Using Number Lines to Compare and Order Rational Numbers

How to using number lines to compare and order rational numbers, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

When numbers are placed on a number line, comparing them becomes visual: the number farther to the right is always larger, and the number farther to the left is always smaller. This rule works for all rational numbers, fractions, decimals, mixed numbers, and negatives alike. This guide shows you exactly how to use that rule to compare and order any set of rational numbers.

What Does “Compare and Order” Mean?

Comparing two numbers means deciding which is greater, which is less, or whether they are equal (using the symbols \(\color{blue}{>}\), \(\color{blue}{<}\), \(\color{blue}{=}\)). Ordering a list of numbers means arranging them from least to greatest (or greatest to least).

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Rules for Comparing Rational Numbers on a Number Line

Rule 1: Right is greater, left is less

On any number line, a number is greater than every number to its left and less than every number to its right. So to compare two numbers, plot both and look at their positions.

Rule 2: Positive numbers are always greater than negative numbers

\(\color{blue}{\frac{1}{4} > -100}\) because every positive number is to the right of every negative number.

Rule 3: Two negative numbers, the one closer to zero is greater

For example, \(\color{blue}{-2 > -5}\) \(\color{blue}{\text{ because } -2}\) is closer to zero (farther right) on the number line.

Comparing fractions and decimals: convert first

When the numbers are in different forms, convert them all to the same form (usually decimals) before plotting or comparing.

• \(\color{blue}{\frac{3}{4} = 0.75}\)
• \(\color{blue}{\frac{4}{5} = 0.8}\)
• So \(\color{blue}{\frac{3}{4} < \frac{4}{5}}\).

Step-by-Step Summary

1. Convert all numbers to decimals (or to fractions with a common denominator).
2. Plot the numbers on a number line.
3. Read off the order: from left to right is least to greatest.
4. Use \(\color{blue}{<}\) or \(\color{blue}{>}\) to write the comparison.

Watch: Comparing and Ordering Rational Numbers (Video Lesson)

Math with Mr. J demonstrates how to compare and order rational numbers using a number line:

Worked Examples

Example 1: Compare \(\color{blue}{\frac{2}{3}}\) and \(\color{blue}{\frac{3}{4}}\). Use \(\color{blue}{<}\) or \(\color{blue}{>}\).

Convert: \(\color{blue}{\frac{2}{3} &\text{ approx }; 0.667}\) and \(\color{blue}{\frac{3}{4} = 0.75}\). On a number line, \(\color{blue}{0.667}\) is to the left of \(\color{blue}{0.75}\), so \(\color{blue}{\frac{2}{3} < \frac{3}{4}}\).

Example 2: Order from least to greatest: \(\color{blue}{-1.5, -\frac{1}{4}, 0.6, -2}\)

Convert all to decimals: \(\color{blue}{-2, -1.5, -0.25, 0.6}\). On the number line from left to right: \(\color{blue}{-2 < -1.5 < -\frac{1}{4} < 0.6}\).

Example 3: Compare \(\color{blue}{-\frac{3}{5}}\) and \(\color{blue}{-\frac{1}{2}}\).

Convert: \(\color{blue}{-\frac{3}{5} = -0.6}\) and \(\color{blue}{-\frac{1}{2} = -0.5}\). \(\color{blue}{\text{ Since } -0.6}\) is to the left: \(\color{blue}{-\frac{3}{5} < -\frac{1}{2}}\).

Example 4: Order from greatest to least: \(\color{blue}{\frac{5}{4}, 1.1, \frac{3}{2}, 1.25}\)

Convert: \(\color{blue}{\frac{5}{4} = 1.25}\), \(\color{blue}{1.1}\), \(\color{blue}{\frac{3}{2} = 1.5}\), \(\color{blue}{1.25}\). Note \(\color{blue}{\frac{5}{4}}\) and \(\color{blue}{1.25}\) are equal. Greatest to least: \(\color{blue}{\frac{3}{2} > \frac{5}{4} = 1.25 > 1.1}\).

More Practice: Ordering Rational Numbers Video

This Khan Academy video covers ordering rational numbers including negative numbers on a number line:

Exercises

1. Compare using \(\color{blue}{<}\) or \(\color{blue}{>}\):  \(\color{blue}{\frac{5}{6}}\) ___ \(\color{blue}{\frac{7}{8}}\)
2. Order least to greatest: \(\color{blue}{-3, -0.5, \frac{1}{4}, -1 \frac{1}{2}}\)
3. Compare: \(\color{blue}{-\frac{4}{5}}\) ___ \(\color{blue}{-\frac{3}{4}}\)
4. Order greatest to least: \(\color{blue}{0.3, \frac{1}{3}, \frac{2}{5}, 0.28}\)
5. Compare: \(\color{blue}{1.4}\) ___ \(\color{blue}{\frac{7}{5}}\)
6. Which is greatest: \(\color{blue}{-\frac{7}{10}, -\frac{3}{5}, -\frac{2}{3}}\)?

Answers

1. \(\color{blue}{\frac{5}{6} &\text{ approx }; 0.833 < 0.875 = \frac{7}{8}}\), so \(\color{blue}{<}\)
2. \(\color{blue}{-3 < -1 \frac{1}{2} < -0.5 < \frac{1}{4}}\)
3. \(\color{blue}{-\frac{4}{5} = -0.8 < -0.75 = -\frac{3}{4}}\), so \(\color{blue}{<}\)
4. \(\color{blue}{\frac{2}{5} = 0.4 > \frac{1}{3} &\text{ approx }; 0.333 > 0.3 > 0.28}\)
5. \(\color{blue}{\frac{7}{5} = 1.4}\), so \(\color{blue}{=}\)
6. \(\color{blue}{-\frac{7}{10} = -0.7}\), \(\color{blue}{-\frac{3}{5} = -0.6}\), \(\color{blue}{-\frac{2}{3} &\text{ approx }; -0.667}\); greatest is \(\color{blue}{-\frac{3}{5}}\)

Frequently Asked Questions

How do you compare and order rational numbers using number lines?

Plot each number on the same number line, then read them from left to right to list them from least to greatest. A number to the right is always greater. For −1.5, 1/2, −3/4 and 2, the order from left to right is −1.5, −3/4, 1/2, 2. Convert fractions to decimals first if that helps you place them.

How do you use number lines to compare fractions?

Divide the space between whole numbers into equal parts that match the denominator, mark each fraction, and see which is farther right. To compare 2/3 and 3/5, you can use a line split into fifteenths, where 2/3 = 10/15 and 3/5 = 9/15. Since 10/15 sits to the right, 2/3 > 3/5.

How do you use number lines for fractions and negative numbers?

Mark 0 in the middle, put positive fractions to the right and negative fractions to the left, using equal-sized steps. For two negatives, the one closer to zero is greater, so −1/4 > −3/4. Every positive number, even 1/4, is greater than every negative number, because it sits to the right of 0.

How do you use number lines to find equivalent fractions?

Draw two or more number lines of the same length from 0 to 1, split into different numbers of equal parts. Fractions that land on the same point are equivalent. On lines divided into halves, quarters and eighths, 1/2, 2/4 and 4/8 all sit at the same spot, so they are equal, and neither is greater than the other.

Related Topics

• Using Number Lines to Represent Rational Numbers
• Word Problems of Comparing and Ordering Rational Numbers
• Absolute Value of Rational Numbers
• Order of Decimals, Mixed Numbers, and Fractions
• Convert Rational Numbers to a Fraction