Unit rates drills

Compute and compare unit rates (per-one quantities) from ratios.

Free, timed SHSAT practice sets on unit rates— every question includes a full worked explanation, and there's no account required.

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How to solve unit rate problems

A unit rate is an amount per one unit. To find it, divide the amount by the quantity it goes with: price per item \(=\) total cost \(\div\) number of items.

Common traps: dividing the wrong way (\(6 \div 4.50\)), dividing by minutes when the question asks per hour, and multiplying two fractions instead of dividing. Quick check: the price of one item must be less than the price of the pack.

Worked examples

Example 1.
Coffee beans cost \(\$6.80\) for 4 pounds. At the same rate, how much do 8 pounds cost?
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Answer:
B

Unit rate = \(\$6.80 \div 4 = \$1.70\) per pound. Scaled up: \(\$1.70 \times 8 = \$13.60\). (Stopping at the unit rate gives $1.70. Setting up the proportion upside down, \(\$6.80 \times 4 \div 8\), gives $3.40. Multiplying $6.80 by 8 without first dividing by 4 gives $54.40.)

Example 2.
A cyclist rides 7 miles in 30 minutes at a constant speed. What is the cyclist's speed, in miles per hour?
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Answer:
A

Convert minutes to hours first: 30 min \(= 30 \div 60 = 0.5\) h. Speed \(= \text{distance} \div \text{time} = 7 \div 0.5 = 14\) mph. (Dividing by 30 minutes without converting to hours gives about 0.233 mph. Dividing the time by the distance, \(0.5 \div 7\), gives about 0.071 mph. Multiplying, \(7 \times 0.5\), gives 3.5 mph.)

Example 3.
A recipe uses \(\frac{3}{4}\) cup of flour for \(\frac{2}{3}\) of a batch. How many cups of flour does one full batch need?
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Answer:
C

Unit rate \(= \frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8}\) cups. (Dividing the other way gives 8/9. Multiplying the two fractions gives 1/2. Flipping both fractions and then multiplying gives 2.)