Word Problems of Comparing and Ordering Rational Numbers

How to solve word problems of comparing and ordering 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.

On the GED Math test, comparing and ordering rational numbers often appears in real-world contexts: Who scored higher? Which temperature was lower? Which item costs less? By translating the words into numbers and then applying comparison rules, you can answer these questions quickly and confidently.

What Are Comparison Word Problems?

A comparison word problem asks you to decide the relationship between two or more quantities that may be expressed as fractions, decimals, mixed numbers, or negative numbers. Key words to look for:

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• Greater, more, higher, larger, farther right, indicate a bigger value.
• Less, fewer, lower, smaller, farther left, indicate a smaller value.
• Order, rank, arrange, sort, indicate you need to list the numbers in sequence.

Strategy for Solving Comparison Word Problems

Step 1: Identify all the numbers

Read the problem carefully and list every quantity mentioned. Note the units. You can only compare numbers with the same units.

Step 2: Convert to a common form

Convert fractions to decimals (or find a common denominator) so every number is in the same form.

Step 3: Compare or order

Use a number line if helpful. Remember: farther \(\color{blue}{\text{ right } = \text{ greater }}\); farther \(\color{blue}{\text{ left } = \text{ lesser }}\).

Step 4: Answer the question completely

Write a complete sentence that directly answers what the problem asked.

Step-by-Step Summary

1. List the quantities from the problem with their units.
2. Convert all numbers to the same form (decimal is usually easiest).
3. Compare or order using <, >, \(\color{blue}{\text{ or } =}\).
4. Write your answer in the context of the original question.

Watch: Comparing Rational Numbers (Video Lesson)

This Khan Academy video demonstrates how to compare rational numbers including fractions, decimals, and negatives:

Worked Examples

Example 1: Three hikers recorded the change in elevation (in miles) for their trails: Ana: \(\color{blue}{-\frac{1}{4}}\), Ben: \(\color{blue}{-\frac{3}{8}}\), Carla: \(\color{blue}{\frac{1}{8}}\). Order their elevation changes from least to greatest.

Convert to decimals: \(\color{blue}{-\frac{1}{4} = -0.25}\), \(\color{blue}{-\frac{3}{8} = -0.375}\), \(\color{blue}{\frac{1}{8} = 0.125}\). From least to greatest: \(\color{blue}{-\frac{3}{8} < -\frac{1}{4} < \frac{1}{8}}\), so Ben < Ana < Carla.

Example 2: In January, the overnight temperature was \(\color{blue}{-5.4^{\circ}F}\) in City A and \(\color{blue}{-5 \frac{3}{8} ^{\circ}F}\) in City B. Which city was colder?

\(\color{blue}{-5 \frac{3}{8} = -5.375}\). Compare: \(\color{blue}{-5.4 < -5.375}\). City A was colder.

Example 3: A recipe calls for \(\color{blue}{\frac{2}{3}}\) cup of sugar. Another recipe calls for \(\color{blue}{\frac{3}{4}}\) cup. Which recipe uses more sugar?

\(\color{blue}{\frac{2}{3} &\text{ approx }; 0.667}\) and \(\color{blue}{\frac{3}{4} = 0.75}\). Since \(\color{blue}{0.75 > 0.667}\), the second recipe uses more sugar.

Example 4: Four students scored \(\color{blue}{\frac{7}{10}, 0.68, \frac{3}{5}, 0.72}\) on a quiz. Rank them from highest to lowest.

Convert: \(\color{blue}{\frac{7}{10} = 0.70}\), \(\color{blue}{0.68}\), \(\color{blue}{\frac{3}{5} = 0.60}\), \(\color{blue}{0.72}\). Highest to lowest: \(\color{blue}{0.72 > 0.70 > 0.68 > 0.60}\).

