5.1 Decimals

Name Decimals

You probably already know quite a bit about decimals based on your experience with money. Suppose you buy a sandwich and a bottle of water for lunch. If the sandwich costs $$3.45$ , the bottle of water costs $$1.25$ , and the total sales tax is $$0.33$ , what is the total cost of your lunch?

A vertical addition problem is shown. The top line shows $3.45 for a sandwich, the next line shows $1.25 for water, and the last line shows $0.33 for tax. The total is shown to be $5.03.

The total is $$5.03.$ Suppose you pay with a $$5$ bill and $3$ pennies. Should you wait for change? No, $$5$ and $3$ pennies is the same as $$5.03.$

Because $100 pennies equals $1,$ each penny is worth $\frac{1}{100}$ of a dollar. We write the value of one penny as $$0.01,$ since $0.01 = \frac{1}{100} .$

Writing a number with a decimal is known as decimal notation. It is a way of showing parts of a whole when the whole is a power of ten. In other words, decimals are another way of writing fractions whose denominators are powers of ten. Just as the counting numbers are based on powers of ten, decimals are based on powers of ten. Table 5.1 shows the counting numbers.

**Table 5.1**
Counting number	Name
$1$	One
$10 equals 10$	Ten
$10 \cdot 10 = 100$	One hundred
$10 \cdot 10 \cdot 10 = 1000$	One thousand
$10 \cdot 10 \cdot 10 \cdot 10 = 10,000$	Ten thousand

How are decimals related to fractions? Table 5.2 shows the relation.

**Table 5.2**
Decimal	Fraction	Name
$0.1$	$\frac{1}{10}$	One tenth
$0.01$	$\frac{1}{100}$	One hundredth
$0.001$	$\frac{1}{1,000}$	One thousandth
$0.0001$	$\frac{1}{10,000}$	One ten-thousandth

When we name a whole number, the name corresponds to the place value based on the powers of ten. In Whole Numbers, we learned to read $10,000$ as ten thousand. Likewise, the names of the decimal places correspond to their fraction values. Notice how the place value names in Figure 5.2 relate to the names of the fractions from Table 5.2.

A chart is shown labeled “Place Value”. There are 12 columns. The columns are labeled, from left to right, Hundred thousands, Ten thousands, Thousands, Hundreds, Tens, Ones, Decimal Point, Tenths, Hundredths, Thousandths, Ten-thousandths, Hundred-thousandths. — **Figure 5.2** This chart illustrates place values to the left and right of the decimal point.

Notice two important facts shown in Figure 5.2.

The “th” at the end of the name means the number is a fraction. “One thousand” is a number larger than one, but “one thousandth” is a number smaller than one.
The tenths place is the first place to the right of the decimal, but the tens place is two places to the left of the decimal.

Remember that $$5. 0 3$ lunch? We read $$5.03$ as five dollars and three cents. Naming decimals (those that don’t represent money) is done in a similar way. We read the number $5.03$ as five and three hundredths.

We sometimes need to translate a number written in decimal notation into words. As shown in Figure 5.3, we write the amount on a check in both words and numbers.

An image of a check is shown. The check is made out to Jane Doe. It shows the number $152.65 and says in words, “One hundred fifty two and 65 over 100 dollars.” — **Figure 5.3** When we write a check, we write the amount as a decimal number as well as in words. The bank looks at the check to make sure both numbers match. This helps prevent errors.

Let’s try naming a decimal, such as 15.68.
We start by naming the number to the left of the decimal.	fifteen______
We use the word “and” to indicate the decimal point.	fifteen and_____
Then we name the number to the right of the decimal point as if it were a whole number.	fifteen and sixty-eight_____
Last, name the decimal place of the last digit.	fifteen and sixty-eight hundredths

The number $15.68$ is read fifteen and sixty-eight hundredths.

