Word Problems Involving Comparing Rates

To compare two rates in a word problem, convert both to the same unit rate and then compare. Dividing each total by its own quantity is what puts them on equal footing.

Step-by-step guide to solving comparing rates word problems

1. Understand the problem: Read the problem carefully and make sure you understand what is being asked. Identify the key information such as the rates, the time, and the units of measurement.
2. Write the rates as fractions: Express each rate as a fraction. For example, if the problem says “a car travels 60 miles in 2 hours,” the rate is 60/2 = 30 miles per hour (mph).
3. Compare the rates: Compare the fractions to see which one is larger or smaller. The larger fraction represents the faster rate.
4. Use cross-multiplication to find the equivalent rates: In some problems, you may need to find equivalent rates to compare them. For example, if one rate is given in miles per hour and the other is given in kilometers per hour, you will need to convert one of them to the other unit. To do this, multiply by a conversion factor, such as 1 mile ≈ 1.609 kilometers. For example, if the problem says “a car travels 60 miles in 2 hours, and another car travels 100 kilometers in 4 hours,” the rates are 60/2 = 30 mph and 100/4 = 25 km/h. Since 1 mile is about 1.609 km, 30 mph is about 48.3 km/h, so the first car is faster.
5. Use the information to answer the question: Once you have identified the key information, use it to answer the question or solve the problem. For example, if the problem asks which car is faster, you would say the first car is faster because it has a faster rate.
6. Check your answer: Make sure your answer is reasonable and make sure you have used the correct units of measurement.

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Word Problems Involving Comparing Rates, Examples 1

Word Problems Involving Comparing Rates, Examples 2

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Worked Examples: Rates in Tables, Graphs and Equations

Example 1: Equation vs. table

Plan A charges C = 0.10m + 15 dollars for m minutes. Plan B charges $20 for 100 minutes and $30 for 300 minutes. Which plan costs more per minute?

Plan A’s rate is the slope, $0.10 per minute. Plan B’s rate is (30 − 20) ÷ (300 − 100) = 10 ÷ 200 = $0.05 per minute. Plan A costs more per minute. Both start at $15, since 20 − 0.05 × 100 = 15.

Example 2: Description vs. graph

Tank A holds 500 gallons and drains 25 gallons per minute. Tank B’s graph passes through (0, 360) and (6, 240). Tank B’s rate is (240 − 360) ÷ 6 = −20 gallons per minute. Tank A drains faster, but Tank B empties first: A takes 500 ÷ 25 = 20 minutes, and B takes 360 ÷ 20 = 18 minutes. A faster rate does not always mean finishing first.

Example 3: Unit price

A 12-ounce box costs $3.60 and a 20-ounce box costs $5.40. That is $0.30 and $0.27 per ounce, so the larger box is the better buy.

Rate Comparison Practice, Answers Included

1. Car A goes 150 miles in 3 hours. Car B’s distance is d = 55t. Which is faster?
2. Jen’s savings follow y = 8x + 20 for x weeks. Sam has $35 after week 1 and $55 after week 3. Who saves faster?
3. Printer A prints 90 pages in 3 minutes. Printer B prints 140 pages in 5 minutes. Which is faster?
4. Candle A is 12 inches and burns 1.5 inches per hour. Candle B is 10 inches after 1 hour and 7 inches after 3 hours. Which burns out first?

Answers: 1) Car B, 55 mph vs. 50 mph. 2) Sam, $10 per week vs. $8. 3) Printer A, 30 vs. 28 pages per minute. 4) Both burn 1.5 inches per hour, but Candle B started at 11.5 inches, so it burns out first, after about 7.7 hours instead of 8.

Frequently Asked Questions

Can you give me some word problems about rates?

Sure. A runner covers 6 miles in 48 minutes; how many minutes per mile is that? (8.) A pump moves 450 liters in 9 minutes; what is the rate? (50 liters per minute.) Store A sells 4 pens for $3 and Store B sells 6 pens for $4.20; which is cheaper per pen? (Store B, $0.70 vs. $0.75.)

How do you compare two linear functions in a word problem?

Find the rate of change and the starting value for each one. The rate is the slope, or how much y changes when x goes up by 1. The starting value is the y-intercept, the amount when x is 0. Compare rates to see which grows or shrinks faster, and starting values to see which begins higher.

How do you find the rate of change from a table?

Pick two rows. Subtract the y-values, subtract the x-values, and divide the first difference by the second. If a table shows 4 hours with $52 and 7 hours with $91, the rate is (91 − 52) ÷ (7 − 4) = 39 ÷ 3 = $13 per hour. Check a third row to confirm the rate stays the same.

How do you compare rates with different units?

Convert one rate so both use the same units, then compare. To compare 30 miles per hour with 40 kilometers per hour, use 1 mile ≈ 1.609 kilometers: 30 mph is about 48.3 km/h, which is faster than 40 km/h. For time, change minutes to hours or hours to minutes first.