1.5 Decimals

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

Name and Write Decimals

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

 $0.1=\frac1{10}$ $0.1$ is “one tenth” $0.01=\frac1{100}$ $0.01$ is “one hundredth” $0.001=\frac1{1000}$ $0.001$ is “one thousandth” $0.0001=\frac1{10000}$ $0.0001$ 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 1.5.1) shows the names of the place values to the left and right of the decimal point.

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

Example 1.5.1

Name the decimal 4.3.

Solution

Step 1: Name the number to the left of the decimal point.

4 is to the left of the decimal point.

$\begin{eqnarray*}&4.3&\\&\text{four ___________}&\\\end{eqnarray*}$

Step 2: Write ‘and’ for the decimal point.

$\begin{eqnarray*}&\text{four and ________}&\end{eqnarray*}$

Step 3: Name the ‘number’ part to the right of the decimal point as if it were a whole number.

3 is to the right of the decimal point.

$\begin{eqnarray*}&\text{four and three _________}&\end{eqnarray*}$

Step 4: Name the decimal place.

Four and three tenths.

Try It

1) Name the decimal: 6.7

Solution

Six and seven tenths.

2) Name the decimal: 5.8

Solution

Five and eight tenths.

We summarize the steps needed to name a decimal below.

HOW TO

Name a Decimal.

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

Example 1.5.2

Name the decimal: -15.571

Solution

Step 1: Name the number to the left of the decimal point.

$\begin{eqnarray*}&\text{negative fifteen _________}&\end{eqnarray*}$

Step 2: Write “and” for the decimal point.

$\begin{eqnarray*}&\text{negative fifteen and _________}&\end{eqnarray*}$

Step 3: Name the number to the right of the decimal point.

$\begin{eqnarray*}&\text{negative fifteen and five hundred seventy one _________}&\end{eqnarray*}$

Step 4: The 1 is in the thousandths place.

Negative fifteen and five hundred seventy-one thousandths.

Try It

3) Name the decimal: -13.461

Solution

Negative thirteen and four hundred sixty-one thousandths.

4) Name the decimal: -2.053.

Solution

Negative two and fifty-three 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.

Example 1.5.3

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

Solution

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 to the left of the decimal point.

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 empty place holders as needed.

Zeros are needed in the tenths place.

Try It

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

Solution

13.68

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

Solution

5.94

We summarize the steps to writing a decimal.

HOW TO

Write a decimal.

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. 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. 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. 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 1.5.4

Round 18.379 to the nearest hundredth.

Solution

Step 1: Locate the given place value and mark it with an arrow.

Step 2: Underline the digit to the right of the given 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, removing all digits to the right of the rounding digit.

Try It

7) Round to the nearest hundredth: 1.047

Solution

1.05

8) Round to the nearest hundredth: 9.173

Solution

9.17

We summarize the steps for rounding a decimal here.

HOW TO

Round Decimals.

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

Example 1.5.5

Round 18.379 to the nearest

a. tenth
b. whole number.

Solution

Round 18.379

a. to the nearest tenth

Step 1: Locate the tenths place with an arrow.

Step 2: Underline the digit to the right of the given place value.

Step 3: Because 7 is greater than or equal to 5, add 1 to the 3.

Step 4: Rewrite the number, deleting all digits to the right of the rounding digit.

$18.4$

Step 5: Notice that the deleted digits were NOT replaced with zeros.

So, $18.379$ rounded to the nearest tenth is $18.4$.

b. to the nearest whole number

Step 1: Locate the ones place with an arrow.

Step 2: Underline the digit to the right of the given place value.

Step 3: Since 3 is not greater than or equal to 5, do not add 1 to the 8.

Step 4: Rewrite the number, deleting all digits to the right of the rounding digit.

So, 18.379 rounded to the nearest whole number is 18.

Try It

9) Round 6.582 to the nearest

a. hundredth
b. tenth
c. whole number.

Solution

a. 6.58
b. 6.6
c. 7

10) Round 15.2175 to the nearest

a. thousandth
b. hundredth
c. tenth.

Solution

a. 15.218
b. 15.22
c. 15.2

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. Write the numbers so the decimal points line up vertically.
2. Use zeros as place holders, as needed.
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 1.5.6

Add: $23.5+41.38$

Solution

Step 1: Write the numbers so the decimal points line up vertically.

$\begin{array}{lllll}\;\;\;23.50\\ \underline{+41.38}\\ \end{array}$

Step 2: Put 0 as a placeholder after the 5 in 23.5. Remember, $\frac5{10}=\frac{50}{100}$ so, $0.5 = 0.50$.

$\begin{array}{lllll}\;\;\;23.50\\ \underline{+41.38}\\ \end{array}$

Step 3: Add the numbers as if they were whole numbers. Then place the decimal point in the sum.