Example 1

Name each decimal: ⓐ $4.3$ ⓑ $2.45$ ⓒ $0.009$ ⓓ $−15.571.$

ⓐ
	4.3
Name the number to the left of the decimal point.	four_____
Write "and" for the decimal point.	four and_____
Name the number to the right of the decimal point as if it were a whole number.	four and three_____
Name the decimal place of the last digit.	four and three tenths

ⓑ
	2.45
Name the number to the left of the decimal point.	two_____
Write "and" for the decimal point.	two and_____
Name the number to the right of the decimal point as if it were a whole number.	two and forty-five_____
Name the decimal place of the last digit.	two and forty-five hundredths

ⓒ
	0.009
Name the number to the left of the decimal point.	Zero is the number to the left of the decimal; it is not included in the name.
Name the number to the right of the decimal point as if it were a whole number.	nine_____
Name the decimal place of the last digit.	nine thousandths

ⓓ
	$minus 15.571$
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 as if it were a whole number.	negative fifteen and five hundred seventy-one_____
Name the decimal place of the last digit.	negative fifteen and five hundred seventy-one thousandths

Write Decimals

Now we will translate the name of a decimal number into decimal notation. We will reverse the procedure we just used.

Let’s start by writing the number six and seventeen hundredths:

	six and seventeen hundredths
The word and tells us to place a decimal point.	___.___
The word before and is the whole number; write it to the left of the decimal point.	6._____
The decimal part is seventeen hundredths. Mark two places to the right of the decimal point for hundredths.	6._ _
Write the numerals for seventeen in the places marked.	6.17

Example 2

Write fourteen and thirty-seven hundredths as a decimal.

	fourteen and thirty-seven hundredths
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.	14. _________
Mark two places to the right of the decimal point for “hundredths”.	14.__ __
Translate the words after “and” and write the number to the right of the decimal point.	14.37
	Fourteen and thirty-seven hundredths is written 14.37.

The second bullet in Step 2 is needed for decimals that have no whole number part, like ‘nine thousandths’. We recognize them by the words that indicate the place value after the decimal – such as ‘tenths’ or ‘hundredths.’ Since there is no whole number, there is no ‘and.’ We start by placing a zero to the left of the decimal and continue by filling in the numbers to the right, as we did above.

Example 3

Write twenty-four thousandths as a decimal.

	twenty-four thousandths
Look for the word "and".	There is no "and" so start with 0 0.
To the right of the decimal point, put three decimal places for thousandths.
Write the number 24 with the 4 in the thousandths place.
Put zeros as placeholders in the remaining decimal places.	0.024
	So, twenty-four thousandths is written 0.024

Before we move on to our next objective, think about money again. We know that $$1$ is the same as $$1.00.$ The way we write $$1 open parenthesis or $1.00 close parenthesis$ depends on the context. In the same way, integers can be written as decimals with as many zeros as needed to the right of the decimal.

$\begin{matrix} 5 = 5.0 & −2 = −2.0 \\ 5 = 5.00 & −2 = −2.00 \\ 5 = 5.000 & −2 = −2.000 \end{matrix}$

$and so on…$

Convert Decimals to Fractions or Mixed Numbers

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that $$5.03$ means $5$ dollars and $3$ cents. Since there are $100$ cents in one dollar, $3$ cents means $\frac{3}{100}$ of a dollar, so $0.03 = \frac{3}{100} .$

We convert decimals to fractions by identifying the place value of the 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 $0.03 .$

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

For our $$5.03$ lunch, we can write the decimal $5.03$ as a mixed number.

$5.03 = 5 \frac{3}{100}$

Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number.

Example 4

Write each of the following decimal numbers as a fraction or a mixed number:

ⓐ $4.09$ ⓑ $3.7$ ⓒ $−0.286$

ⓐ
	4.09
There is a 4 to the left of the decimal point. Write "4" as the whole number part of the mixed number.	$The number 4 is followed by a fraction with an empty square numerator and an empty square denominator, suggesting an incomplete mathematical expression or a prompt to fill in the blanks.$
Determine the place value of the final digit.
Write the fraction. Write 9 in the numerator as it is the number to the right of the decimal point.	$A mathematical expression showing a mixed number where the whole number is 4, the numerator of the fraction is 9, and the denominator is represented by an empty rectangle or box, indicating a missing value.$
Write 100 in the denominator as the place value of the final digit, 9, is hundredth.	$A mixed number is displayed, consisting of the whole number 4, followed by the fraction 9/100, where 9 is the numerator and 100 is the denominator. The expression represents four and nine hundredths.$
The fraction is in simplest form.	$The image shows the conversion of the decimal number 4.09 into a mixed fraction, expressed as 'So, 4.09 = 4 9/100' on a white background.$

Did you notice that the number of zeros in the denominator is the same as the number of decimal places?