More Practice: Rational Number Word Problems Video

This Khan Academy video works through rational number word problems involving ratios:

Exercises

1. A submarine dove to \(\color{blue}{-120 \frac{3}{4}}\) feet and a whale dove to \(\color{blue}{-120.8}\) feet. Which went deeper?
2. Order from least to greatest: \(\color{blue}{0.45, \frac{4}{9}, 0.5, \frac{2}{5}}\)
3. Two runners finished a race in \(\color{blue}{11 \frac{3}{4}}\) seconds and \(\color{blue}{11.7}\) seconds. Who was faster?
4. Three stocks changed in value: \(\color{blue}{-\frac{1}{5}, -0.15, -\frac{1}{4}}\). Order from greatest to least change (least \(\color{blue}{\text{ negative } = \text{ greatest }}\)).
5. Which is greater: \(\color{blue}{-\frac{7}{8}}\) or \(\color{blue}{-0.9}\)?
6. Order from least to greatest: \(\color{blue}{\frac{3}{8}, 0.4, \frac{5}{12}, 0.35}\)

Answers

1. \(\color{blue}{-120 \frac{3}{4} = -120.75}\) and \(\color{blue}{-120.8}\); since \(\color{blue}{-120.8 < -120.75}\), the whale went deeper.
2. Convert all: \(\color{blue}{\frac{2}{5} = 0.4}\), \(\color{blue}{\frac{4}{9} &\text{ approx }; 0.444}\), \(\color{blue}{0.45}\), \(\color{blue}{0.5}\). Order: \(\color{blue}{\frac{2}{5} < \frac{4}{9} < 0.45 < 0.5}\).
3. \(\color{blue}{11.7 < 11.75}\); the runner in 11.7 s was faster.
4. \(\color{blue}{-0.15 > -\frac{1}{5} = -0.2 > -\frac{1}{4} = -0.25}\)
5. \(\color{blue}{-\frac{7}{8} = -0.875 > -0.9}\), so \(\color{blue}{-\frac{7}{8}}\) is greater.
6. \(\color{blue}{0.35 < \frac{3}{8} = 0.375 < 0.4 < \frac{5}{12} &\text{ approx }; 0.417}\)

Two Ways to Compare Rational Numbers

Use decimals when the problem already has decimals in it, such as money or measurements. Use a common denominator when the fractions have small denominators that are easy to combine.

Comparing two negative numbers

This is where most mistakes happen. With negatives, the number closer to zero is the greater one. Compare −1/2 and −1/5: as decimals they are −0.5 and −0.2. Since −0.2 is closer to zero, −1/5 > −1/2. A quick check: picture a thermometer. Minus 2 degrees is warmer than minus 5 degrees.

Sign rules to check first

• A positive number is always greater than a negative number.
• Zero is greater than every negative number and less than every positive number.
• If both numbers are negative, the one with the smaller absolute value is greater.

Frequently Asked Questions

How do you compare rational numbers?

Look at the signs first, since any positive number is greater than any negative number. If the signs match, write both numbers in the same form, either decimals or fractions with a common denominator, and compare. For example, 5/8 = 0.625, which is greater than 0.6. On a number line, the number farther to the right is always greater.

What is an example of a rational number word problem?

Here is one: Monday’s low was −4.5°F and Tuesday’s low was −4¼°F. Which day was colder? Change −4¼ to −4.25. Both are negative, so the number farther from zero is smaller, and −4.5 < −4.25. Monday was colder. Writing the final answer in words, not just the inequality, shows you answered the question that was asked.

How do you order rational numbers from least to greatest?

Convert every number to a decimal, put the negatives first starting with the one farthest from zero, then list the positives from smallest to largest. To order 3/5, −1.2, 0.55 and −5/4, rewrite them as 0.6, −1.2, 0.55 and −1.25. The order is −5/4, −1.2, 0.55, 3/5. A quick number line sketch helps you check.

How do you solve comparing numbers word problems?

List every quantity, make sure the units match, convert to one form, compare, then answer in a full sentence. Suppose Maya ran 2 3/10 miles and Leo ran 2.25 miles. Since 2 3/10 = 2.3 and 2.3 > 2.25, Maya ran farther, by 0.05 mile. Key words like more, less, lower and farther tell you which comparison the question wants.

Is comparing and ordering rational numbers on the TEAS test?

Yes. The ATI TEAS math section includes comparing and ordering rational numbers, which means fractions, decimals, percents and negative numbers, often in short real-world items. The method is the same as for any test: convert everything to decimals, place negatives before positives, and check the order on a quick number line before choosing an answer.

Related Topics

• Using Number Lines to Compare and Order Rational Numbers
• Absolute Value of Rational Numbers
• Solve Word Problems of Absolute Value and Integers
• Convert Rational Numbers to a Fraction
• Using Number Lines to Represent Rational Numbers