 Elementary Algebra 2e

# 1.7Decimals

### Learning Objectives

By the end of this section, you will be able to:

• Name and write decimals
• Round decimals
• Multiply and divide decimals
• Convert decimals, fractions, and percents
Be Prepared 1.7

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Decimals.

### Name and Write Decimals

Decimals are another way of writing fractions whose denominators are powers of 10.

$0.1=1100.1is “one tenth”0.01=11000.01is “one hundredth”0.001=11,0000.001 is “one thousandth”0.0001=110,0000.0001 is “one ten-thousandth”0.1=1100.1is “one tenth”0.01=11000.01is “one hundredth”0.001=11,0000.001 is “one thousandth”0.0001=110,0000.0001 is “one ten-thousandth”$

Notice that “ten thousand” is a number larger than one, but “one ten-thousandth” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.

When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values. Figure 1.14 shows the names of the place values to the left and right of the decimal point.

Figure 1.14 Place value of decimal numbers are shown to the left and right of the decimal point.

### Example 1.91

#### How to Name Decimals

Name the decimal 4.3.

Try It 1.181

Name the decimal: $6.7.6.7.$

Try It 1.182

Name the decimal: $5.8.5.8.$

We summarize the steps needed to name a decimal below.

### How To

#### Name a Decimal.

1. Step 1. Name the number to the left of the decimal point.
2. Step 2. Write “and” for the decimal point.
3. Step 3. Name the “number” part to the right of the decimal point as if it were a whole number.
4. Step 4. Name the decimal place of the last digit.

### Example 1.92

Name the decimal: $−15.571.−15.571.$

Try It 1.183

Name the decimal: $−13.461.−13.461.$

Try It 1.184

Name the decimal: $−2.053.−2.053.$

When we write a check we write both the numerals and the name of the number. Let’s see how to write the decimal from the name.

### Example 1.93

#### How to Write Decimals

Write “fourteen and twenty-four thousandths” as a decimal.

Try It 1.185

Write as a decimal: thirteen and sixty-eight thousandths.

Try It 1.186

Write as a decimal: five and ninety-four thousandths.

We summarize the steps to writing a decimal.

### How To

#### Write a decimal.

1. Step 1. Look for the word “and”—it locates the decimal point.
• Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
• If there is no “and,” write a “0” with a decimal point to its right.
2. Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
3. Step 3. Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
4. Step 4. Fill in zeros for place holders as needed.

### Round Decimals

Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.

### Example 1.94

#### How to Round Decimals

Round 18.379 to the nearest hundredth.

Try It 1.187

Round to the nearest hundredth: $1.047.1.047.$

Try It 1.188

Round to the nearest hundredth: $9.173.9.173.$

We summarize the steps for rounding a decimal here.

### How To

#### Round Decimals.

1. Step 1. Locate the given place value and mark it with an arrow.
2. Step 2. Underline the digit to the right of the place value.
3. Step 3. Is this digit greater than or equal to 5?
• Yes—add 1 to the digit in the given place value.
• No—do not change the digit in the given place value.
4. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.

### Example 1.95

Round 18.379 to the nearest tenth whole number.

Try It 1.189

Round $6.5826.582$ to the nearest hundredth tenth whole number.

Try It 1.190

Round $15.217515.2175$ to the nearest thousandth hundredth tenth.

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

### How To

1. Step 1. Write the numbers so the decimal points line up vertically.
2. Step 2. Use zeros as place holders, as needed.
3. Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.

### Example 1.96

Add: $23.5+41.38.23.5+41.38.$

Try It 1.191

Add: $4.8+11.69.4.8+11.69.$

Try It 1.192

Add: $5.123+18.47.5.123+18.47.$

### Example 1.97

Subtract: $20−14.65.20−14.65.$

Try It 1.193

Subtract: $10−9.58.10−9.58.$

Try It 1.194

Subtract: $50−37.42.50−37.42.$

### Multiply and Divide Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first convert them to fractions and then multiply.

So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side. Look for a pattern! Convert to fractions. Multiply. Convert to decimals. Notice, in the first example, we multiplied two numbers that each had one digit after the decimal point and the product had two decimal places. In the second example, we multiplied a number with one decimal place by a number with two decimal places and the product had three decimal places.

