Word Problems Involving Rates and Ratios

Rates and ratios show up constantly in everyday situations — speed, pricing, mixing, and scaling. Whether a problem tells you a car’s speed and asks for distance, or gives you a price per unit and asks for total cost, the strategy is always the same: set up the right proportion and solve. This guide breaks it down clearly.

What Are Rates and Ratios in Word Problems?

A ratio compares two quantities of the same kind (e.g., 3 red balls to 5 blue balls). A rate compares two quantities of different kinds (e.g., 60 miles per hour). Word problems combine real-world context with these mathematical relationships, asking you to find a missing quantity.

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How to Solve Rate and Ratio Word Problems

Step 1 — Identify the rate or ratio given

Find the comparison stated in the problem. Write it as a fraction: \(\color{blue}{\text{ quantity } \frac{A}{\text{ quantity }} B}\).

Step 2 — Set up a proportion

Place the known values and the unknown \(\color{blue}{x}\) into a proportion where matching units are in the same positions:

\(\color{blue}{\frac{a}{b} = \frac{c}{x}}\)   or   \(\color{blue}{\frac{a}{b} = \frac{x}{d}}\)

Step 3 — Solve by cross-multiplying

Cross-multiply: \(\color{blue}{a \times x = b \times c}\), then divide both sides by the coefficient of \(\color{blue}{x}\).

Step 4 — Check with units

Label each number with its unit. The units should cancel to give the unit of the answer.

Step-by-Step Summary

1. Read the problem; underline the rate or ratio and the target question.
2. Write the known rate as a fraction with consistent units.
3. Set up the proportion with \(\color{blue}{x}\) for the unknown.
4. Cross-multiply and solve for \(\color{blue}{x}\).
5. Check that your answer makes sense (units match; value is reasonable).

Watch: Ratio and Proportion Word Problems

The Organic Chemistry Tutor works through a variety of ratio and proportion word problems step by step:

Worked Examples

Example 1: A car travels 240 miles in 4 hours. How far does it travel in 7 hours at the same speed?

Rate: \(\color{blue}{\frac{240}{4} = 60 \text{ mph }}\). Distance in 7 hours: \(\color{blue}{60 \times 7 = 420 \text{ miles }}\).
Answer: 420 miles

Example 2: 3 pens cost $4.50. How much do 7 pens cost?

Proportion: \(\color{blue}{\frac{3}{4.50} = \frac{7}{x}}\). Cross-multiply: \(\color{blue}{3x = 4.50 \times 7 = 31.50}\). So \(\color{blue}{x = 31.50 \div 3 = 10.50}\).
Answer: $10.50

Example 3: 5 workers can finish a job in 8 days. How many workers are needed to finish the same job in 4 days?

This is an inverse proportion (more \(\color{blue}{\text{ workers } = \text{ fewer }}\) days). \(\color{blue}{\text{ Workers } \times \text{ days }}\) stays constant: \(\color{blue}{5 \times 8 = 40}\). Workers for 4 days: \(\color{blue}{40 \div 4 = 10}\).
Answer: 10 workers

Example 4: The ratio of cats to dogs at a shelter is 5 : 3. If there are 24 dogs, how many cats are there?

Proportion: \(\color{blue}{\frac{5}{3} = \frac{x}{24}}\). Cross-multiply: \(\color{blue}{3x = 120}\). So \(\color{blue}{x = 40}\).
Answer: 40 cats

More Practice: How to Solve Proportions

Math with Mr. J reviews the full strategy for solving proportions, including ratio word problems:

Exercises

1. A train travels 180 miles in 3 hours. How far does it travel in 5 hours?
2. 8 apples cost $2.40. How much do 15 apples cost?
3. The ratio of boys to girls in a class is 3 : 4. If there are 28 girls, how many boys are there?
4. A faucet fills a tank at the rate of 6 gallons per minute. How long does it take to fill a 90-gallon tank?
5. A recipe makes 24 cookies using 3 eggs. How many eggs are needed for 40 cookies?
6. A map scale is 1 inch : 25 miles. Two cities are 4.5 inches apart on the map. What is the real distance?

Answers

1. \(\color{blue}{300 \text{ miles }}\)
2. \(\color{blue}{$4.50}\)
3. \(\color{blue}{21 \text{ boys }}\)
4. \(\color{blue}{15 \text{ minutes }}\)
5. \(\color{blue}{5 \text{ eggs }}\)
6. \(\color{blue}{112.5 \text{ miles }}\)

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Frequently Asked Questions

What is the difference between a direct and an inverse proportion?

In a direct proportion, both quantities increase or decrease together (e.g., more hours \(\color{blue}{\text{ worked } = \text{ more }}\) pay). In an inverse proportion, one increases as the other decreases (e.g., more \(\color{blue}{\text{ workers } = \text{ fewer }}\) days to finish). Set up the equation differently for each type.

How do I decide which number goes in the numerator?

Be consistent with units. If the rate is miles per hour, the numerator is always miles and the denominator is always hours in both fractions of the proportion.

Do I always need to cross-multiply?

Not always. If you can find the scale factor by inspection (e.g., one denominator is exactly twice the other), scaling is faster. Use cross-multiplication when the relationship is not an obvious whole-number multiple.

Related Topics

• Equivalent Rates
• Word Problems Involving Equivalent Ratio
• Finding Equivalent Ratio
• Ratio Tables
• Word Problems Involving Comparing Ratio