way to approach different types of problems.
• T est Taker pages provide math tips and problem-solving approaches that good test takers use to take standardized tests.
Key Features Skill Preview The Skill Preview can help you determine what skills you already have and identify which areas you most need to concentrate on. Unit Preview The beginning of each unit features questions and topics to write about or discuss with classmates. Talking about math is key to building your understanding. Core Connections Throughout each unit, you will work with topics that connect math ideas to real life and to other math concepts. These Core Connections add depth to your learning and correspond to the national Common Core State Standards (CCSS). Special Problems These specially labeled problems require an in-depth exploration of math ideas. You may be asked to explain your reasoning or to find multiple solutions. These problems will prepare you for the short-response items on the GED test. Mixed Reviews and Unit Reviews Periodic checkups will help you see how well you understand and can apply the material. Unit Reviews also allow you to practice the types of questions that you will see on the GED test, including:
• Multiple-choice questions
• Fill-in-the-blank questions
• Drop-down and matching questions
• Short-response questions
GED Simulated Test At the end of the book is a half-length simulated GED Mathematics Test that focuses on the topics from this book only. You can use this final assessment to judge how well you have mastered the skills and strategies presented in this book. © New Readers Press. All rights reserved.
If you are taking the TASC or HiSET® test, go to newreaderspress.com. On the Math Sense 2 product page, you can download a free practice test. Glossary This list of terms defines key math words and ideas. Tool Kit These resource pages provide helpful information you can use as you work through the book.
Introduction 5
Skill Preview This survey of math skills will help you and your teacher decide what you need to study to get the most out of this book. It will show you how much you already know and what you need to learn. Do as much as you can of each section below. If you can’t do all of the problems in a section, go ahead to the next section and do all of the problems that you can.
Part 1: Whole Numbers Solve the following problems. 1. 7,605 + 4,986
3. 3,004 – 1,839
5. 8,605 × 52
2. 714 × 8
4. 14,392 ÷ 7
6. 25g 4, 555
? mi .
334 mi . Denver
Hays
Kansas City
8. Educe has two job offers. The first job pays $27,000 per year. The second pays $2,375 per month. Which job pays more? Explain your answer.
9. Ticket prices for an amusement park are shown on the sign below. How much will a family of 5 (2 adults and 3 children) pay for admission to the park?
Magic Land Park Admission
6 Skill Preview
Adults ......... $43.00 Children ..... $26.00
10. Yuki kept a log of the hours she worked for the last 4 weeks. Find the total hours she worked during that period.
Week 1 Week 2 Week 3 Week 4
45 42 38 46
11. Alex puts $125 per month in a savings account. At that rate, how much will he put into the account over 18 months?
12. A department store recorded 25,916 sales in a month. If there were 31 days in the month, how many sales did the store average per day? © New Readers Press. All rights reserved.
7. Nasser and Farah are driving from Denver to Kansas City, a distance of 606 miles. They drive 334 miles to Hays, Kansas, on the first day of the trip. How much farther do they have left to drive?
Part 2: Decimals Write each of the following values using a decimal point. 13. nine hundredths
15. forty dollars and five cents
14. two dollars and ninety-eight cents
16. thirty-five thousandths
Solve the following problems. 17. 100 – 84.7
20. 36 ÷ 0.08
18. 54 × 0.006
21. 6.3 × 8
19. 8.3 + 9.56 + 10.8
22. 1.8 ÷ 5
23. A wheel assembly is made from 2 wheels, each weighing 2.6 pounds, and a steel axle, which weighs 1.8 pounds. What is the total weight of the assembly?
2 .6 lb 2 .6 lb
1 .8 lb
24. Nick owes $42.78 for new windshield wipers. He hands the cashier three $20 bills. How much does he receive in change?
25. Tanya saw the sign below at an electronics store. She bought 3 DVDs and 4 video games at the prices shown. How much was her purchase before sales tax?
BARGAIN BIN ALL DVDs All Video Games....
$12.89 $10.69
26. If an airplane travels 2.5 hours at an average speed of 630 miles per hour, how many total miles will it travel?
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Skill Preview 7
Division by Two or More Digits When you divide by two or more digits, you use both educated guessing and estimation to help you find the answer.
Dividing by a Large Number Example Mike needs to save $4,316 in 1 year (52 weeks). How much will he need to save each week? Estimate: $4,500 ÷ 50 = $90 Step 1 52 won’t go into 4 or 43. Estimate how many times 52 goes into 431. Using basic division facts, look for compatible numbers. Using lead digits from 52 and 431, think, “How many 5s are in 40?” Try 8 times 52.
Step 2 Multiply: 52 × 8 = 416 Subtract. Bring down the 6. Using lead digits from 52 and 156, think, “How many 5s are in 15?” Try 3 times 52. 83 52g 4316 - 416 4156
83 52g 4316
Step 3 Multiply: 52 × 3 = 156 Subtract. 83 52g 4316 – 4160 0156 –0 156 0000
Check: Your answer of $83 is close to the estimate of $90. Divide.
