Elementary Algebra 2e · free preview

1.7 Decimals

Learning objectives: Name and write decimals; Round decimals; Add and subtract decimals; Multiply and divide decimals; Convert decimals, fractions, and percents

1. Learning objectives

Learning objectives

Name and write decimals

Round decimals

Add and subtract decimals

Multiply and divide decimals

Convert decimals, fractions, and percents

2. Name and Write Decimals

Name and Write Decimals

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

\begin{array}{rllccccl} 0.1 & = & \frac{1}{10} & & & & & 0.1\,\mathrm{is "one tenth"} \\ 0.01 & = & \frac{1}{100} & & & & & 0.01\,\mathrm{is "one hundredth"} \\ 0.001 & = & \frac{1}{1,000} & & & & & \mathrm{0.001 is "one thousandth"} \\ 0.0001 & = & \frac{1}{10,000} & & & & & \mathrm{0.0001 is "one ten-thousandth"} \\ \end{array}

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.

3. Example 1.91

Example 1.91

How to Name Decimals

Name the decimal 4.3.

We summarize the steps needed to name a decimal below.

Name a Decimal.

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

4. Example 1.92

Example 1.92

Name the decimal: $-15.571.$

Solution


	$-15.571$
Name the number to the left of the decimal point.	negative fifteen \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
Write “and” for the decimal point.	negative fifteen and \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
Name the number to the right of the decimal point.	negative fifteen and five hundred seventy-one \_\_\_\_\_\_\_\_\_\_
The 1 is in the thousandths place.	negative fifteen and five hundred seventy-one thousandths

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.

5. Example 1.93

Example 1.93

How to Write Decimals

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

We summarize the steps to writing a decimal.

Write a decimal.

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.
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.
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.
Step 4. Fill in zeros for place holders as needed.

6. Round Decimals

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.

7. Example 1.94

Example 1.94

How to Round Decimals

Round 18.379 to the nearest hundredth.

We summarize the steps for rounding a decimal here.

Round Decimals.

Step 1. Locate the given place value and mark it with an arrow.
Step 2. Underline the digit to the right of the place value.
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.
Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.

8. Example 1.95

Example 1.95

Round 18.379 to the nearest ⓐ tenth ⓑ whole number.

Solution

Round 18.379

ⓐ to the nearest tenth Locate the tenths place with an arrow. Underline the digit to the right of the given place value. Because 7 is greater than or equal to 5, add 1 to the 3. Rewrite the number, deleting all digits to the right of the rounding digit. Notice that the deleted digits were NOT replaced with zeros. So, 18.379 rounded to the nearest tenth is 18.4.

ⓑ to the nearest whole number Locate the ones place with an arrow. Underline the digit to the right of the given place value. Since 3 is not greater than or equal to 5, do not add 1 to the 8. Rewrite the number, deleting all digits to the right of the rounding digit. So, 18.379 rounded to the nearest whole number is 18.

9. Add and Subtract Decimals

Add and Subtract Decimals

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.

Add or Subtract Decimals.

Step 1. Write the numbers so the decimal points line up vertically.
Step 2. Use zeros as place holders, as needed.
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.

10. Example 1.96

Example 1.96

Add: $23.5+41.38.$

Solution


Write the numbers so the decimal points line up vertically.	$\underset{\mathrm{\_\_\_\_\_\_}}{\begin{array}{l} \,23.5 \\ +41.38 \\ \end{array}}$
Put 0 as a placeholder after the 5 in 23.5. Remember, $\frac{5}{10}=\frac{50}{100}\,\mathrm{so}\,0.5=0.50$ .	$\underset{\mathrm{\_\_\_\_\_\_}}{\begin{array}{l} \,23.50 \\ +41.38 \\ \end{array}}$
Add the numbers as if they were whole numbers. Then place the decimal point in the sum.	$\begin{array}{l} \underset{\mathrm{\_\_\_\_\_\_}}{\begin{array}{l} \,23.50 \\ +41.38 \\ \end{array}} \\ \,64.88 \\ \end{array}$

Table

1.28

11. Example 1.97

Example 1.97

Subtract: $20-14.65.$

Solution


	$20-14.65$
Write the numbers so the decimal points line up vertically. Remember, 20 is a whole number, so place the decimal point after the 0.	$\underset{\mathrm{\_\_\_\_\_\_}}{\begin{array}{l} \,20. \\ -14.65 \\ \end{array}}$
Put in zeros to the right as placeholders.	$\begin{array}{l} \,20.00 \\ \underset{\mathrm{\_\_\_\_\_\_}}{-14.65} \\ \end{array}$
Subtract and place the decimal point in the answer.	$\begin{array}{l} \underset{\mathrm{\_\_\_\_\_\_\_\_\_\_}}{\begin{array}{l} \,\overset{1}{2}\overset{\overset{9}{10}}{0}.\overset{\overset{9}{10}}{0}\overset{10}{0} \\ -\,1\,4\,.\,6\,5 \\ \end{array}} \\ \,5\,.\,3\,5 \\ \end{array}$

Table

1.29

12. Multiply and Divide Decimals

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!

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.

Multiply Decimals.

Step 1. Determine the sign of the product.
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.
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.
Step 4. Write the product with the appropriate sign.