ⓑ
	3.7
There is a 3 to the left of the decimal point. Write "3" as the whole number part of the mixed number.	$The number 3 is shown next to a horizontal fraction line, with an empty square box above the line and another empty square box below the line, against a white background.$
Determine the place value of the final digit.
Write the fraction. Write 7 in the numerator as it is the number to the right of the decimal point.	$A mathematical expression showing the number 3 next to a fraction bar with the number 7 above it and an empty square below it, representing an incomplete mixed number or fraction.$
Write 10 in the denominator as the place value of the final digit, 7, is tenths.
The fraction is in simplest form.

ⓒ
	−0.286
There is a 0 to the left of the decimal point. Write a negative sign before the fraction.	$A mathematical expression featuring a negative sign followed by a fraction, where both the numerator and denominator are depicted as empty square placeholders.$
Determine the place value of the final digit and write it in the denominator.
Write the fraction. Write 286 in the numerator as it is the number to the right of the decimal point. Write 1,000 in the denominator as the place value of the final digit, 6, is thousandths.	$A negative fraction is displayed, with the number 286 in the numerator and 1000 in the denominator.$
We remove a common factor of 2 to simplify the fraction.	$The image displays the negative fraction -143/500, a mathematical expression showing a division of 143 by 500 with a negative sign preceding it. It is presented on a plain white background.$

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Order Decimals

Which is larger, $0.04$ or $0.40 ?$

If you think of this as money, you know that $$0.40$ (forty cents) is greater than $$0.04$ (four cents). So,

$0.40 greater than 0.04$

In previous chapters, we used the number line to order numbers.

$\begin{matrix} a < b ‘ a is less than b ’ when a is to the left of b on the number line \\ a > b ‘ a is greater than b ’ when a is to the right of b on the number line \end{matrix}$

Where are $0.04$ and $0.40$ located on the number line?

A number line is shown with 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 labeled. There is a red dot between 0.0 and 0.1 labeled as 0.04. There is another red dot at 0.4.

We see that $0.40$ is to the right of $0.04 .$ So we know $0.40 greater than 0.04.$

How does $0.31$ compare to $0.308 ?$ This doesn’t translate into money to make the comparison easy. But if we convert $0.31$ and $0.308$ to fractions, we can tell which is larger.

	$0.31$	$0.308$
Convert to fractions.	$\frac{31}{100}$	$\frac{308}{1000}$
We need a common denominator to compare them.	$A fraction with a numerator of 31 multiplied by 10 and a denominator of 100 multiplied by 10. The number 10 is highlighted in red in both the numerator and denominator.$	$\frac{308}{1000}$
	$\frac{310}{1000}$	$\frac{308}{1000}$

Because $310 greater than 308,$ we know that $\frac{310}{1000} > \frac{308}{1000} .$ Therefore, $0.31 greater than 0.308.$

Notice what we did in converting $0.31$ to a fraction—we started with the fraction $\frac{31}{100}$ and ended with the equivalent fraction $\frac{310}{1000} .$ Converting $\frac{310}{1000}$ back to a decimal gives $0.310 .$ So $0.31$ is equivalent to $0.310 .$ Writing zeros at the end of a decimal does not change its value.

$\frac{31}{100} = \frac{310}{1000} and 0.31 = 0.310$

If two decimals have the same value, they are said to be equivalent decimals.

$0.31 equals 0.310$

We say $0.31$ and $0.310$ are equivalent decimals.

Remember, writing zeros at the end of a decimal does not change its value.

Example 7

Order the following decimals using $less than or >:$

ⓐ $0.64 __ 0.6$
ⓑ $0.83 __ 0.803$

ⓐ
	$0.64 __ 0.6$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.6.	$0.64 __ 0.60$
Compare the numbers to the right of the decimal point as if they were whole numbers.	$64 greater than 60$
Order the numbers using the appropriate inequality sign.	$0.64 greater than 0.60$ $0.64 greater than 0.6$

ⓑ
	$0.83 __ 0.803$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of 0.83.	$0.830 __ 0.803$
Compare the numbers to the right of the decimal point as if they were whole numbers.	$830 greater than 803$
Order the numbers using the appropriate inequality sign.	$0.830 greater than 0.803$ $0.83 greater than 0.803$

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because $−2$ lies to the right of $−3$ on the number line, we know that $−2 greater than −3.$ Similarly, smaller numbers lie to the left on the number line. For example, because $−9$ lies to the left of $−6$ on the number line, we know that $−9 less than −6.$

A number line is shown with integers from negative 10 to 0. Blue dots are placed on negative nine and negative six. Red dots are placed at negative two and negative three.

