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4.15: Comparison of Rational Numbers

Difficulty Level: At Grade Created by: CK-12
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Christine is working on a craft project and she needs ribbon. She heads to the fabric store where they are having a sale on ribbon remnants (leftover scraps of ribbon). Everything in the ribbon remnants bin is only $2. She wants to buy one ribbon remnant but wants to get the longest piece of ribbon she can. She sees some ribbon is \begin{align*}\frac{2}{3}\end{align*}23 of a yard, some ribbon is \begin{align*}\frac{7}{8}\end{align*}78 of a yard, and some ribbon is 0.8 of a yard. Which ribbon should Christine buy if she wants the most ribbon?

In this concept, you will learn how to order rational numbers on a number line and how to compare rational numbers using inequality symbols.

Comparing Rational Numbers

The rational numbers are the set of numbers that can be written as a ratio of two integers. Integers, fractions, repeating decimals, and terminating decimals are all rational numbers.

Sometimes you will want to compare different rational numbers or put them in order from smallest to largest. A number line can help you to do this. On a number line, the further a number is to the right, the larger its value. The further a number is to the left, the smaller its value.

Once you know how rational numbers are related, you can use the following inequality symbols to describe their relationship:

  • > means is greater than.
  • < means is less than.
  • = means is equal to.

Let's look at an example.

Choose the inequality symbol that goes in the blank to make this statement true.

\begin{align*}-2.5 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -5\end{align*}2.55

First, draw a number line from -5 to 5. Place the numbers -2.5 and -5 on that number line. -2.5 will be between -2 and -3 on the number line.

Since -2.5 is further to the right on the number line compared to -5, -2.5 is greater than -5.

The answer is \begin{align*} -2.5 > -5\end{align*}2.5>5.

Let's look at another example.

Order these rational numbers from least to greatest.

\begin{align*}\frac{4}{5} \quad 0.6 \quad 1 \quad 0.\overline{6}\end{align*}

When comparing rational numbers it can help to have all your numbers in the same form. It is usually easiest to compare numbers when they are in decimal form. Convert \begin{align*}\frac{4}{5}\end{align*} to its decimal form.

\begin{align*}\frac{4}{5}=0.8 \end{align*}

Next, plot all four numbers on a number line. When dealing with decimals it can help to draw a number line that is divided into tenths. So, draw a number line from 0 to 1 and divide it into tenths.

Notice that \begin{align*}0.\overline{6}\end{align*} appears between the 0.6 and the 0.7 on the number line.

Now, with the four numbers plotted on the number line you can see that the smallest number is 0.6 because it is furthest to the left. The largest number is 1 because it is furthest to the right.

\begin{align*}0.6< 0.\overline{6} < \frac{4}{5} <1\end{align*}

The answer is that ordered from least to greatest, the numbers are \begin{align*}0.6, 0. \overline{6}, \frac{4}{5}, 1\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Christine, who was at the fabric store.

She is trying to figure out which ribbon to buy. She wants the most ribbon for her money. Her three options are:

  • \begin{align*}\frac{2}{3}\end{align*} of a yard of ribbon for $2
  • \begin{align*}\frac{7}{8}\end{align*} of a yard of ribbon for $2
  • 0.8 of a yard of ribbon for $2

Christine needs to decide which piece of ribbon is the longest. She is trying to compare the rational numbers \begin{align*}\frac{2}{3}, \frac{7}{8}\end{align*} and 0.8 to figure out which number is the biggest.

To help Christine, you should first make all numbers be in the same form. It is often easiest to compare numbers in decimal form, so you should convert the fractions to decimals. You can also write the decimals so they each have the same number of digits after the decimal point to make the comparison easier.

\begin{align*}\frac{2}{3}=0.66 \overline{6}\end{align*}

\begin{align*}\frac{7}{8}=0.875\end{align*}

\begin{align*}0.8=0.800\end{align*}

Looking at it this way, you can see that 0.875 is the largest number and \begin{align*}0.66 \overline{6}\end{align*} is the smallest number. If Christine wants to get the most amount of ribbon, she should buy the ribbon that is \begin{align*}\frac{7}{8}\end{align*} of a yard.

The answer is that Christine should buy the \begin{align*}\frac{7}{8}\end{align*} of a yard of ribbon if she wants the most ribbon.

Example 2

Compare the following rational numbers.

\begin{align*}-\frac{4}{5}\end{align*} and -0.25

First, put both rational numbers into the same form. Let's convert \begin{align*} -\frac{4}{5}\end{align*} to a decimal. Divide 4 by 5 to find that \begin{align*}\frac{4}{5}=0.8\end{align*}. So you have

\begin{align*}-\frac{4}{5}=-0.8\end{align*}

Next, plot both numbers on a number line. Draw a number line from -1 to 0 and divide it into tenths.