13. Example 1.98

Example 1.98

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

Solution

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.

Multiply a Decimal by a Power of Ten.

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

14. Example 1.99

Example 1.99

Multiply 5.63 ⓐ by 10 ⓑ by 100 ⓒ by 1,000.

Solution

By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.

ⓐ

ⓑ

ⓒ

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 $10.$ 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:

\begin{array}{c} \frac{0.8}{0.4} \\ \frac{0.8(10)}{0.4(10)} \\ \frac{8}{4} \\ \end{array}

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:

\underset{\mathrm{dividend}}{a}\mathrm{÷}\underset{\mathrm{divisor}}{b}=\underset{\mathrm{quotient}}{c}\,\underset{\mathrm{divisor}}{b}\begin{array}{r} \underset{\mathrm{quotient}}{c} \\ \underset{\mathrm{dividend}}{a} \\ \end{array}

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

Divide Decimals.

Step 1. Determine the sign of the quotient.
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.
Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
Step 4. Write the quotient with the appropriate sign.

15. Example 1.100

Example 1.100

Divide: $-25.65\mathrm{÷}(-0.06).$

Solution

Remember, you can “move” the decimals in the divisor and dividend because of the Equivalent Fractions Property.

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).

16. Example 1.101

Example 1.101

Divide: $\textdollar 3.99\mathrm{÷}24.$

Solution

17. Convert Decimals, Fractions, and Percents

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.

0\,0.03=\frac{3}{100}

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.

Convert a Decimal to a Proper Fraction.

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

18. Example 1.102

Example 1.102

Write 0.374 as a fraction.

Solution

Did you notice that the number of zeros in the denominator of $\frac{374}{1,000}$ is the same as the number of decimal places in 0.374?

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 $\frac{4}{5}$ can be written $4\mathrm{÷}5$ or $54.$ This leads to the following method for converting a fraction to a decimal.

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.

19. Example 1.103

Example 1.103

Write $-\,\frac{5}{8}$ as a decimal.

Solution

Since a fraction bar means division, we begin by writing $\frac{5}{8}$ as $85.$ Now divide.

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.

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.

20. Example 1.104

Example 1.104

Write $\frac{43}{22}$ as a decimal.

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

21. Example 1.105

Example 1.105

Simplify: $\frac{7}{8}+6.4.$

Solution

First we must change one number so both numbers are in the same form. We can change the fraction to a decimal, or change the decimal to a fraction. Usually it is easier to change the fraction to a decimal.

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


	$6\%$	$78\%$	$135\%$
Write as a ratio with denominator 100.	$\frac{6}{100}$	$\frac{78}{100}$	$\frac{135}{100}$
Change the fraction to a decimal by dividing the numerator by the denominator.	$0.06$	$0.78$	$1.35$

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.

22. Example 1.106

Example 1.106

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

Solution

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.


	$0.83$	$1.05$	$0.075$
Write as a fraction.	$\frac{83}{100}$	$1\frac{5}{100}$	$\frac{75}{1000}$
The denominator is 100.		$\frac{105}{100}$	$\frac{7.5}{100}$
Write the ratio as a percent.	$83\%$	$105\%$	$7.5\%$

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.

23. Example 1.107

Example 1.107

Convert each decimal to a percent: ⓐ 0.51 ⓑ 1.25 ⓒ 0.093.

Solution

24. Section 1.7 Exercises

Section 1.7 Exercises

Name and Write Decimals

In the following exercises, write as a decimal.

In the following exercises, name each decimal.

Round Decimals

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

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

Add and Subtract Decimals

In the following exercises, add or subtract.

Multiply and Divide Decimals

In the following exercises, multiply.

In the following exercises, divide.

Convert Decimals, Fractions and Percents

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

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

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

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

Everyday Math

Writing Exercises

ⓐ 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?

25. Practice 1

Name the decimal: $6.7.$

Name the decimal: $5.8.$

Name the decimal: $-13.461.$

26. Practice 2

Name the decimal: $-2.053.$

27. Practice 3

Add: $5.123+18.47.$

Multiply: $-4.5(6.107).$

Multiply: $-10.79(8.12).$

28. Practice 4

Divide: $\$6.99\mathrm{÷}36.$

29. Practice 5

Divide: $\$4.99\mathrm{÷}12.$

Write 0.234 as a fraction.

Write 0.024 as a fraction.

30. Practice 6

Write $-\,\frac{7}{8}$ as a decimal.

Write $-\,\frac{3}{8}$ as a decimal.

Write $\frac{27}{11}$ as a decimal.

31. Practice 7

Write $\frac{51}{22}$ as a decimal.

32. Practice 8

Negative eleven and nine ten-thousandths

33. Practice 9

$-17\mathrm{.9}\,$

$21.76-30.99$

$-16.53-24.38$

34. Practice 10

$-38.69+31.47$

$72.5-100$

$2.51-7.4$

35. Practice 11

$(-4.3)(2.71)$

$(55.2)(1000)$

$\$117.25\mathrm{÷}48$

36. Practice 12

$1.44\mathrm{÷}(-0.3)$

$-\,\frac{310}{25}$

$\frac{15}{11}$

37. Practice 13

$\frac{15}{111}$

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.

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.

38. Practice 14

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?

How does knowing about US money help you learn about decimals?

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.

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