Comparing Fractions and Decimals

Learning Outcomes

Locate decimals on the number line
Order decimals and fractions

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.

Exercises

Locate $0.4$ on a number line.

Solution
The decimal $0.4$ is equivalent to ${\Large\frac{4}{10}}$ , so $0.4$ is located between $0$ and $1$ . On a number line, divide the interval between $0$ and $1$ into $10$ equal parts and place marks to separate the parts.

Label the marks $0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0$ . We write $0$ as $0.0$ and $1$ as $1.0$ , so that the numbers are consistently in tenths. Finally, mark $0.4$ 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 at 0.4.

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2. Locate $0.6$ on a number line.

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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,

$0.40>0.04$

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

[latex]\begin{array}{}\\ ab\text{ , }a\text{ is greater than }b\text{ when }a\text{ is to the right of }b\text{ on the number line}\hfill \end{array}[/latex]

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 $0.40>0.04$ .

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.	${\Large\frac{31}{100}}$	${\Large\frac{308}{1000}}$
We need a common denominator to compare them.	${\Large\frac{31\cdot\color{red}{10}}{100\cdot\color{red}{10}}}$	${\Large\frac{308}{1000}}$
	${\Large\frac{310}{1000}}$	${\Large\frac{308}{1000}}$

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

Notice what we did in converting $0.31$ to a fraction—we started with the fraction $\Large\frac{31}{100}$ and ended with the equivalent fraction $\Large\frac{310}{1000}$ . Converting $\Large\frac{310}{1000}$ 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.

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

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

$0.31=0.310$

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

Equivalent Decimals

Two decimals are equivalent decimals if they convert to equivalent fractions.

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

Order decimals

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.
Compare the numbers to the right of the decimal point as if they were whole numbers.
Order the numbers using the appropriate inequality sign.

example

Order the following decimals using $<\text{ or }\text{>}$ :

$0.64$ ____ $0.6$
$0.83$ ____ $0.803$

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Solution

1.
	$0.64$ ____ $0.6$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of $0.6$ .	$0.64$ ____ $0.6$
Compare the numbers to the right of the decimal point as if they were whole numbers.	$64>60$
Order the numbers using the appropriate inequality sign.	$0.64>0.60$ $0.64>0.6$

2.
	$0.83$ ____ $0.803$
Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of $0.83$ .	$0.83$ ____ $0.803$
Compare the numbers to the right of the decimal point as if they were whole numbers.	$830>803$
Order the numbers using the appropriate inequality sign.	$0.830>0.803$ $0.83>0.803$

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In the following video lesson, we show how to order decimals using inequality notation by comparing place values, and by using fractions.

Order Decimals and Fractions

In an earlier lesson, we compared two decimals and determined which was larger. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals.

example

Order $({\Large\frac{3}{8}})$ and $(0.4)$ using $<$ or $\text{>.}$

Solution

	$({\Large\frac{3}{8}})$ $(0.4)$
Convert $\Large\frac{3}{8}$ to a decimal.	$(0.375)$ $(0.4)$
Compare $0.375$ to $0.4$	$0.375<0.4$
Rewrite with the original fraction.	${\Large\frac{3}{8}}<0.4$

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Watch this video to see more examples of ordering fractions and decimals.

Module 13: F-Distribution and One-Way ANOVA

Learning Outcomes

Locate Decimals On The Number Line

Exercises

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Order Decimals

Equivalent Decimals

Order decimals

example

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Order Decimals and Fractions

example

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Candela Citations