We multiply the numbers just as we do whole numbers, temporarily ignoring the decimal point. We then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product.

The rules for multiplying positive and negative numbers apply to decimals, too, of course!

When multiplying two numbers,

• if their signs are the same the product is positive.
• if their signs are different the product is negative.

When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. Finally, we write the product with the appropriate sign.

### How To

#### Multiply Decimals.

1. Step 1. Determine the sign of the product.
2. Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
3. Step 3. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
4. Step 4. Write the product with the appropriate sign.

### Example 1.98

Multiply: $(−3.9)(4.075).(−3.9)(4.075).$

Try It 1.195

Multiply: $−4.5(6.107).−4.5(6.107).$

Try It 1.196

Multiply: $−10.79(8.12).−10.79(8.12).$

In many of your other classes, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc.). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.

### How To

#### Multiply a Decimal by a Power of Ten.

1. Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
2. Step 2. Add zeros at the end of the number as needed.

### Example 1.99

Multiply 5.63 by 10 by 100 by 1,000.

Try It 1.197

Multiply 2.58 by 10 by 100 by 1,000.

Try It 1.198

Multiply 14.2 by 10 by 100 by 1,000.

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.

To divide decimals, determine what power of 10 to multiply the denominator by to make it a whole number. Then multiply the numerator by that same power of $10.10.$ Because of the equivalent fractions property, we haven’t changed the value of the fraction! The effect is to move the decimal points in the numerator and denominator the same number of places to the right. For example:

$0.80.40.8(10)0.4(10)840.80.40.8(10)0.4(10)84$

We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.

We review the notation and vocabulary for division:

$adividend÷bdivisor=cquotientbdivisorcquotientadividendadividend÷bdivisor=cquotientbdivisorcquotientadividend$

We’ll write the steps to take when dividing decimals, for easy reference.

### How To

#### Divide Decimals.

1. Step 1. Determine the sign of the quotient.
2. Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places—adding zeros as needed.
3. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
4. Step 4. Write the quotient with the appropriate sign.

### Example 1.100

Divide: $−25.65÷(−0.06).−25.65÷(−0.06).$

Try It 1.199

Divide: $−23.492÷(−0.04).−23.492÷(−0.04).$

Try It 1.200

Divide: $−4.11÷(−0.12).−4.11÷(−0.12).$

A common application of dividing whole numbers into decimals is when we want to find the price of one item that is sold as part of a multi-pack. For example, suppose a case of 24 water bottles costs $3.99. To find the price of one water bottle, we would divide$3.99 by 24. We show this division in Example 1.101. In calculations with money, we will round the answer to the nearest cent (hundredth).

### Example 1.101

Divide: $3.99÷24.3.99÷24.$

Try It 1.201

Divide: $6.99÷36.6.99÷36.$

Try It 1.202

Divide: $4.99÷12.4.99÷12.$

### Convert Decimals, Fractions, and Percents

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal 0.03 the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.

$00.03=310000.03=3100$

Notice, when the number to the left of the decimal is zero, we get a fraction whose numerator is less than its denominator. Fractions like this are called proper fractions.

The steps to take to convert a decimal to a fraction are summarized in the procedure box.

### How To

#### Convert a Decimal to a Proper Fraction.

1. Step 1. Determine the place value of the final digit.
2. Step 2. Write the fraction.
• numerator—the “numbers” to the right of the decimal point
• denominator—the place value corresponding to the final digit

### Example 1.102

Write 0.374 as a fraction.

Try It 1.203

Write 0.234 as a fraction.

Try It 1.204

Write 0.024 as a fraction.

We’ve learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar means division. So $4545$ can be written $4÷54÷5$ or $54.54.$ This leads to the following method for converting a fraction to a decimal.

### How To

#### Convert a Fraction to a Decimal.

To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

### Example 1.103

Write $−58−58$ as a decimal.

Try It 1.205

Write $−78−78$ as a decimal.

Try It 1.206

Write $−38−38$ as a decimal.

When we divide, we will not always get a zero remainder. Sometimes the quotient ends up with a decimal that repeats. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly. A bar is placed over the repeating block of digits to indicate it repeats.

### Repeating Decimal

A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.

A bar is placed over the repeating block of digits to indicate it repeats.

### Example 1.104

Write $43224322$ as a decimal.

