Ratios, rates, proportions & units
These questions reward one habit above all: tracking units. When you keep units attached to numbers and let them cancel, the right setup reveals itself — and the common traps (flipping a rate, using the wrong conversion) become obvious.
What College Board tests
Setting up and solving proportions, computing and comparing unit rates, and converting between units (including compound units like km/h). Many word problems are really one proportion in disguise.
The core ideas
Keep units written next to every number. If the units don't cancel to what the question asks for, the setup is wrong — fix it before computing.
Four worked examples in SAT format. Read the approach, try it yourself, then tap Show the full solution.
1 · Solve a proportion
A recipe uses 3 pounds of flour to make 12 servings. At the same rate, how many pounds of flour are needed to make 20 servings?
Approach Set two equal ratios, keeping the same units in the same positions: pounds over servings on both sides. Then cross-multiply and solve.
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Answer: B
Why the other choices are wrong
A Rounds the rate down rather than solving exactly.
C Sets up the ratio upside-down on one side (servings over pounds).
D Doubles the original 3 → 6 → 8 by adding, ignoring the true rate.
2 · Compare unit rates
Brand A sells 18 ounces of coffee for $4.50. Brand B sells 12 ounces for $3.30. Which brand costs less per ounce, and by how much?
- Brand A, by $0.025 per ounce
- Brand B, by $0.025 per ounce
- Brand A, by $0.05 per ounce
- They cost the same per ounce
Approach A unit rate is price divided by amount — dollars per one ounce. Compute it for each brand, then compare. Whichever is smaller is the better buy.
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Answer: A
A unit rate is the price for one ounce, so the target is a fraction with 1 ounce in the denominator. "Per one ounce" is exactly what "unit rate" means — divide the price by the number of ounces, and the result carries units of dollars per ounce:
The unit check confirms the comparison is valid: for both brands, so they're measured in the same units. And the rate works the other way too — buying 18 oz of Brand A costs , recovering the original price. Brand A's lower per-ounce price makes it the better buy.
Why the other choices are wrong
B Right difference, wrong brand — Brand B is the more expensive one.
C Doubles the true difference.
D The two unit rates differ, so they aren't equal.
3 · Convert compound units
A train travels at a constant speed of 90 kilometers per hour. What is this speed in meters per second? (1 km = 1000 m, 1 hour = 3600 s)
Approach Multiply by conversion fractions chosen so the unwanted units cancel. Put kilometers and hours where they'll divide out, leaving meters on top and seconds on the bottom.
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Answer: B
Why the other choices are wrong
A Divides by 6 somewhere instead of carrying both conversions through.
C Converts hours but forgets to convert kilometers (or vice versa).
D Leaves the value unchanged — no conversion applied.
4 · Apply a rate
A machine produces 240 parts in 8 hours, working at a constant rate. How many parts does it produce in 5 hours?
Approach Find the rate in parts per one hour, then multiply by the number of hours. The "hours" will cancel, leaving parts.
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Answer: B
Why the other choices are wrong
A Uses a rate of 24/hr (mis-divides 240 by 10).
C Uses 4 hours' worth at the wrong rate.
D Forgets to scale down from 8 hours to 5.