Digital SAT · Ratios, Rates & Proportions

How to Solve Ratios and Proportions on the Digital SAT

A ratio compares two amounts; a proportion says two ratios are equal. Line them up with the same category on top in both, cross-multiply, and solve. The one thing to watch is whether the question wants part-to-part, comparing two groups, or part-to-whole, comparing a group to the total, since those give different answers from the same numbers.

Written from Perfect1600’s analysis of every ratios and proportions question in our bank·Method checked against the current Bluebook test

per test: less than 1 per test
typical difficulty: Medium to hard
practice questions: 57

How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

same order

Line them up

Same category on top in both ratios, then cross-multiply.

46%

Do it on Desmos

About half are a proportion the calculator solves in a line.

read it

Part or whole

Whether the answer is part-to-part or part-to-whole is the deciding call.

How to recognize ratios and proportions questions

Two quantities are compared, or a scaled version of a relationship is given.
A ratio is stated and a missing amount at the same ratio is asked.
The words ratio, proportion, per, or for every appear.
It may ask for a part-to-part or a part-to-whole ratio.

Why students miss these

The trouble is the setup, not the cross-multiplication. People stack the ratios inconsistently, blur part-to-part with part-to-whole, or build the total wrong. Keep the same category on top across both ratios, and pause to check whether the whole includes every part, and the arrangement stays honest.

The step-by-step method

1
State the known ratio
Write it with clear categories, like boys to girls or miles to gallons.
2
Line up the proportion
Put the second ratio in the same order, unknown in the matching slot.
3
Cross-multiply and solve
Multiply across the equals sign and solve for the unknown.
4
Part or whole
Confirm whether the answer should be part-to-part or part-to-whole, and build the total if needed.

Solving it on Desmos

Type the proportion. Enter the two equal ratios as an equation with the unknown as x.
Read the value. Desmos returns the number that makes the ratios equal.
Form the whole if needed. For a part-to-whole answer, compute the total and build that ratio.

Full Desmos walkthrough for ratios and proportions→

When to use it: The calculator handles the cross-multiplication and division, handy on grid-ins. Arranging the proportion and choosing part-to-part versus part-to-whole are the calls you make.

See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Easy example

The ratio of the length of segment

XY

to the length of segment

ZV

is 6 to 1. If

XY

is 102 inches long, what is the length, in inches, of

ZV

17

96

108

612

A: Correct. From $\frac{6}{1} = \frac{102}{ZV}$ , $6 \cdot ZV = 102$ , so $ZV = 17$ .

B: Incorrect. This subtracts 6 from 102 instead of dividing.

C: Incorrect. This adds 6 to 102 instead of dividing.

D: Incorrect. This multiplies 102 by 6 instead of dividing.

Explanation

Solve $\frac{6}{1} = \frac{102}{ZV}$ : $ZV = 17$ .

Medium example

A class contains 10 boys and 15 girls. What is the ratio of the number of boys to the total number of students in the class?

2 to 5

3 to 5

2 to 3

5 to 2

A: Correct. The total is $10 + 15 = 25$ , so boys to total is $\frac{10}{25} = \frac{2}{5}$ .

B: Incorrect. This is the ratio of girls to the total, not boys to total.

C: Incorrect. This is the ratio of boys to girls, not boys to total.

D: Incorrect. This inverts the ratio of boys to total.

Explanation

Boys to total: $\frac{10}{25} = \frac{2}{5}$ , or 2 to 5.

Hard example

A traveler will convert 4,000 rupees to US dollars at a rate of 1 rupee = 0.012 US dollars. The exchange service charges a 3% fee on the converted dollar amount.

How many US dollars will the traveler receive after the 3% fee is applied?

46.56

48.00

333.33

480.00

A: Correct. Converting gives $4000 \times 0.012 = 48.00$ ; after the 3% fee, $48.00 \times 0.97 = 46.56$ .

B: Incorrect. This is the amount before the 3% fee is subtracted.

C: Incorrect. This divides by the rate instead of multiplying.

D: Incorrect. This misplaces the decimal, using a rate of 0.12.

Explanation

Convert: $48.00$ ; apply fee: $48.00 \times 0.97 = 46.56$ .

Detailed explanation

Multiplying $4{,}000$ rupees by $0.012$ gives $48.00$ dollars. The 3% fee removes $0.03 \times 48.00 = 1.44$ , leaving $48.00 - 1.44 = 46.56$ . Skipping the fee gives 48.00, and misplacing the decimal or inverting the rate gives the larger values.

The common traps

Pattern	What it does	The tell
Inconsistent stacking	Put categories in different orders in the two ratios.	Keep the same category on top across both.
Part versus whole	Gave part-to-part when part-to-whole was asked.	Check whether the total includes both parts.
Wrong total	Built the whole by adding the parts incorrectly.	The whole is the sum of every part in the group.

Try it: two real questions

Question 1easy

At an animal shelter, the ratio of cats to dogs is 4 to 3. If the shelter has 12 cats, how many dogs does it have?

Question 2easy

A car travels 120 miles using 4 gallons of gasoline. At this rate, how many miles can the car travel using 7 gallons of gasoline?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Ratios, Rates & Proportions

Rates and Unit Conversion

Percentages

Area & Volume

Scale Factor for Area and Volume

Common questions

How do you solve a proportion on the SAT?

Set the two ratios equal with matching categories in the same positions, cross-multiply, and solve. In Desmos, type the two ratios as an equation and read the value.

What is the difference between part-to-part and part-to-whole ratios?

Part-to-part compares two groups, like boys to girls. Part-to-whole compares one group to the total, like boys to all students. The question tells you which.

How do I set up a proportion correctly?

Write one ratio with clear categories, then the second in the same order with the unknown in the matching spot, keeping the same units on top in both.

Can Desmos solve ratio problems?

For the arithmetic, yes. Type the proportion as an equation with x and read the solution. You still set up the ratio and decide part versus whole.

Practice ratios and proportions the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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What the question bank shows

How to recognize ratios and proportions questions

Why students miss these

The step-by-step method

State the known ratio

Line up the proportion

Cross-multiply and solve

Part or whole

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do you solve a proportion on the SAT?

What is the difference between part-to-part and part-to-whole ratios?

How do I set up a proportion correctly?

Can Desmos solve ratio problems?

Practice ratios and proportions the way it is tested