$\begin{array}{lllll}\;\;\;23.50\\ \underline{+41.38}\\ \;\;\;64.88 \end{array}$

Try It

11) Add: $4.8+11.69$

Solution

16.49

12) Add: $5.123+18.47$

Solution

23.593

Example 1.5.7

Subtract: $20-14.65$

Solution

Step 1: Write the numbers so the decimal points line up vertically.

$\begin{array}{lllll}\;\;\;20.\\ \underline{-14.65}\\ \end{array}$

Step 2: Put in zeros to the right as placeholders.

Remember, 20 is a whole number, so place the decimal point after the 0.

$\begin{array}{lllll}\;\;\;20.00\\ \underline{-14.65}\end{array}$

Step 3: Subtract and place the decimal point in the answer.

$\begin{array}{lllll}\;\;\;20.00\\ \underline{-14.65}\\\;\;\;\;5.35\end{array}$

Try It

13) Subtract $10-9.58$.

Solution

0.42

14) Subtract: $50-37.42$.

Solution

12.58

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!

 $(0.\underset{1\;place}{\underbrace3)}\;\;\;\;\;\;\;\;(0.\underset{1\;place}{\underbrace7)}$ $(0.\underset{1\;place}{\underbrace2)}\;\;\;\;\;\;\;\;(0.\underset{2\;places}{\underbrace{46})}$ Convert to fractions. $\frac3{10}\cdot\frac7{10}$ $\frac2{10}\cdot\frac{46}{100}$ Multiply. $\frac{21}{100}$ $\frac{92}{1000}$ Convert to decimals. $(0.\underset{2\;places}{\underbrace{21})}$ $(0.\underset{3\;places}{\underbrace{096})}$

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. Determine the sign of the product.
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. 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. Write the product with the appropriate sign.

Example 1.5.8

Multiply: $(-3.9)(4.075)$

Solution

Step 1: The signs are different. The product will be negative.
Step 2: Write in vertical format, lining up the numbers on the right.

\begin{align*} 4.075&\\ \underline{\times\;\;3.9}&\\ \end{align*}

Step 3: Multiply.

\begin{align*}4.075&\\ \underline{\times\;\;3.9}&\\158925&\end{align*}

Step 4: Add the number of decimal places in the factors $(1 + 3)$.

$(-3.\underbrace{9}_{1\;place})$   $(4.\underbrace{075}_{3\;places})$

Step 5: Place the decimal point 4 places from the right.

\begin{align*}4.075\\\underline{\times\;\;3.9}\\15.\underbrace{8925}&\\ \;\;\;\;\text{4 places}\end{align*}

Step 6: The signs are different, so the product is negative.

$(−3.9)(4.075) = −15.8925$

Try It

15) Multiply: $-4.5(6.107)$

Solution

-27.4815

16) Multiply: $-10.79(8.12)$

Solution

-87.6148

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

Example 1.5.9

Multiply 5.63

a. by 10
b. by 100
c. 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.

a.

Step 1: There is 1 zero in 10, so move the decimal point 1 place to the right.

$5.63(10)$

b.

Step 1: There are 2 zeros in 100, so move the decimal point 2 places to the right.

$5.63(100)$

c.

Step 1: There are 3 zeros in 1,000, so move the decimal point 3 places to the right.

$5.63(1000)$

Step 2: A zero must be added at the end.

$5,630$

Try It

17) Multiply 2.58

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

Solution

a. 25.8
b. 258
c. 2,580

18) Multiply 14.2
a. by 10
b. by 100
c. by 1,000

Solution

a. 142
b. 1,420
c. 14,200

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:

$\frac{0.8}{0.4}$

$\frac{0.8(10)}{0.4(10)}$

$\frac{8}{4}$

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:

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

HOW TO

Divide Decimals.

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

Example 1.5.10

Divide: $-25.56\div (-0.06)$

Solution

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

Step 1: The signs are the same.

The quotient is positive.

Step 2: Make the divisor a whole number by “moving” the decimal point all the way to the right.
Step 3: “Move” the decimal point in the dividend the same number of places.

Step 4: Divide.

Place the decimal point in the quotient above the decimal point in the dividend.

Step 5: Write the quotient with the appropriate sign.

$-25.65\div (-0.06)=427.5$

Try It

19) Divide: $-23.492\div (-0.04)$

Solution

687.3

20) Divide: $-4.11\div (-0.12)$

Solution

34.25

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 1.5.13. In calculations with money, we will round the answer to the nearest cent (hundredth).

Example 1.5.11

Divide: $\3.99\div 24$.

Solution

Step 1: Place the decimal point in the quotient above the decimal point in the dividend.
Step 2: Divide as usual.

When do we stop? Since this division involves money, we round it to the nearest cent (hundredth.) To do this, we must carry the division to the thousandths place.

Step 3: Round to the nearest cent.

