Word Problems Involving Comparing Ratio

Comparing ratios in word problems is a skill that appears in everyday life — which deal is a better value? Which team has a better win rate? Which mixture is stronger? On the GED Math test, you need a systematic method to set up the comparison correctly and reach a reliable answer.

What Does “Comparing Ratios” Mean?

When you compare two ratios, you decide which is larger, which is smaller, or whether they are equal. For example, is the ratio \(\color{blue}{3 : 5}\) greater than, less than, or equal to \(\color{blue}{2 : 3}\)?

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Converting both ratios to the same form (fractions with a common denominator, or decimals) makes the comparison straightforward.

Three Methods for Comparing Ratios

Method 1: Convert to decimals

Divide the first term by the second in each ratio to get a decimal. Then compare the decimals.

\(\color{blue}{3 : 5 = 3 \div 5 = 0.60}\) and \(\color{blue}{2 : 3 = 2 \div 3 &\text{ approx }; 0.667}\). Since \(\color{blue}{0.667 > 0.60}\), the ratio \(\color{blue}{2 : 3}\) is greater.

Method 2: Convert to fractions with a common denominator

Write each ratio as a fraction, find the LCD, and compare numerators.

\(\color{blue}{\frac{3}{5}}\) and \(\color{blue}{\frac{2}{3}}\). \(\color{blue}{\text{ LCD } = 15}\). \(\color{blue}{\frac{3}{5} = \frac{9}{15}}\) and \(\color{blue}{\frac{2}{3} = \frac{10}{15}}\). Since \(\color{blue}{10 > 9}\), \(\color{blue}{\frac{2}{3} > \frac{3}{5}}\).

Method 3: Cross-multiply

For \(\color{blue}{\frac{a}{b}}\) vs \(\color{blue}{\frac{c}{d}}\): compute \(\color{blue}{a \times d}\) and \(\color{blue}{b \times c}\). The larger product corresponds to the larger ratio.

\(\color{blue}{\frac{3}{5}}\) vs \(\color{blue}{\frac{2}{3}}\): \(\color{blue}{3 \times 3 = 9}\) and \(\color{blue}{5 \times 2 = 10}\). Since \(\color{blue}{10 > 9}\), \(\color{blue}{\frac{2}{3} > \frac{3}{5}}\).

Step-by-Step Summary

1. Write both ratios as fractions.
2. Choose a comparison method (decimal, common denominator, or cross-multiply).
3. Compare and write the result using \(\color{blue}{<}\), \(\color{blue}{>}\), or \(\color{blue}{=}\).
4. Answer the question in context — which deal, team, or mixture is better?

Watch: Ratio Problems with Tables Example 2 (Video Lesson)

Khan Academy compares two ratio tables and interprets them to solve a word problem:

Worked Examples

Example 1: Store A sells 3 notebooks for $5. Store B sells 5 notebooks for $7. Which store has the better unit price?

Store A: \(\color{blue}{$5 \div 3 &\text{ approx }; $1.667}\) per notebook. Store B: \(\color{blue}{$7 \div 5 = $1.40}\) per notebook. Store B is cheaper — better unit price.

Example 2: Team A won 8 games out of 12 played. Team B won 9 games out of 15 played. Which team has a better win ratio?

Team A: \(\color{blue}{\frac{8}{12} = \frac{2}{3} &\text{ approx }; 0.667}\). Team B: \(\color{blue}{\frac{9}{15} = \frac{3}{5} = 0.60}\). Team A has the better win ratio.

Example 3: A punch recipe A uses 2 cups of juice to 5 cups of water. Recipe B uses 3 cups of juice to 7 cups of water. Which recipe is stronger (higher juice-to-water ratio)?

Recipe A: \(\color{blue}{\frac{2}{5} = 0.40}\). Recipe B: \(\color{blue}{\frac{3}{7} &\text{ approx }; 0.429}\). Recipe B has a slightly higher juice ratio — it is stronger.

Example 4: A car covered 120 miles on 4 gallons. A truck covered 210 miles on 7 gallons. Which vehicle has better fuel efficiency?

Car: \(\color{blue}{120 \div 4 = 30}\) mpg. Truck: \(\color{blue}{210 \div 7 = 30}\) mpg. They are equal.

More Practice: Ratio Word Problem Examples Video

Khan Academy solves a variety of ratio word problem examples including comparisons:

Exercises

1. Juice X: 4 oz concentrate per 12 oz water. Juice Y: 3 oz concentrate per 8 oz water. Which is stronger?
2. Maria reads 45 pages in 3 hours. Juan reads 52 pages in 4 hours. Who reads faster?
3. Bag A has 5 apples per 8 fruits total. Bag B has 7 apples per 11 fruits total. Which bag has a higher proportion of apples?
4. Price: Cereal A — 18 oz for $3.60. Cereal B — 12 oz for $2.52. Which is the better deal?
5. A student scored 42 out of 60 on Test 1 and 35 out of 50 on Test 2. On which test did the student do better?

Answers

1. X: \(\color{blue}{4 \div 12 &\text{ approx }; 0.333}\). Y: \(\color{blue}{3 \div 8 = 0.375}\). Juice Y is stronger.
2. Maria: \(\color{blue}{45 \div 3 = 15}\) pages/hour. Juan: \(\color{blue}{52 \div 4 = 13}\) pages/hour. Maria reads faster.
3. A: \(\color{blue}{\frac{5}{8} = 0.625}\). B: \(\color{blue}{\frac{7}{11} &\text{ approx }; 0.636}\). Bag B has a slightly higher proportion of apples.
4. A: \(\color{blue}{\(3.60 \div 18 = \)\frac{0.20}{\text{ oz }}}\). B: \(\color{blue}{\(2.52 \div 12 = \)\frac{0.21}{\text{ oz }}}\). Cereal A is the better deal.
5. Test 1: \(\color{blue}{\frac{42}{60} = 0.70}\). Test 2: \(\color{blue}{\frac{35}{50} = 0.70}\). Equal performance.

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Frequently Asked Questions

What is the quickest method for comparing two ratios?

Convert both to decimals. One division per ratio, then a simple decimal comparison. This is usually faster than finding a common denominator, especially for non-obvious denominators.

Does the order matter when I cross-multiply to compare ratios?

Yes. For \(\color{blue}{\frac{a}{b}}\) vs \(\color{blue}{\frac{c}{d}}\), compute \(\color{blue}{a \times d}\) (first ratio’s numerator times second ratio’s denominator) and compare it to \(\color{blue}{b \times c}\) (first ratio’s denominator times second ratio’s numerator). The larger product belongs to the larger ratio.

How do I handle a comparing ratio problem where I need to find a missing value first?

Find the missing value using the ratio (scale factor or cross-multiplication), then compare the completed ratios using any of the three methods above.

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