1.1 Ordering and Rounding Decimals

Ordering and Rounding Decimals

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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 shows $3.45 for a sandwich, $1.25 for water, and +$0.33 for tax. The bottom total is $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 1.1 shows the counting numbers.

**Table 1.1**
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 1.2 shows the relation.

**Table 1.2**
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. For example, we 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 1.1 relate to the names of the fractions from Table 1.2.

A chart titled “Place Value” has 12 columns. Eleven headings read, from left to right: Hundred thousands, Ten thousands, Thousands, Hundreds, Tens, Ones, Tenths, Hundredths, Thousandths, Ten-thousandths, and Hundred-thousandths. Between Ones and Tenths is an unlabeled column with a decimal point in its bottom cell. — **Figure 1.1** This chart illustrates place values to the left and right of the decimal point.

Notice two important facts shown in Figure 1.1.

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 $$5.03$ lunch? We read $$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 1.2, 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 1.2** 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: ⓐ $4.3$ ⓑ $2.45$ ⓒ $0.009$ ⓓ $−15.571.$

Show solution

ⓐ
	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

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,

$0.40 greater than 0.04$

We can also use 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?

Number line labelled 0.0 through 1.0 in increments of 0.1. One marked point between 0.0 and 0.1 is labelled 0.04; another marked point is 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 2

Order the following decimals using $< or > :$

ⓐ $0.64 __ 0.6$
ⓑ $0.83 __ 0.803$

Show solution

ⓐ
	$0.64 __ 0.6$
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 helps to look first at negative integers. 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 $−9 less than −6.$

Number line labelled with the integers from −10 through 0. Marked points are at −9, −6, −3, and −2.

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 3

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

Show solution

	$−0.1 __ −0.8$
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. 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 1.3 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 1.3** ⓐ 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 the method below.

Example 4

Round $18.379$ to the nearest hundredth.

Show solution


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 5

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

Show solution

ⓐ Round 18.379 to the nearest tenth.

Locate the tenths place and mark it with an arrow.
Inspect the digit immediately to the right of the tenths digit: 7. Publisher note: the original illustration underlines the tenths digit 3; use the following digit 7 to decide whether to round up.
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.

Practice Makes Perfect

Name Decimals

In the following exercises, name each decimal.

$5.5$

Show solution

five and five tenths

$7.8$

$5.01$

Show solution

five and one hundredth

$14.02$

$8.71$

Show solution

eight and seventy-one hundredths

$2.64$

$0.002$

Show solution

two thousandths

$0.005$

$0.381$

Show solution

three hundred eighty-one thousandths

$0.479$

$−17.9$

Show solution

negative seventeen and nine tenths

$−31.4$

Order Decimals

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

$0.9 __ 0.6$

Show solution

$0.7 __ 0.8$

$0.37 __ 0.63$

Show solution

$0.86 __ 0.69$

$0.6 __ 0.59$

Show solution

$0.27 __ 0.3$

$0.91 __ 0.901$

Show solution

$0.415 __ 0.41$

$−0.5 __ −0.3$

Show solution

$−0.1 _ −0.4$

$−0.62 _ −0.619$

Show solution

$−7.31 _ −7.3$

Round Decimals

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

$0.67$

Show solution

0.7

$0.49$

$2.84$

Show solution

2.8

$4.63$

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

$0.845$

Show solution

0.85

$0.761$

$5.7932$

Show solution

5.79

$3.6284$

$0.299$

Show solution

0.30

$0.697$

$4.098$

Show solution

4.10

$7.096$

$5.781$

Show solution

ⓐ 5.78
ⓑ 5.8
ⓒ 6

$1.638$

$63.479$

Show solution

ⓐ 63.48
ⓑ 63.5
ⓒ 63

$84.281$

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