1.8: Decimals

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Learning Objectives

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

Name and write decimals
Round decimals
Add and subtract decimals
Multiply and divide decimals
Convert decimals, fractions, and percents

Note

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

Name and Write Decimals

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

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

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 $\PageIndex{1}$ shows the names of the place values to the left and right of the decimal point.

A table is shown with the title Place Value. From left to right the row reads “Hundred thousands,” “Ten thousands,” “Thousands,” “Hundreds,” “Tens,” and “Ones.” Then there is a blank cell and below it is a decimal point. To the right of this, the cells read “Tenths,” “Hundredths,” “Thousandths,” “Ten-thousandths,” and “Hundred-thousandths.” — Figure $\PageIndex{1}$: Place value of decimal numbers are shown to the left and right of the decimal point.

Exercise $\PageIndex{1}$

Name the decimal $4.3$.

Answer: $A table is given with four steps. Additionally, the number 4.3 is given. The first step reads “Step 1. Name the number to the left of the decimal point.” To the right of this, it is noted that “4 is to the left of the decimal point.” To the right of this, it reads “four” followed by a large blank space.$ $The second step reads “Step 2. Write ‘and’ for the decimal point.” To the right of this it reads “four and” followed by a blank space.$ $The third step reads “Step 3. Name the ‘number’ part to the right of the decimal point as if it were a whole number.” To the right of this, it reads “3 is to the right of the decimal point.” To the right of this, it reads “four and three” followed by a blank.$ $Finally, the last step reads “Step 4. Name the decimal place.” To the right of this, it reads “four and three tenths.”$

Exercise $\PageIndex{2}$

Name the decimal $6.7$.

Answer: six and seven tenths

Exercise $\PageIndex{3}$

Name the decimal $5.8$.

Answer: five and eight tenths

We summarize the steps needed to name a decimal below.

NAME A DECIMAL.

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

Exercise $\PageIndex{4}$

Name the decimal: $−15.571$.

Answer

	$−15.571$
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

Exercise $\PageIndex{5}$

Name the decimal: $−13.461$.

Answer: negative thirteen and four hundred sixty-one thousandths

Exercise $\PageIndex{6}$

Name the decimal: $−2.053$.

Answer: 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.

Exercise $\PageIndex{7}$: How to Write decimals

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

Answer: $A table is given with four steps. The first step reads “Step 1. Look for the work ‘and’ – it locates the decimal point. Place a decimal point under the word ‘and’. Translate the words before ‘and’ into the whole number and place it to the left of the decimal point.” To the right of this, we have the words “fourteen and twenty-four thousandths.” Below this word, we have “fourteen and twenty-four thousandths” with the word “and” underlined. Below this word, we have a small blank space separated from a larger blank space by a decimal point. Under this, we have 14 in the small blank space followed by the decimal point and the larger blank space.$ $The second step reads “Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.” To the right of this it reads “The last word is thousandths.” To the right of this there is the number 14 followed by a decimal point and three small blank spaces. Under the blank spaces, the words “tenths,” “hundredths,” and “thousandths” are written.$ $The third step reads “Step 3. Translate the words after ‘and’ into the number to the right of the decimal point. Write the number in the spaces – putting the final digit in the last place.” To the right of this, we have 14 followed by a decimal followed by a blank space followed by 2 and 4 on the other two previously blank spaces.$ $Finally, the last step reads “Step 4. Fill in zeros for empty place holders as needed.” To the right of this, it reads “Zeros are needed in the tenths place.” To the right of this, we have 14 followed by a decimal point followed by 0, 2, and 4, respectively, on the blank spaces. Below this, we have “fourteen and twenty-four thousandths is written 14.024.”$

Exercise $\PageIndex{8}$

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

Answer: 13.068

Exercise $\PageIndex{9}$

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

Answer: 5.094

We summarize the steps to writing a decimal.

WRITE A DECIMAL.

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.
Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
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.
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.

Exercise $\PageIndex{10}$

Round 18.379 to the nearest hundredth.

Answer: $A table is given with four steps. The first step reads “Step 1: Locate the given place value and mark it with an arrow.” To the right of this, we have the number 18.379; above it, are the words hundreds place, which has an arrow pointing to the 7.$ $The second step reads “Step 2. Underline the digit to the right of the given place value.” To the right of this, we have 18.379 with the 9 underlined.$ $The third step reads “Step 3. Is this digit greater than or equal to 5? Below this reads, “Yes: add 1 to the digit in the given place value.” Below this reads, “No: do not change the digit in the given place value.” To the right of this, it says “Because 9 is greater than or equal to ” To the right of this, we have the number 18.379 with the 9 marked “delete” and the 7 marked “add 1.”$ $Finally, the last step reads “Step 4. Rewrite the number, removing all digits to the right of the rounding digit.” To the right of this, we have 18.38 followed by “18.38 is 18.379 rounded to the nearest hundredth.”$

Exercise $\PageIndex{11}$

Round to the nearest hundredth: 1.047.