Notice that -0.25 appears between the -0.2 and the -0.3 on the number line.

Now, with the two numbers plotted on the number line you can see that \begin{align*}-\frac{4}{5}\end{align*} is the smaller number because is further to the left. The larger number is -0.25.

The answer is that \begin{align*}-\frac{4}{5}<-0.25\end{align*}.

Example 3

Compare the following rational numbers.

-0.7 and \begin{align*}-\frac{7}{10}\end{align*}

First, put both rational numbers into the same form. Let's convert \begin{align*}-\frac{7}{10}\end{align*} to a decimal. Divide 7 by 10 to find that \begin{align*}\frac{7}{10}=0.7\end{align*}. So you have

\begin{align*}-\frac{7}{10}=-0.7\end{align*}

Next, you would usually draw a number line to help you to compare the numbers, but you can already see that the two numbers are equal! So drawing the number line is not necessary this time.

The answer is that \begin{align*}-0.7=-\frac{7}{10}\end{align*}.

Example 4

Compare the following rational numbers.

0.34 and \begin{align*}\frac{1}{2}\end{align*}

First, put both rational numbers into the same form. Let's convert \begin{align*}\frac{1}{2}\end{align*} to a decimal.

\begin{align*}\frac{1}{2}=0.5\end{align*}

Next, plot both numbers on a number line. Draw a number line from 0 to 1 and divide it into tenths.

Notice that 0.34 appears between the 0.3 and the 0.4 on the number line.

Now, with the two numbers plotted on the number line you can see that 0.34 is the smaller number because is further to the left. The larger number is \begin{align*}\frac{1}{2}\end{align*}.

The answer is that \begin{align*}0.34< \frac{1}{2}\end{align*}.

Example 5

Compare the following rational numbers.

\begin{align*}7.77 \overline{7}\end{align*} and \begin{align*}7 \frac{7}{10}\end{align*}

First, put both rational numbers into the same form. Let's convert \begin{align*}7 \frac{7}{10}\end{align*} to a decimal. You know that \begin{align*} \frac{7}{10}=0.7\end{align*}, so you have

\begin{align*}7 \frac{7}{10}=7.7\end{align*}.

Next you could plot both numbers on a number line, but this time that might not be as easy or as helpful. Sometimes when comparing numbers in decimal form it can be helpful for each number to have the same number of digits after the decimal point.

You can rewrite 7.7 by adding zeros at the end without changing the value of the number.

\begin{align*}7.7=7.70 \overline{0}\end{align*}

Now you want to compare \begin{align*}7.70 \overline{0}\end{align*} with \begin{align*}7.77 \overline{7}\end{align*}. Focus on the digits after the decimal point. You know that 777 is greater than 700. This means that \begin{align*}7.77 \overline{7}\end{align*} is greater than \begin{align*}7.70 \overline{0}\end{align*}.

The answer is that \begin{align*}7.77 \overline{7} > 7 \frac{7}{10}\end{align*}.

Review

Choose the inequality symbol (>, <, or =) that goes in the blank to make each statement true.

  1. \begin{align*}1 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} 1 \frac{1}{10}\end{align*} 
  2. \begin{align*}-2 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} 1 \frac{1}{3}\end{align*} 
  3. \begin{align*}\frac{2}{5} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} 0.3\end{align*} 
  4. \begin{align*}-.34 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \frac{-3}{10}\end{align*} 
  5. \begin{align*}5.6 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} 5.7\end{align*} 
  6. \begin{align*}-8 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -12\end{align*} 
  7. \begin{align*}-\frac{3}{4} \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -.75\end{align*} 
  8. \begin{align*}18.4 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} 18.2\end{align*} 
  9. \begin{align*}-.356 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -1 \frac{1}{10}\end{align*} 
  10. \begin{align*}5.678888 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} 6 \frac{5}{10}\end{align*} 
  11. \begin{align*}-.509 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \frac{-509}{1000}\end{align*} 
  12. \begin{align*}.87 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \frac{8}{10}\end{align*} 
  13. \begin{align*}-4 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} -14\end{align*} 
  14. \begin{align*}1.212 \underline{\;\;\;\;\;\;\;\;\;\;\;\;} 1.232\end{align*} 

Place each rational number on the number line below. Then list these rational numbers in order from greatest to least.

  1.  \begin{align*}\frac{1}{2} \qquad 0.9 \qquad 0 \qquad 0. \overline{9}\end{align*} 

 

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.15.

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Difficulty Level:
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Date Created:
Dec 02, 2015
Last Modified:
Aug 11, 2016
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