Problem Solving & Data Analysis · SAT Math

Ratios and Proportions on the Digital SAT

Ratios, proportions, and unit rates are the most-tested topic in the Digital SAT's problem-solving and data-analysis domain. The math is mostly arithmetic — the difficulty comes from setting up the right proportion. This lesson covers ratio basics, cross-multiplication, unit rates, unit conversions, and 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~9 min read

What is a ratio?

A ratio compares two quantities. It can be written three ways — they all mean the same thing:

  • 3 to 5
  • 3 : 5
  • 3 / 5

A ratio doesn't tell you the actual amounts — just the relationship between them. A ratio of 3 : 5 could mean 3 and 5, or 6 and 10, or 30 and 50.

Part-to-part vs part-to-whole. If a bag has 3 red and 5 blue marbles, the part-to-part ratio of red to blue is 3 : 5. The part-to-whole ratio of red to total is 3 : 8 (since 3 + 5 = 8). The SAT often switches between these — read carefully.

Setting up a proportion

A proportion is two ratios set equal: a/b = c/d. To solve, cross-multiply: a × d = b × c.

Example. If 3 apples cost $2, how much do 12 apples cost?

Set up the proportion: 3 apples / $2 = 12 apples / $x.

Cross-multiply: 3x = 24 → x = 8. So $8.

Key rule: when you set up a proportion, the same unit must appear in the same position on both sides. Apples on top, dollars on bottom — or apples on left, dollars on right. Mixing positions gives a wrong answer.

Rates and unit rates

A rate is a ratio that compares two quantities with different units (miles per hour, dollars per pound, miles per gallon).

A unit rate is a rate with a denominator of 1 (60 miles per 1 hour). Unit rates make comparisons easier and are often the goal of an SAT problem.

Example. If a car travels 240 miles on 8 gallons of gas, its unit rate is 240 / 8 = 30 miles per gallon.

Unit conversions

The SAT regularly tests unit conversions — converting between feet/inches, kilometers/miles, hours/minutes, etc. The technique is dimensional analysis: multiply by conversion ratios so unwanted units cancel.

Example. Convert 5 feet per second to miles per hour. Use that 1 mile = 5280 feet and 1 hour = 3600 seconds.

(5 ft/s) × (1 mi / 5280 ft) × (3600 s / 1 hr) = 18000 / 5280 ≈ 3.41 mph.

Notice the ft on top cancels the ft on bottom; the s on bottom cancels the s on top. Only mi/hr remains.

Setup rule: arrange every conversion ratio so the unit you want to cancel appears in the opposite position (top vs bottom) from where it currently is.

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Solving with Desmos

Desmos can handle ratio and proportion problems faster than scratch paper. Three techniques:

1. Solve a proportion with cross-multiplication

For 3/8 = x/24, type 3/8 = x/24 directly into Desmos. It auto-shows x = 9. No mental cross-multiplication needed.

2. Unit conversions via the calculator

For "convert 5 ft/s to mph," type 5 * 3600 / 5280. Desmos returns ~3.41. Eliminates unit-cancellation errors.

3. Test answer choices with substitution

If a proportion problem asks "what is x?" with four choices, plug each into Desmos as x = [choice] and see which makes both sides of the proportion equal.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Setting up the proportion upside down

If 3 apples cost $2, the proportion is 3/2 = 12/x, NOT 3/2 = x/12. Match units position-for-position on both sides.

Confusing part-to-part with part-to-whole

A 3:5 ratio of red to blue means part-to-part. The fraction that's red is 3/8, not 3/5 — because the whole is 3 + 5 = 8.

Forgetting to convert units before computing

If the question gives time in minutes but the rate is per hour, convert one of them first. 30 mph × 30 minutes ≠ 30 × 30 = 900 miles. It's 30 × 0.5 = 15 miles.

Misreading 'X to Y' as 'X out of Y'

A ratio of 3 to 5 means 3 for every 5 of the other thing. It does NOT mean 3 out of 5. Different setups.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

If 3/x = 12/20, what is the value of x?

  1. 3
  2. 5
  3. 7
  4. 9

Click to reveal solution →

Answer: (B) 5

Cross-multiply: 3 · 20 = 12 · x → 60 = 12x → x = 5.

2

The ratio of teachers to students at a school is 1 to 18. If there are 540 students, how many teachers are there?

  1. 20
  2. 27
  3. 30
  4. 36

Click to reveal solution →

Answer: (C) 30

Set up: 1/18 = T/540. Cross-multiply: 540 = 18T → T = 30.

3

A bag contains red and blue marbles in a ratio of 4 to 7. If there are 33 marbles total, how many are red?

  1. 11
  2. 12
  3. 14
  4. 21

Click to reveal solution →

Answer: (B) 12

Total parts = 4 + 7 = 11. Each part is 33 / 11 = 3 marbles. Red gets 4 parts: 4 × 3 = 12.

4

A car travels at a steady speed of 65 miles per hour. How many miles does it travel in 2 hours and 30 minutes?

  1. 130
  2. 150
  3. 162.5
  4. 195

Click to reveal solution →

Answer: (C) 162.5

Convert 2 hours 30 minutes to 2.5 hours.

65 mph × 2.5 hr = 162.5 miles.

5

A recipe calls for 3 cups of flour for every 5 cookies. How many cups of flour are needed to make 35 cookies?

  1. 18
  2. 21
  3. 24
  4. 27

Click to reveal solution →

Answer: (B) 21

Set up: 3/5 = x/35. Cross-multiply: 105 = 5x → x = 21 cups.

6

A 200-gram serving of cereal contains 8 grams of protein. How many grams of protein are in a 350-gram serving?

  1. 12
  2. 14
  3. 16
  4. 18

Click to reveal solution →

Answer: (B) 14

Proportion: 8/200 = p/350. Cross-multiply: 2800 = 200p → p = 14.

7

A scale on a map shows that 2 inches represents 50 miles. If two cities are 7 inches apart on the map, how many miles apart are they in reality?

Click to reveal solution →

Answer: 175 miles

Proportion: 2/50 = 7/x. Cross-multiply: 2x = 350 → x = 175 miles.

8

A faucet leaks 3 liters of water every 8 minutes. At this rate, how many liters leak in 2 hours?

  1. 30
  2. 45
  3. 60
  4. 75

Click to reveal solution →

Answer: (B) 45

Convert 2 hours to 120 minutes.

Proportion: 3/8 = x/120. Cross-multiply: 8x = 360 → x = 45 liters.

Frequently asked questions

What is a ratio in math?

+
A ratio compares two quantities of the same type. It can be written as 3 to 5, 3:5, or 3/5 — all three notations mean the same thing. A ratio tells you the relationship between quantities, not the actual amounts.

How do I solve a proportion?

+
Cross-multiply. If a/b = c/d, then a × d = b × c. Solve for the unknown variable. On the Digital SAT, Desmos can solve proportions directly — just type the equation as written.

What's the difference between a ratio and a rate?

+
A ratio compares two quantities of the same type (apples to oranges). A rate compares two quantities of different types with units attached (miles per hour, dollars per pound). A unit rate is a rate with a denominator of 1.

Part-to-part vs part-to-whole — what's the difference?

+
Part-to-part compares one part to another (3 red marbles for every 5 blue marbles). Part-to-whole compares a part to the total (3 red marbles out of 8 total). Read SAT questions carefully — they often switch between these.

How do I handle unit conversions on the SAT?

+
Use dimensional analysis: multiply by conversion ratios written so unwanted units cancel out. Arrange each ratio so the unit you want to remove appears in the opposite position (top vs bottom) from where it currently is.

Keep going

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