Answer: 1.05

Exercise $\PageIndex{12}$

Round to the nearest hundredth: 9.173.

Answer: 9.17

We summarize the steps for rounding a decimal here.

ROUND DECIMALS.

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

Exercise $\PageIndex{13}$

Round 18.379 to the nearest

tenth
whole number.

Answer

Round 18.379

1. 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.

2. 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.

Exercise $\PageIndex{14}$

Round 6.582 to the nearest

hundredth
tenth
whole number.

Answer

6.58
6.6
7

Exercise $\PageIndex{15}$

Round 15.2175 to the nearest

thousandth
hundredth
tenth.

Answer

15.218
15.22
15.2

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.

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

Exercise $\PageIndex{16}$

Add: $23.5+41.38$.

Answer: \[\text{Write the numbers so that the decimal points line up vertically.} \quad \begin{array} {r} { 23.50 } \\ { + 41.38 } \\ \hline \end{array}\]
\[\text{Put 0 as a placeholder after the 5 in 23.5. Remember, } \frac{5}{10} = \frac{50}{100}, \text{ so } 0.5 = 0.50 \quad \begin{array} {r} { 23.50 } \\ { + 41.38 } \\ \hline \end{array}\]
\[\text{Add the numbers as if they were whole numbers . Then place the decimal point in the sum.} \quad \begin{array} {r} { 23.50 } \\ { + 41.38 } \\ \hline 64.88 \end{array}\]

Exercise $\PageIndex{17}$

Add: $4.8+11.69$.

Answer: $16.49$

Exercise $\PageIndex{18}$

Add: $5.123+18.47$.

Answer: $23.593$

Exercise $\PageIndex{19}$

Subtract: $20−14.65$.

Answer: \[\begin{array} {ll} {\text{Write the numbers so that the decimal points line up vertically.}} &{ \begin{align} {20 - 14.65} \\ {20.} \\ {-14.65} \\ \hline \end{align}} \\ {\text{Remember, 20 is a whole number, so place the decimal point after the 0.}} &{} \end{array}\]
\[\begin{array} {ll} {\text{Put zeros to the right as placeholders.}} &{ \begin{align} {20.00} \\ {-14.65} \\ \hline \end{align}} \end{array}\]
\[\begin{array} {ll} {\text{Write the numbers so that the decimal points line up vertically.}} &{ \begin{align} {\tiny{9} \quad \tiny{9}\qquad} \\ {\small{1} \not{\small{10}} \not{\small10}\not{\small10}}\\ {\not{2}\not{0.}\not{0}\not{0}} \\ {-14.65} \\ \hline \\{5.35} \end{align}} \end{array}\]

Exercise $\PageIndex{20}$

Subtract: $10−9.58$.

Answer: 0.42

Exercise $\PageIndex{21}$

Subtract: $50−37.42$.

Answer: 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!

Table $\PageIndex{1}$
	$.$
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.

MULTIPLY DECIMALS.

Determine the sign of the product.
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.
Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
Write the product with the appropriate sign.

Exercise $\PageIndex{22}$

Multiply: $(−3.9)(4.075)$.

Answer

	$(−3.9)(4.075)$
The signs are different. The product will be negative.
Write in vertical format, lining up the numbers on the right.	$.$
Multiply.	$.$
Add the number of decimal places in the factors $(1 + 3)$. $.$ Place the decimal point 4 places from the right.	$.$
The signs are different, so the product is negative.	$(−3.9)(4.075) = −15.8925$

Exercise $\PageIndex{23}$

Multiply: $−4.5(6.107)$.

Answer: $−27.4815$

Exercise $\PageIndex{24}$

Multiply: −10.79(8.12).

Answer: $−87.6148$

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.

MULTIPLY A DECIMAL BY A POWER OF TEN.

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

Exercise $\PageIndex{25}$

Multiply 5.63

by 10
by 100
by 1,000.

Answer

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.

