4.15: Comparison of Rational Numbers
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
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.
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
Let's look at another example.
Order these rational numbers from least to greatest.
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
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
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.
The answer is that ordered from least to greatest, the numbers are
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:
23 of a yard of ribbon for $278 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
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.
Looking at it this way, you can see that 0.875 is the largest number and
The answer is that Christine should buy the
Example 2
Compare the following rational numbers.
First, put both rational numbers into the same form. Let's convert
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
The answer is that
Example 3
Compare the following rational numbers.
0.7 and
First, put both rational numbers into the same form. Let's convert
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
Example 4
Compare the following rational numbers.
0.34 and
First, put both rational numbers into the same form. Let's convert
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
The answer is that
Example 5
Compare the following rational numbers.
First, put both rational numbers into the same form. Let's convert
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.
Now you want to compare
The answer is that
Review
Choose the inequality symbol (>, <, or =) that goes in the blank to make each statement true.
1−−−−−−1110 −2−−−−−−113 25−−−−−−0.3 −.34−−−−−−−310 5.6−−−−−−5.7 −8−−−−−−−12 −34−−−−−−−.75 18.4−−−−−−18.2 −.356−−−−−−−1110 5.678888−−−−−−6510 −.509−−−−−−−5091000 .87−−−−−−810 −4−−−−−−−14 1.212−−−−−−1.232
Place each rational number on the number line below. Then list these rational numbers in order from greatest to least.

120.900.9¯¯¯
Review (Answers)
To see the Review answers, open this PDF file and look for section 4.15.
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In this concept, you will learn how to order rational numbers on a number line and how to compare rational numbers using inequality symbols.