1. 42g 84
2. 52g 700
36g 540
26g 1000
19g 152
18g 2250
23g 322
55g 13695
93g 3384
13g 11100
Sometimes a zero is needed in the answer to keep the correct place value. You can use an estimate to check place value.
Putting Zeros in Answers
Step 1 16 ÷ 8 = 2 203 8g 1630 – 1600 00
Step 2 Bring down the 3. Since 8 won’t go into 3, put a 0 in the answer above the 3. 20 3 8g 1630 – 1600 003
Step 3 Bring down the 0 and divide 30 by 8. 203 R6 8g 1630 – 1600 0030 – 24 00 0006
Lei needs 204 metal racks, which is close to your estimate of 200.
24 Unit 1
Step 4 Decide what to do with the remainder. Lei will completely fill 203 racks. She will need one more rack to hold the remaining 6 toys. 203 + 1 = 204
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Example Lei has 1,630 toy figures to display. She can fit 8 toys on a metal rack. How many racks will she need? Estimate: 1,600 ÷ 8 = 200
Divide. Watch out for zeros. 3. 12g 2, 460 4. 15g 9, 078
9g 20, 736
25g 201, 025
26g 22, 360
30g 7, 625
38g 3, 040
140g 84, 280
Solve the problems below. Estimate an answer first. 5. Mikail earned $1,632 for 4 weeks of work. He worked 160 hours during the month. How much did Mikail earn per week? Estimate: $1,600 ÷ 4 = Exact: $1,632 ÷ 4 = 6. Renée needs to cut 20 equal lengths of ribbon from a roll containing 2,100 centimeters. What will be the length of each of the 20 ribbons?
7. Joan scored 252 points during a basketball season. She played in 18 games. On average, how many points did she score per game? 8. A total of 220,460 cars crossed the Lincoln Toll Bridge over the course of a year. What is the average number of cars that crossed the bridge in a day? (Hint: There are 365 days in a year.)
2,100 cm
Use the graph to answer the questions below. 9. Taxes are withheld from the couple’s paychecks at an even rate throughout the year. How much do they pay in taxes in 1 month? 10. Next year the Ruizes would like to save twice as much, although they expect their income will be the same. They decide to cut back on food, transportation, clothing, and other expenses to save money. If they cut the same dollar amount on each of the 4, how much will they spend on clothing next year? 11. Explain The amount the Ruizes spend on taxes is nearly triple the expense of one other item. Which item is it? Explain how you found your answer.
Marisol and Edwin Ruiz Annual Expenses Total Income — $57,055 Clothing $2,375 Savings $2,000
Other $5,200 Housing $16,800
Healthcare $3,900 Food $6,400 Transportation $7,780
Taxes $12,600
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Answers start on page 177. Unit 1 25
Rounding Decimals 2.7
Look at the number line. Is the decimal 2.7 closer to 2 or to 3? The decimal 2.7 rounds to the whole number 3.
2
2.5
3
Rounding to the nearest whole number means to figure out which whole number the decimal is closest to. You often round to the nearest whole number when you shop. Example At the drug store, you buy items costing $3.99, $5.29, and $7.89. You have a $20 bill. Do you have enough money? You can use rounding to estimate the amount of money the items cost. Round each item to the nearest dollar. $3.99 rounds to $4 $5.29 rounds to $5 $7.89 rounds to $8
Add: $4 $5 + $8 $17
Even adding about $2 for sales tax, you should have enough money.
To round a decimal to a certain place value, look at the decimal to the right of the desired place.
Rounding a Decimal to a Given Place Value Example Round 13.648 and 13.6712 to the nearest tenth. Step 1 Identify the place you need to round to. Many students find it useful to underline this digit. 13.648
13.6712
Step 2 Look at the digit immediately to the right of the underlined digit.
If the digit to the right is less than 5, leave the underlined digit as is. Drop all the remaining digits to the right.
13.648 4 is less than 5. The digit doesn’t change.
If the digit to the right is 5 or more, add 1 to the underlined digit and drop the remaining digits.
13.6712 7 is more than 5. Add 1 to the 6 and drop the rest.
13.648 rounds to 13.6
13.6712 rounds to 13.7
After dividing $9 by 7, a calculator display reads:
9÷7
To round to 2 decimal places, you need 3 places. The thousandths place is equal to 5, so round up and drop the remaining digits.
1.285 rounds to 1.29
42 Unit 2
1.285714286
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Knowing how to round is crucial to interpreting calculator results.
Round to the nearest whole number. 1. 12.8
1.6
5.08
2. 20.5 20.4 20.099 For each calculator display, round to the nearest tenth, hundredth, and thousandth.