ⓐ

	$5.63(10)$
There is 1 zero in 10, so move the decimal point 1 place to the right.	$.$

ⓑ

	$5.63(100)$
There are 2 zeros in 100, so move the decimal point 2 places to the right.	$.$

ⓒ


There are 3 zeros in 1,000, so move the decimal point 3 places to the right.	$.$
A zero must be added at the end.	$.$

Exercise $\PageIndex{26}$

Multiply 2.58

by 10
by 100
by 1,000.

Answer

25.8
258
2,580

Exercise $\PageIndex{27}$

Multiply 14.2

by 10
by 100
by 1,000.

Answer

142
1,420
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:

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

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:

\[\begin{array} {ll} {} &{\underset{\text{quotient}}{c}} \\ {\underset{\text{dividend}}{a} \div \underset{\text{divisor}}{b} = \underset{\text{quotient}}{c}} & {\underset{\text{divisor}}{b})\overline{\underset{\text{dividend}}{a}}} \end{array}\]

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

DIVIDE DECIMALS.

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

Exercise $\PageIndex{28}$

Divide: $−25.65\div (−0.06)$.

Answer

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

	$−25.65\div (−0.06)$
The signs are the same.	The quotient is positive.
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.	$.$
Divide. Place the decimal point in the quotient above the decimal point in the dividend.	$.$
Write the quotient with the appropriate sign.	$−25.65\div (−0.06) = 427.5$

Exercise $\PageIndex{29}$

Divide: $−23.492\div (−0.04)$.

Answer: 687.3

Exercise $\PageIndex{30}$

Divide: $−4.11\div(−0.12)$.

Answer: 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 Exercise $\PageIndex{31}$. In calculations with money, we will round the answer to the nearest cent (hundredth).

Exercise $\PageIndex{31}$

Divide: $$3.99\div 24$.

Answer

.99 divided by 24 is given. A long division problem is set up with 24 dividing 3.99. A table is given with directions on the left and the mathematical steps on the right. The first step reads “Place the decimal point in the quotient above the decimal point in the dividend. 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.” To the right of this, we have a long division problem set up with 24 dividing 3.990. The quotient is given as 0.166. To show the work, below 3.990 it reads 24, solid horizontal line, 159, 144, solid horizontal line, 150, 144, solid horizontal line, and finally 6. The fifth step reads “Round to the nearest cent.” To the right of this, we have $0.166 is approximately equal to $0.17 and hence >.99 divided by 24 is $0.17.">

	$$3.99\div 24$
Place the decimal point in the quotient above the decimal point in the dividend.
Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth.) To do this, we must carry the division to the thousandths place.	$.$
Round to the nearest cent.	$$0.166\approx $0.17$ $$3.99\div 2\approx $0.17$

Exercise $\PageIndex{32}$

Divide: $$6.99\div 36$.

Answer: $$0.19$

Exercise $\PageIndex{33}$

Divide: $$4.99\div 12$.

Answer: $$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.

\[00.03 = \frac { 3 } { 100 }\]

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.

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

Exercise $\PageIndex{34}$

Write 0.374 as a fraction.

Answer

	0.374
Determine the place value of the final digit.	$.$
Write the fraction for 0.374: The numerator is 374. The denominator is 1,000.	$\dfrac{374}{1000}$
Simplify the fraction.	$\dfrac{2\cdot 187}{2\cdot 500}$
Divide out the common factors.	$\dfrac{187}{500}$ so, $0.374=\dfrac{187}{500}$

Did you notice that the number of zeros in the denominator of $\dfrac{374}{1000}$ is the same as the number of decimal places in 0.374?

Exercise $\PageIndex{35}$

Write 0.234 as a fraction.

Answer: $\dfrac{117}{500}$

Exercise $\PageIndex{36}$

Write 0.024 as a fraction.

Answer: $\dfrac{3}{125}$

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 $\dfrac{4}{5}$ can be written $4\div 5$ or $5)\overline{4}$. 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.

Exercise $\PageIndex{37}$

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

Answer

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

$This is a long division problem with 8 dividing 5.000 and 0.625 as the quotient. Below 5.000 we have 48, a solid horizontal line, 20, 16, a solid horizontal line, 40, 40, and a final horizontal line. So five eighths equals 0.625.$

Exercise $\PageIndex{38}$

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

Answer: −0.875

Exercise $\PageIndex{39}$

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

Answer: −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.