If we zoomed in on the interval between $0$ and $−1,$ we would see in the same way that $−0.2 greater than −0.3 and −0.9 less than −0.6.$

Example 8

Use $< or >$ to order. $−0.1 __ −0.8.$

	$−0.1 __ −0.8$
Write the numbers one under the other, lining up the decimal points.	$−0.1$ $−0.8$
They have the same number of digits.
Since $−1 greater than −8, −1$ tenth is greater than $−8$ tenths.	$−0.1 greater than −0.8$

Round Decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at $$3.279$ per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay $$3.28$ , because the final price would be rounded to the nearest cent. In Whole Numbers, we saw that we round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round $$2.72$ to the nearest dollar. Is it closer to $$2$ or to $$3 ?$ What if we wanted to round $$2.72$ to the nearest ten cents; is it closer to $$2.70$ or to $$2.80 ?$ The number lines in Figure 5.4 can help us answer those questions.

In part a, a number line is shown with 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3. There is a dot between 2.7 and 2.8 labeled as 2.72. In part b, a number line is shown with 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, and 2.80. There is a dot at 2.72. — **Figure 5.4** ⓐ We see that $2.72$ is closer to $3$ than to $2 .$ So, $2.72$ rounded to the nearest whole number is $3 .$
ⓑ We see that $2.72$ is closer to $2.70$ than $2.80 .$ So we say that $2.72$ rounded to the nearest tenth is $2.7 .$

Can we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.

Example 9

Round $18.379$ to the nearest hundredth.


Locate the hundredths place and mark it with an arrow.
Underline the digit to the right of the 7.
Because 9 is greater than or equal to 5, add 1 to the 7.
Rewrite the number, deleting all digits to the right of the hundredths place.
	18.38 is 18.379 rounded to the nearest hundredth.

Example 10

Round $18.379$ to the nearest ⓐ tenth ⓑ whole number.

ⓐ Round 18.379 to the nearest tenth.

Locate the tenths place and mark it with an arrow.
Underline the digit to the right of the tenths digit.
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 tenths place.
	So, 18.379 rounded to the nearest tenth is 18.4.

ⓑ Round 18.379 to the nearest whole number.

Locate the ones place and mark it with an arrow.
Underline the digit to the right of the ones place.
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 ones place.
	So 18.379 rounded to the nearest whole number is 18.

Key Concepts

Name a decimal number.
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 number from its name.
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.
Convert a decimal number to a fraction or mixed number.
1. Look at the number to the left of the decimal.
  If it is zero, the decimal converts to a proper fraction.
  If it is not zero, the decimal converts to a mixed number.
  Write the whole number.
2. Determine the place value of the final digit.
3. Write the fraction. numerator—the ‘numbers’ to the right of the decimal point denominator—the place value corresponding to the final digit
4. Simplify the fraction, if possible.
Order decimals.
1. Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match.
2. Compare the numbers to the right of the decimal point as if they were whole numbers.
3. Order the numbers using the appropriate inequality sign.
Round a decimal.
1. Locate the given place value and mark it with an arrow.
2. Underline the digit to the right of the given 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, removing all digits to the right of the given place value.

Practice Makes Perfect

Name Decimals

In the following exercises, name each decimal.

$5.5$

five and five tenths

$7.8$

$5.01$

five and one hundredth

$14.02$

$8.71$

eight and seventy-one hundredths

$2.64$

$0.002$

two thousandths

$0.005$

$0.381$

three hundred eighty-one thousandths

$0.479$

$−17.9$

negative seventeen and nine tenths

$−31.4$

Write Decimals

In the following exercises, translate the name into a decimal number.