Try It 1.207

Write $27112711$ as a decimal.

Try It 1.208

Write $51225122$ as a decimal.

Sometimes we may have to simplify expressions with fractions and decimals together.

### Example 1.105

Simplify: $78+6.4.78+6.4.$

Try It 1.209

Simplify: $38+4.9.38+4.9.$

Try It 1.210

Simplify: $5.7+1320.5.7+1320.$

A percent is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent.

### Percent

A percent is a ratio whose denominator is 100.

Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator.

 $6%6%$ $78%78%$ $135%135%$ Write as a ratio with denominator 100. $61006100$ $7810078100$ $135100135100$ Change the fraction to a decimal by dividing the numerator by the denominator. $0.060.06$ $0.780.78$ $1.351.35$
Table 1.30

Do you see the pattern? To convert a percent number to a decimal number, we move the decimal point two places to the left. ### Example 1.106

Convert each percent to a decimal: 62% 135% 35.7%.

Try It 1.211

Convert each percent to a decimal: $9%9%$ $87%87%$ 3.9%.

Try It 1.212

Convert each percent to a decimal: 3% 91% 8.3%.

Converting a decimal to a percent makes sense if we remember the definition of percent and keep place value in mind.

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent.

 $0.830.83$ $1.051.05$ $0.0750.075$ Write as a fraction. $8310083100$ $1510015100$ $751000751000$ The denominator is 100. $105100105100$ $7.51007.5100$ Write the ratio as a percent. $83%83%$ $105%105%$ $7.5%7.5%$
Table 1.31

Recognize the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign. ### Example 1.107

Convert each decimal to a percent: 0.51 1.25 0.093.

Try It 1.213

Convert each decimal to a percent: 0.17 1.75 0.0825.

Try It 1.214

Convert each decimal to a percent: 0.41 2.25 0.0925.

### Section 1.7 Exercises

#### Practice Makes Perfect

Name and Write Decimals

In the following exercises, write as a decimal.

531.

Twenty-nine and eighty-one hundredths

532.

Sixty-one and seventy-four hundredths

533.

Seven tenths

534.

Six tenths

535.

Twenty-nine thousandth

536.

Thirty-five thousandths

537.

Negative eleven and nine ten-thousandths

538.

Negative fifty-nine and two ten-thousandths

In the following exercises, name each decimal.

539.

5.5

540.

14.02

541.

8.71

542.

2.64

543.

0.002

544.

0.479

545.

$−17.9−17.9$

546.

$−31.4−31.4$

Round Decimals

In the following exercises, round each number to the nearest tenth.

547.

0.67

548.

0.49

549.

2.84

550.

4.63

In the following exercises, round each number to the nearest hundredth.

551.

0.845

552.

0.761

553.

0.299

554.

0.697

555.

4.098

556.

7.096

In the following exercises, round each number to the nearest hundredth tenth whole number.

557.

5.781

558.

1.6381

559.

63.479

560.

$84.28184.281$

In the following exercises, add or subtract.

561.

$16.92+7.5616.92+7.56$

562.

$248.25−91.29248.25−91.29$

563.

$21.76−30.9921.76−30.99$

564.

$38.6+13.6738.6+13.67$

565.

$−16.53−24.38−16.53−24.38$

566.

$−19.47−32.58−19.47−32.58$

567.

$−38.69+31.47−38.69+31.47$

568.

$29.83+19.7629.83+19.76$

569.

$72.5−10072.5−100$

570.

$86.2−10086.2−100$

571.

$15+0.7315+0.73$

572.

$27+0.8727+0.87$

573.

$91.95−(−10.462)91.95−(−10.462)$

574.

$94.69−(−12.678)94.69−(−12.678)$

575.

$55.01−3.755.01−3.7$

576.

$59.08−4.659.08−4.6$

577.

$2.51−7.42.51−7.4$

578.

$3.84−6.13.84−6.1$

Multiply and Divide Decimals

In the following exercises, multiply.

579.

$(0.24)(0.6)(0.24)(0.6)$

580.

$(0.81)(0.3)(0.81)(0.3)$

581.

$(5.9)(7.12)(5.9)(7.12)$

582.

$(2.3)(9.41)(2.3)(9.41)$

583.