Exercise $\PageIndex{40}$

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

Answer: $The number 43/22 is given. The direction is given to “Divide 43 by 22.” A long division problem is given with 22 dividing 43.00000 with 1.95454 as the quotient. Below 43.00000 we have 22, a solid horizontal line, 210, 198, a solid horizontal line, 120, 110, a horizontal line, 100, 88, a solid horizontal line, 120, 110, a solid horizontal line, 100, 88, a solid horizontal line, and then three dots. It is noted that the 120 repeats and that the 100 repeats. This is further explicated as “The pattern repeats, so the numbers in the quotient will repeat as well. At the end, we are given the statement that 43/22 equals 1.954 with a small horizontal line over the 54.$

Exercise $\PageIndex{41}$

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

Answer: $2.\overline{45}$

Exercise $\PageIndex{42}$

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

Answer: $2.3\overline{18}$

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

Exercise $\PageIndex{43}$

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

Answer

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.

		$\dfrac{7}{8}+6.4$
Change $\dfrac{7}{8}$ to a decimal.	$.$
Add.		$0.875+6.4$
		$7.275$
		So, $\dfrac{7}{8}+6.4 = 7.275$

Exercise $\PageIndex{44}$

Simplify: $\dfrac{3}{8}+4.9$.

Answer: $5.275$

Exercise $\PageIndex{45}$

Simplify: $5.7 + \dfrac{13}{20}$.

Answer: $6.35$

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.

\[\begin{array} {llll} {} &{\text{6%}} &{\text{78%}} &{\text{135%}} \\ {\text { Write as a ratio with denominator } 100. } &{\dfrac{6}{100}} &{\dfrac{78}{100}} &{\dfrac{135}{100}} \\ { \text { Change the fraction to a decimal by dividing}} &{0.06} &{0.78} &{1.35}\\ {\text{the numerator by the denominator.}} &{} &{} &{} \end{array}\]

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

The first part of this figure shows 6% with an arrow drawn from between the 6 and the percentage sign to the space to the left of 6 and then to the space further to the left of that space. Below this, the number 0.06 is given. The second part of this figure shows 78% with an arrow drawn from between the 8 and the percentage sign to the space between the 7 and the 8 and then to the space to the left of the 7. Below this, the number 0.78 is given. The third part of this figure shows 2.7% with an arrow drawn from the decimal point to the space to the left of the 2 and then to the space further to the left of that space. Below this, the number 0.027 is given. The fourth part of this figure shows 135% with an arrow drawn from between the 5 and the percentage sign to the space between 3 and 5 and then to the space between 1 and 3. Below this, the number 1.35 is given. — Figure $\PageIndex{2}$

Exercise $\PageIndex{46}$

Convert each percent to a decimal:

62%
135%
35.7%.

Answer

1.
	$.$
Move the decimal point two places to the left.	0.62

2.
	$.$
Move the decimal point two places to the left.	1.35

3.
	$.$
Move the decimal point two places to the left.	0.057

Exercise $\PageIndex{47}$

Convert each percent to a decimal:

9%
87%
3.9%.

Answer

0.09
0.87
0.039

Exercise $\PageIndex{48}$

Convert each percent to a decimal:

3%
91%
8.3%.

Answer

0.03
0.91
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.

\[\begin{array} {llll} {} &{0.83} &{1.05} &{0.075} \\ {\text {Write as a fraction }} &{\frac{83}{100}} &{\small{1}\frac{5}{100}} &{\frac{75}{1000}} \\ { \text {The denominator is 100.}} &{} &{\frac{105}{100}} &{\frac{7.5}{100}}\\ {\text{Write the ratio as a percent.}} &{\text{83%}} &{\text{105%}} &{\text{7.5%}} \end{array}\]

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.

The first part of this figure shows 0.05 with an arrow drawn from the decimal point to the space between 0 and 5 and then to the space after 5. Below this, the number 5% is given. The second part of this figure shows 0.83 with an arrow drawn from the decimal point to the space between 8 and 3 and then to the space after 3. Below this, the number 83% is given. The third part of this figure shows 1.05 with an arrow drawn from the decimal point to the space between 0 and 5 and then to the space after 5. Below this, the number 105% is given. The fourth part of this figure shows 0.075 with an arrow drawn from the decimal point to the space between 0 and 7 and then to the space between 7 and 5. Below this, the number 7.5% is given. The fifth part of this figure shows 0.3 with an arrow drawn from the decimal point to the space after 3 and then to space further to the right of that 3. Below this, the number 30% is given. — Figure $\PageIndex{3}$

Exercise $\PageIndex{49}$

Convert each decimal to a percent:

0.51
1.25
0.093.