Eight and three hundredths

8.03

Nine and seven hundredths

Twenty-nine and eighty-one hundredths

29.81

Sixty-one and seventy-four hundredths

Seven tenths

0.7

Six tenths

One thousandth

0.001

Nine thousandths

Twenty-nine thousandths

0.029

Thirty-five thousandths

Negative eleven and nine ten-thousandths

−11.0009

Negative fifty-nine and two ten-thousandths

Thirteen and three hundred ninety-five ten thousandths

13.0395

Thirty and two hundred seventy-nine thousandths

Convert Decimals to Fractions or Mixed Numbers

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

$1.99$

$1 \frac{99}{100}$

$5.83$

$15.7$

$15 \frac{7}{10}$

$18.1$

$0.239$

$\frac{239}{1000}$

$0.373$

$0.13$

$\frac{13}{100}$

$0.19$

$0.011$

$\frac{11}{1000}$

$0.049$

$−0.00007$

$- \frac{7}{100000}$

$−0.00003$

$6.4$

$6 \frac{2}{5}$

$5.2$

$7.05$

$7 \frac{1}{20}$

$9.04$

$4.006$

$4 \frac{3}{500}$

$2.008$

$10.25$

$10 \frac{1}{4}$

$12.75$

$1.324$

$1 \frac{81}{250}$

$2.482$

$14.125$

$14 \frac{1}{8}$

$20.375$

Locate Decimals on the Number Line

In the following exercises, locate each number on a number line.

$0.8$

There is a number line shown with integers from negative 4 to 4. There is a red dot between 0 and 1 labeled 0.8.

$0.3$

$−0.2$

There is a number line shown with integers from negative 4 to 4. There is a red dot between negative 1 and 0 labeled negative 0.2.

$−0.9$

$3.1$

This is an image of a number line. It spans from negative 5 on the left to 5 on the right. To the right of 0 are tick marks with the numbers 1, 2, 3, 4, 5 on the number line. To the left of the zero are tick marks with the numbers negative 1, negative 2, negative 3, negative 4, and negative 5. A point is plotted at 3.1.

$2.7$

$−2.5$

There is a number line shown with integers from negative 4 to 4. There is a red dot between negative 3 and negative 2 labeled negative 2.5.

$−1.6$

Order Decimals

In the following exercises, order each of the following pairs of numbers, using $less than or >.$

$0.9 __ 0.6$

$0.7 __ 0.8$

$0.37 __ 0.63$

$0.86 __ 0.69$

$0.6 __ 0.59$

$0.27 __ 0.3$

$0.91 __ 0.901$

$0.415 __ 0.41$

$−0.5 __ −0.3$

$−0.1 _ −0.4$

$−0.62 _ −0.619$

$−7.31 _ −7.3$

Round Decimals

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

$0.67$

0.7

$0.49$

$2.84$

2.8

$4.63$

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

$0.845$

0.85

$0.761$

$5.7932$

5.79

$3.6284$

$0.299$

0.30

$0.697$

$4.098$

4.10

$7.096$

$5.781$

ⓐ 5.78
ⓑ 5.8
ⓒ 6

$1.638$

$63.479$

ⓐ 63.48
ⓑ 63.5
ⓒ 63

$84.281$

Everyday Math

Salary Increase Danny got a raise and now makes $$58,965.95$ a year. Round this number to the nearest:

ⓐ dollar

ⓑ thousand dollars

ⓐ $58,966
ⓑ $59,000
ⓒ $60,000

New Car Purchase Selena’s new car cost $$23,795.95.$ Round this number to the nearest:

ⓐ dollar

ⓑ thousand dollars

Sales Tax Hyo Jin lives in San Diego. She bought a refrigerator for $$1624.99$ and when the clerk calculated the sales tax it came out to exactly $$142.186625.$ Round the sales tax to the nearest ⓐ penny ⓑ dollar.

ⓐ $142.19
ⓑ $142

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 ⓑ dollar.

Writing Exercises

How does your knowledge of money help you learn about decimals?

Answers will vary.

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

Jim ran a $100-meter$ race in $12.32 seconds.$ Tim ran the same race in $12.3 seconds.$ Who had the faster time, Jim or Tim? How do you know?

Tim had the faster time. 12.3 is less than 12.32, so Tim had the faster time.

Gerry saw a sign advertising postcards marked for sale at $“ 10 for 0.99 ¢.”$ What is wrong with the advertised price?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment chart for students to gauge their understanding of decimals, with categories: Confidently, With some help, and No-I don't get it! Tasks include naming, writing, converting, locating, ordering, and rounding decimals. — **Figure 5.5**

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.