Word Problems Involving Comparing Ratio

Comparing ratios in word problems is a skill you use 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 \(\color{blue}{2 : 3}\)?

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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Turning the Words into Ratios

Most mistakes happen before any math starts, when the words get turned into numbers. These phrases tell you how to set up each ratio.

Common Mistakes When Comparing Ratios

• Flipping the order. If one ratio is juice to water, the other must be juice to water too, not water to juice.
• Mixing part-to-part with part-to-whole. Class A has 12 girls and 18 boys. Class B has 15 girls out of 35 students. To compare, write both as girls out of the whole class: 12/30 = 0.4 and 15/35 ≈ 0.429. Class B has the larger share of girls, even though Class A’s ratio 12 : 18 might look like it should be compared to 15 : 35 directly.
• Comparing raw numbers. Team B winning 9 games does not beat Team A winning 8 if Team B played more games. Compare the rates, not the counts.
• Rounding too early. When two decimals are close, keep three decimal places or cross-multiply to be sure.

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.

How do you write a ratio from a word problem?

Find the two quantities the question asks about and write them in the order the question names them. If it asks for cost per notebook, put cost on top and notebooks on the bottom. Ignore numbers that the question does not use, and make sure both ratios you compare follow the same order and the same units.

How can I tell if two ratios are equal?

Cross-multiply. Write the ratios as fractions a/b and c/d and compare a × d with b × c. If the products match, the ratios are equal. For example, 120 miles on 4 gallons and 210 miles on 7 gallons give 120 × 7 = 840 and 4 × 210 = 840, so both are 30 miles per gallon.

What is a unit rate and how does it help compare ratios?

A unit rate tells you how much of one quantity there is for exactly one of the other, such as dollars per item or miles per gallon. Once both ratios are unit rates, you only compare two numbers. For example, $5 for 3 notebooks is about $1.67 each and $7 for 5 notebooks is $1.40 each, so the second deal is cheaper.

Can you compare ratios that use different units?

Yes, but convert first so both ratios use the same units. If one runner covers 2 miles in 20 minutes and another covers 3 miles in half an hour, change half an hour to 30 minutes. Then 2/20 = 0.1 and 3/30 = 0.1 miles per minute, so the two runners have the same pace.

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