Answer

1.
	$.$
Move the decimal point two places to the right.	$51%$

2.
	$.$
Move the decimal point two places to the right.	$125%$

3.
	$.$
Move the decimal point two places to the right.	$9.3%$

Exercise $\PageIndex{50}$

Convert each decimal to a percent:

0.17
1.75
0.0825

Answer

17%
175%
8.25%

Exercise $\PageIndex{51}$

Convert each decimal to a percent:

0.41
2.25
0.0925.

Answer

41%
225%
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.
Add or Subtract Decimals
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.

		\(\dfrac{7}{8}+6.4\)
Change \(\dfrac{7}{8}\) to a decimal.	$.$
Add.		\(0.875+6.4\)
		\(7.275\)
		So, \(\dfrac{7}{8}+6.4 = 7.275\)

	\(−15.571\)
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

	\((−3.9)(4.075)\)
The signs are different. The product will be negative.
Write in vertical format, lining up the numbers on the right.	$.$
Multiply.	$.$
Add the number of decimal places in the factors \((1 + 3)\). $.$ Place the decimal point 4 places from the right.	$.$
The signs are different, so the product is negative.	\((−3.9)(4.075) = −15.8925\)

	\(5.63(10)\)
There is 1 zero in 10, so move the decimal point 1 place to the right.	$.$

	\(5.63(100)\)
There are 2 zeros in 100, so move the decimal point 2 places to the right.	$.$

	\(−25.65\div (−0.06)\)
The signs are the same.	The quotient is positive.
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.	$.$
Divide. Place the decimal point in the quotient above the decimal point in the dividend.	$.$
Write the quotient with the appropriate sign.	\(−25.65\div (−0.06) = 427.5\)

	\($3.99\div 24\)
Place the decimal point in the quotient above the decimal point in the dividend.
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.	$.$
Round to the nearest cent.	\($0.166\approx $0.17\) \($3.99\div 2\approx $0.17\)

2.
	$.$
Move the decimal point two places to the right.	\(125%\)

3.
	$.$
Move the decimal point two places to the right.	\(9.3%\)

Learning Objectives

Note

Name and Write Decimals

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

NAME A DECIMAL.

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\): How to Write decimals

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

WRITE A DECIMAL.

Round Decimals

Exercise \(\PageIndex{10}\)

Exercise \(\PageIndex{11}\)

Exercise \(\PageIndex{12}\)

ROUND DECIMALS.

Exercise \(\PageIndex{13}\)

Exercise \(\PageIndex{14}\)

Exercise \(\PageIndex{15}\)

Add and Subtract Decimals

ADD OR SUBTRACT DECIMALS.

Exercise \(\PageIndex{16}\)

Exercise \(\PageIndex{17}\)

Exercise \(\PageIndex{18}\)

Exercise \(\PageIndex{19}\)

Exercise \(\PageIndex{20}\)

Exercise \(\PageIndex{21}\)

Multiply and Divide Decimals

MULTIPLY DECIMALS.

Exercise \(\PageIndex{22}\)

Exercise \(\PageIndex{23}\)

Exercise \(\PageIndex{24}\)

MULTIPLY A DECIMAL BY A POWER OF TEN.

Exercise \(\PageIndex{25}\)

Exercise \(\PageIndex{26}\)

Exercise \(\PageIndex{27}\)

DIVIDE DECIMALS.

Exercise \(\PageIndex{28}\)

Exercise \(\PageIndex{29}\)

Exercise \(\PageIndex{30}\)

Exercise \(\PageIndex{31}\)

Exercise \(\PageIndex{32}\)

Exercise \(\PageIndex{33}\)

Convert Decimals, Fractions, and Percents

CONVERT A DECIMAL TO A PROPER FRACTION.

Exercise \(\PageIndex{34}\)

Exercise \(\PageIndex{35}\)

Exercise \(\PageIndex{36}\)

CONVERT A FRACTION TO A DECIMAL.

Exercise \(\PageIndex{37}\)

Exercise \(\PageIndex{38}\)

Exercise \(\PageIndex{39}\)

REPEATING DECIMAL

Exercise \(\PageIndex{40}\)

Exercise \(\PageIndex{41}\)

Exercise \(\PageIndex{42}\)

Exercise \(\PageIndex{43}\)

Exercise \(\PageIndex{44}\)

Exercise \(\PageIndex{45}\)

PERCENT

Exercise \(\PageIndex{46}\)

Exercise \(\PageIndex{47}\)

Exercise \(\PageIndex{48}\)

Exercise \(\PageIndex{49}\)

Exercise \(\PageIndex{50}\)

Exercise \(\PageIndex{51}\)

Key Concepts