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Ratios, Rates & Proportional Relationships for the SAT: Complete Study Guide + Free Practice Problems
Everything the SAT tests about ratios, rates, and proportional relationships 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 boys to girls in a class. 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 | 3 red marbles to 5 blue marbles | Same units, often written 3:5 |
| Rate | 60 miles per 2 hours | Different units (miles per hour) |
| Proportion | 3/5 = 9/15 | Two equal ratios or rates |
A ratio of 3:5 does not mean there are only 8 total items — it means the quantities exist in that proportion. The actual total could be 8, 80, or 800; you need more information to know which.
Setting up and simplifying ratios
Ratios can be written three ways — a:b, a to b, or a/b — and all three mean the same thing. Always simplify a ratio the same way you'd simplify a fraction, by dividing both parts by their greatest common factor.
A recipe uses flour and sugar in a ratio of 5:2. If a baker uses 20 cups of flour, how much sugar is needed?
Part-to-part ratios and part-to-whole ratios are different. A ratio of 5:2 flour to sugar means the whole recipe is divided into 7 parts, not 5 or 2 — watch for questions that ask for a fraction of the total, not just one part compared to the other.
Unit rates
A unit rate expresses a rate per single unit of the other quantity — miles per 1 hour, dollars per 1 item, words per 1 minute. Finding a unit rate means dividing to make the second quantity equal to 1.
A car travels 315 miles using 9 gallons of gas. What is the car's fuel efficiency in miles per gallon?
When comparing two unit rates to find which is the better deal (price per ounce, cost per item), always convert both options to the same unit rate before comparing — comparing totals directly can be misleading if package sizes differ.
Solving proportions
When two ratios are set equal to each other, cross multiplication turns the proportion into a simple linear equation.
Solve for x: 4/9 = x/45
Scale factors & scale drawings
A scale factor is a ratio comparing a scale drawing (a map, blueprint, or model) to the real, full-size object. Every length in the drawing relates to the real length by that same constant ratio.
A map has a scale of 1 inch = 25 miles. Two cities are 3.5 inches apart on the map. What is the actual distance?
When a scale factor applies to area instead of length, the area scales by the square of the scale factor, not the scale factor itself. Doubling every length in a drawing quadruples the area, not doubles it.
Proportional relationship word problems
Most SAT ratio and rate questions are word problems. The key skill is translating the situation into a proportion with matching units in matching positions.
- Keep units aligned. If the first ratio is miles over hours, the second ratio must also be miles over hours — not hours over miles.
- Watch for two-step problems. Some questions require finding a unit rate first, then multiplying it by a new quantity.
- Identify what's constant. A proportional relationship has a constant ratio between the two quantities — check that the relationship in the problem is truly proportional before setting up a proportion.
A printer produces 150 pages in 6 minutes. At this constant rate, how many pages does it produce in 25 minutes?
Common SAT traps
- Part-to-part vs. part-to-whole: a ratio of 3:4 means 3 out of 7 total parts, not 3 out of 4.
- Flipped proportions: setting up a/b = d/c instead of a/b = c/d will give a completely wrong answer — keep matching quantities in matching positions.
- Scaling area or volume like length: area scales by the square of the scale factor; volume scales by the cube.
- Mixing units: convert units (minutes to hours, inches to feet) before setting up any ratio if the problem mixes them.
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 →