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Ratios, Rates & Proportions for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about ratios, rates, and proportions in one place: setting up ratios, unit rates, solving proportions, scale factors, and word problems — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhat are ratios, rates & proportions?
A ratio compares two quantities of the same kind, like the ratio of red to blue marbles. A rate compares two quantities with different units, like miles per hour. A proportion is a statement that two ratios or rates are equal.
| Term | Example | Units |
|---|---|---|
| Ratio | 4 red pens to 7 blue pens | Same units, written 4:7 |
| Rate | 90 miles per 3 hours | Different units (miles per hour) |
| Proportion | 4/7 = 12/21 | Two equal ratios or rates |
A ratio of 4:7 does not mean there are only 11 total items — it means the quantities exist in that proportion. The actual total could be 11, 110, or any multiple; more information is needed to know which.
Setting up and simplifying ratios
A trail mix uses nuts and raisins in a ratio of 3:2. If a batch uses 18 cups of nuts, how many cups of raisins are needed?
Part-to-part ratios and part-to-whole ratios are different. A ratio of 3:2 means the whole is divided into 5 parts total, not 3 or 2 — watch for questions asking for a fraction of the whole rather than one part compared to the other.
Unit rates
A printer prints 240 pages in 8 minutes. What is the unit rate in pages per minute?
When comparing two options to find the better deal, always convert both to the same unit rate before comparing — comparing raw totals directly can be misleading when package sizes differ.
Solving proportions
Solve for x: 5/8 = x/40
Scale factors & scale drawings
A blueprint has a scale of 1 inch = 6 feet. A wall measures 4.5 inches on the blueprint. What is the actual length?
When a scale factor applies to area instead of length, area scales by the square of the scale factor. Doubling every length in a drawing quadruples the area, not doubles it.
Word problems
- Keep units aligned. If the first ratio is miles over hours, the second ratio must match that same order.
- Watch for two-step problems. Some questions require finding a unit rate first, then scaling it to a new quantity.
- Confirm the relationship is truly proportional before setting up a proportion — the ratio between the two quantities must stay constant.
A factory produces 180 units in 4 hours at a constant rate. How many units does it produce in 10 hours?
Common PSAT traps
- Part-to-part vs. part-to-whole: a ratio of 3:2 means 3 out of 5 total parts, not 3 out of 2.
- Flipped proportions: keep matching quantities in matching positions in both ratios of a proportion.
- Scaling area or volume like length: area scales by the square of the scale factor; volume scales by the cube.
- Mixing units: convert units before setting up any ratio that mixes them (like minutes and hours).
Test day strategy for ratios, rates & proportions
| Question signal | Fastest approach |
|---|---|
| "For every ___, there are ___" | Set up a ratio, then scale it using multiplication or a proportion |
| "Per," "each," or "every" in a rate | Divide to find the unit rate first |
| Two equal fractions with a missing variable | Cross multiply and solve the resulting linear equation |
| Map, blueprint, or model given | Set up scale = drawing / actual, keeping units matched |
| Area or volume of a scaled figure | Square the scale factor for area, cube it for volume |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Ratios, Rates & Proportions 1
Setting up ratios, unit rates, and solving proportions.
Start quiz → Quiz 2Ratios, Rates & Proportions 2
Scale factors, scale drawings, and word problems.
Start quiz →