$(−4.3)(2.71)(−4.3)(2.71)$

584.

$(−8.5)(1.69)(−8.5)(1.69)$

585.

$(−5.18)(−65.23)(−5.18)(−65.23)$

586.

$(−9.16)(−68.34)(−9.16)(−68.34)$

587.

$(0.06)(21.75)(0.06)(21.75)$

588.

$(0.08)(52.45)(0.08)(52.45)$

589.

$(9.24)(10)(9.24)(10)$

590.

$(6.531)(10)(6.531)(10)$

591.

$(55.2)(1000)(55.2)(1000)$

592.

$(99.4)(1000)(99.4)(1000)$

In the following exercises, divide.

593.

$4.75÷254.75÷25$

594.

$12.04÷4312.04÷43$

595.

$117.25÷48117.25÷48$

596.

$109.24÷36109.24÷36$

597.

$0.6÷0.20.6÷0.2$

598.

$0.8÷0.40.8÷0.4$

599.

$1.44÷(−0.3)1.44÷(−0.3)$

600.

$1.25÷(−0.5)1.25÷(−0.5)$

601.

$−1.75÷(−0.05)−1.75÷(−0.05)$

602.

$−1.15÷(−0.05)−1.15÷(−0.05)$

603.

$5.2÷2.55.2÷2.5$

604.

$6.5÷3.256.5÷3.25$

605.

$11÷0.5511÷0.55$

606.

$14÷0.3514÷0.35$

Convert Decimals, Fractions and Percents

In the following exercises, write each decimal as a fraction.

607.

0.04

608.

0.19

609.

0.52

610.

0.78

611.

1.25

612.

1.35

613.

0.375

614.

0.464

615.

0.095

616.

0.085

In the following exercises, convert each fraction to a decimal.

617.

$17201720$

618.

$13201320$

619.

$114114$

620.

$174174$

621.

$−31025−31025$

622.

$−28425−28425$

623.

$15111511$

624.

$18111811$

625.

$1511115111$

626.

$2511125111$

627.

$2.4+582.4+58$

628.

$3.9+9203.9+920$

In the following exercises, convert each percent to a decimal.

629.

1%

630.

2%

631.

63%

632.

71%

633.

150%

634.

250%

635.

21.4%

636.

39.3%

637.

7.8%

638.

6.4%

In the following exercises, convert each decimal to a percent.

639.

0.01

640.

0.03

641.

1.35

642.

1.56

643.

3

644.

4

645.

0.0875

646.

0.0625

647.

2.254

648.

2.317

#### Everyday Math

649.

Salary Increase Danny got a raise and now makes $58,965.95 a year. Round this number to the nearest dollar thousand dollars ten thousand dollars. 650. New Car Purchase Selena’s new car cost$23,795.95. Round this number to the nearest dollar thousand dollars ten thousand dollars.

651.

Sales Tax Hyo Jin lives in San Diego. She bought a refrigerator for $1,624.99 and when the clerk calculated the sales tax it came out to exactly$142.186625. Round the sales tax to the nearest penny and dollar.

652.

Sales Tax Jennifer bought a $1,038.99 dining room set for her home in Cincinnati. She calculated the sales tax to be exactly$67.53435. Round the sales tax to the nearest penny and dollar.

653.

Paycheck Annie has two jobs. She gets paid $14.04 per hour for tutoring at City College and$8.75 per hour at a coffee shop. Last week she tutored for 8 hours and worked at the coffee shop for 15 hours. How much did she earn? If she had worked all 23 hours as a tutor instead of working both jobs, how much more would she have earned?

654.

Paycheck Jake has two jobs. He gets paid $7.95 per hour at the college cafeteria and$20.25 at the art gallery. Last week he worked 12 hours at the cafeteria and 5 hours at the art gallery. How much did he earn? If he had worked all 17 hours at the art gallery instead of working both jobs, how much more would he have earned?

#### Writing Exercises

655.

656.

Explain how you write “three and nine hundredths” as a decimal.

657.

Without solving the problem “44 is 80% of what number” think about what the solution might be. Should it be a number that is greater than 44 or less than 44? Explain your reasoning.

658.

When the Szetos sold their home, the selling price was 500% of what they had paid for the house 30 years ago. Explain what 500% means in this context.

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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