Comparing ratios & rates drills

Compare two ratios or rates given across tables, graphs, equations, or words.

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

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How to compare ratios and rates

To compare two rates, change each one to a unit rate (the amount for one unit) in the same units. Then compare or subtract.

Read the order carefully: "How much greater is A than B?" means A minus B.

Common traps:

Worked examples

Example 1.
Relationship A: A car uses gas at a constant rate. 1 gallon of gas corresponds to 7 miles driven, and \(y = kx\). Relationship B: A cyclist rides at a constant rate. \(4y = 6x\), and \(y = kx\). How much greater is the constant of proportionality of Relationship A than Relationship B?
Show answer and explanation
Answer:
C

\(k_A = 7 \div 1 = 7\); \(k_B = 6 \div 4 = 1.5\). Difference \(= 7 - 1.5 = 5.5\). (Adding instead of subtracting, \(7 + 1.5\), gives 8.5. Stopping at \(k_A\) gives 7. Using \(4 \div 6\) for B gives about 6.333.)

Example 2.
Relationship A: A typist types at a constant rate. 6 minutes correspond to 27 words typed, and \(y = kx\). Relationship B: A student reads at a constant rate. \(6y = 3x\), and \(y = kx\). How much greater is the constant of proportionality of Relationship A than Relationship B?
Show answer and explanation
Answer:
A

\(k_A = 27 \div 6 = 4.5\); \(k_B = 3 \div 6 = 0.5\). Difference \(= 4.5 - 0.5 = 4\). (Adding instead of subtracting, \(4.5 + 0.5\), gives 5. Stopping at \(k_A\) gives 4.5. Using \(6 \div 3\) for B gives 2.5.)

Example 3.
Brand A's birdseed costs \(\$38.00\) for 8 pounds. Brand B's birdseed costs \(\$21.30\) for 6 pounds. How much more per pound, in dollars, does Brand A cost than Brand B?
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Answer:
B

Unit prices: A \(= \$38.00 \div 8 = \$4.75\), B \(= \$21.30 \div 6 = \$3.55\). Difference \(= \$4.75 - \$3.55 = \$1.20\). (Subtracting in the reversed order gives -$1.20. Subtracting the total prices instead of the prices per pound gives $16.70. Stopping at Brand A's price per pound gives $4.75.)