Word Problems of Comparing and Ordering Rational Numbers

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. \(\color{blue}{0.35 < \frac{3}{8} = 0.375 < 0.4 < \frac{5}{12} &\text{ approx }; 0.417}\)

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Frequently Asked Questions

How do I handle mixed units in a comparison problem?

Convert all quantities to the same unit before comparing. For instance, if one distance is in feet and another in inches, convert both to inches (or both to feet) before writing an inequality.

What do I do when fractions have different denominators?

Find a common denominator or convert each fraction to a decimal by dividing numerator by denominator. Decimals are often quicker for comparison.

Does a more negative number mean a worse or better outcome?

It depends on context. A more negative temperature is colder (worse if you want warmth). A more negative financial change means a larger loss. Always read the problem to decide what “better” or “worse” means.

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