$\begin{eqnarray*}0.166&\approx&0.17\\\3.99\div24&\approx&\0.17\end{eqnarray*}$

Try It

21) Divide: $\6.99\div 36$

Solution

0.19

22) Divide: $\4.99\div 12$

Solution

0.42

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.

$0.03=\frac{3}{100}$

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.

HOW TO

Convert a Decimal to a Proper Fraction.

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

Example 1.5.12

Write $0.374$ as a fraction.

Solution

Step 1: Determine the place value of the final digit.

Step 2: Write the fraction for 0.374:

• The numerator is 374.
• The denominator is 1,000.

$\frac{374}{1000}$

Step 3: Simplify the fraction.

$\frac{2\;\times\;187}{2\;\times\;500}$

Step 4: Divide out the common factors.

$\frac{187}{500}$

So, $0.374=\frac{187}{500}$.

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 $0.374$?

Try It

23) Write $0.234$ as a fraction.

Solution

$\frac{117}{500}$

24)Write $0.024$ as a fraction.

Solution

$\frac{3}{125}$

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 $4\div5$ or 5$\overline)4$ . 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 1.5.13

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

Solution

Step 1: Since a fraction bar means division, we begin by writing $\frac{5}{8}$ as $8\overline)5$

Step 2: Now divide.

Try It

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

Solution

$-0.875$

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

Solution

$-0.375$

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 1.5.14

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

Solution

Try It

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

Solution

$2.\bar{45}$

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

Solution

$2.3\bar{18}$

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

Example 1.5.15

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.

Step 1: Change $\frac{7}{8}$ to a decimal.

$0.875+6.4 = 7.275$

So, $\frac{7}{8}+6.4=7.275$

Try It

29) Simplify: $\frac{3}{8}+4.9$

Solution

$5.275$

30) Simplify: $5.7+\frac{13}{20}$

Solution

$6.35$

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.

 $6\%$ $78\%$ $135\%$ 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. $0.06$ $0.78$ $1.35$

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 1.5.16

Convert each percent to a decimal:

a. $62\%$
b. $135\%$
c. $35.7\%$

Solution

a.

Step 1: Move the decimal point two places to the left.

$0.62$

b.

Step 1: Move the decimal point two places to the left.

$1.35$

c.

Step 1: Move the decimal point two places to the left.

$0.057$

Try It

31) Convert each percent to a decimal:
a. $9\%$
b. $87\%$
c. $3.9\%$

Solution

a. $0.09$
b. $0.87$
c. $0.039$

32) Convert each percent to a decimal:
a. $3\%$
b. $91\%$
c. $8.3\%$

Solution

a. $0.03$
b. $0.91$
c. $0.083$

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

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 1.5.17

Convert each decimal to a percent:

a. $0.51$
b. $1.25$
c. $0.093$

Solution

a.

Step 1: Move the decimal point two places to the right.

$51\%$

b.

Step 1: Move the decimal point two places to the right.

$125\%$

c.

Step 1: Move the decimal point two places to the right.

$9.3\%$

Try It

33) Convert each decimal to a percent:
a. $0.17$
b. $1.75$
c. $0.0825$

Solution

a. $17\%$
b. $175\%$
c. $8.25\%$

34) Convert each decimal to a percent:
a. $0.41$
b. $2.25$
c. $0.0925$

Solution

a. $41\%$
b. $225\%$
c. $9.25\%$

Key Concepts

• Name a Decimal
1. Name the number to the left of the decimal point.
2. Write ”and” for the decimal point.
3. Name the “number” part to the right of the decimal point as if it were a whole number.
4. Name the decimal place of the last digit.
• Write a Decimal
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. 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. 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. Fill in zeros for place holders as needed.
• Round a Decimal
1. Locate the given place value and mark it with an arrow.
2. Underline the digit to the right of the place value.
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. Rewrite the number, deleting all digits to the right of the rounding digit.
1. Write the numbers so the decimal points line up vertically.
2. Use zeros as place holders, as needed.
3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
• Multiply Decimals
1. Determine the sign of the product.
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. Place the decimal point. The number of decimal places in the product is the sum of the decimal places in the factors.
4. Write the product with the appropriate sign.
• Multiply a Decimal by a Power of Ten
1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
2. Add zeros at the end of the number as needed.
• Divide Decimals
1. Determine the sign of the quotient.
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. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
4. Write the quotient with the appropriate sign.
• Convert a Decimal to a Proper Fraction
1. Determine the place value of the final digit.
2. Write the fraction: numerator—the ‘numbers’ to the right of the decimal point; denominator—the place value corresponding to the final digit.
• Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator

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?

Glossary

decimal
A decimal is another way of writing a fraction whose denominator is a power of ten.
percent
A percent is a ratio whose denominator is 100.
repeating decimal
A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.
definition