3. 69÷16
tenth
hundredth
4.3125
4. 5÷12 0.416666667
5. 5÷32 0.15625
thousandth
Choose the correct answer for each problem. 6. The rainfall for a 3-month period was 8.51 inches. To the nearest inch, how many inches of rain fell? A. 7
9. A credit card has a thickness of 0.076 centimeters. What is the thickness to the nearest tenth centimeter? A. 1.0 B. 0.1
B. 8
C. 0.8
C. 9
D. 0.08
D. 10 7. John is 1.905 meters tall. What is his height to the nearest tenth meter? A. 1.0 B. 1.8 C. 1.9
10. Seven friends are splitting the cost of a meal. Tom plans to divide the total by 7 and then round to the nearest cent. For how many decimal places will he need to continue the division to be able to round as planned? A. 2
D. 2.0
B. 3 C. 4
8. When rounded to the nearest cent, which of these amounts rounds to $15.60?
D. 5
A. $15.465
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B. $15.591 C. $15.593 D. $15.595
Answers start on page 180. Unit 2 43
Dividing Decimals When dividing a decimal by a whole number, be sure you place the decimal point in the correct place.
Dividing a Decimal by a Whole Number Example Andrew cut a 16.1-foot board into 4 equal pieces. How long is each piece? Step 1 Estimate first.
16.1 rounds to 16. 16 ÷ 4 = 4
Step 2 Set up the problem. Place the decimal point directly above the decimal point in the dividend. .1 4 g 6.1 divisor dividend
Step 3 Divide as you would whole numbers.
Step 4 Continue dividing. Add zeros if necessary and bring them down.
4 .0 4g 16.1 16.1 16.1
4.025 4g 16.100 16.100 16.100 16. 080 16.020 16.0 20 00.00 0
Check: The answer 4.025 is close to the estimate of 4. When dividing a number by a decimal, first change the divisor to a whole number. Then divide as shown above.
Dividing BY a Decimal Example Marisol has a roll of lace that is 90 inches long. How many strips of lace 6.25 inches long can she cut from the roll?
90 rounds to 100. 6.25 is close to 5. 100 ÷ 5 = 20 Your estimate tells you to expect an answer that begins in the tens column.
Step 2 Move the decimal point in the divisor to the right until the divisor is a whole number. 6. 2 5.g 90
A whole number is understood to have a decimal point after it.
Step 3 Move the decimal point in the dividend the same number of places.
g
625. 90. 0 0. Add zeros so you can move the decimal point enough places.
Step 4 Bring the decimal point up from its new position. Divide as usual. 14.4 625.g 9000.0 6250 2750 2500.0 0250.0 0250.0 0000. 0
Check: Marisol could cut 14 strips of lace. Ignore the remaining 0.4 because the problem asks how many whole strips she can cut. You know you placed the decimal point correctly because your answer 14 is reasonably close to your estimate of 20.
54 Unit 2
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Step 1 Estimate first.
Solve the following division problems. 1. 6g 6.36
3g .075
.4g 3.4
1.2g 98.4
2. 1.792 ÷ 5.6
21.15 ÷ 9
.16 ÷ 32
6.84 ÷ .36
Solve the following word problems.
Problems 3 and 4 refer to finding an average. To find an average, add a group of numbers, then divide the total by the number of items in the group. Example Find the average of 2.6, 4.6, and 1.5. 2.6 + 4.6 + 1.5 = 8.7 8.7 ÷ 3 = 2.9
4. Craig ran 4 days last week. His distances were 8.25, 6.5, 7.25, and 10.5 kilometers. What was his average distance per day for the 4 days?
5. Four concert tickets cost $154.68. How much does each ticket cost?
6. One serving of potato chips is 1.75 ounces. The entire can of chips holds 15.75 ounces. How many servings are in the can?
The average is 2.9.
3. Rainfall for the last 3 months was 6.2, 8.5, and 3.9 inches. What was the average rainfall per month for the 3-month period?
s p Chi
1 serving = 1.75 oz
15.75 oz
Use the table for problems 7 and 8. Monday
7.75 hours
Tuesday
6.5 hours
Wednesday
7.25 hours
Thursday
8.0 hours
Friday
7.25 hours
9. Carla has $60 to spend on art supplies. Acrylic paint is on sale for $1.85 per tube. How many tubes of paint can she buy? (Hint: Carla can’t buy part of a tube of paint.)
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7. Evan’s work schedule for next week is shown above. What is the average number of hours he is scheduled to work per day?
10. A land developer wants to subdivide a 14.8-acre piece of land into lots to build homes. If each lot is 0.62 acre in size, how many lots are possible?
8. Explain Evan’s gross pay (before deductions) for the week will be 463.05. How much does he earn per hour? Explain. Answers start on page 181. Unit 2 55