Math

Ratios and Proportions

5 min readEasy5-question drill

Ratios and proportions show up in nearly every SAT word problem — recipes, maps, mixtures, similar triangles. The trick is keeping units lined up.

Where this lesson fits

Before

Linear Equations→

Ratios and Proportions

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→Percentages →Unit Conversion

A ratio compares two quantities by division. 3 cups of flour to 2 cups of sugar is the ratio $3:2$ or $\frac{3}{2}$ .

A proportion says two ratios are equal. $\frac{3}{2} = \frac{x}{8}$ — solve by cross-multiplying.

The cross-multiplication shortcut. If $\frac{a}{b} = \frac{c}{d}$ , then $a \cdot d = b \cdot c$ . Solve $\frac{3}{2} = \frac{x}{8}$ :

$3 \cdot 8 = 2 \cdot x \Rightarrow x = 12$

Watch the units. When setting up a proportion, the units must match across both fractions. 2 inches per 5 miles on a map; how many miles is 8 inches?

$\frac{2 \text{ in}}{5 \text{ mi}} = \frac{8 \text{ in}}{x \text{ mi}}$

Inches over miles on both sides — units match. Cross-multiply: $2x = 40 \Rightarrow x = 20$ miles.

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Setting up a proportion — the unit-matching rule

Problem	Right setup	Wrong setup
3 cups flour : 2 cups sugar; need 9 cups flour	$\frac{3 \text{ flour}}{2 \text{ sugar}} = \frac{9 \text{ flour}}{x \text{ sugar}}$	$\frac{3}{2} = \frac{x}{9}$ (units flipped)
2 inches : 5 miles on a map; map is 8 inches	$\frac{2 \text{ in}}{5 \text{ mi}} = \frac{8 \text{ in}}{x \text{ mi}}$	$\frac{2}{5} = \frac{x}{8}$ (units flipped)
20 students : 1 teacher; 80 students total	$\frac{20 \text{ st}}{1 \text{ tchr}} = \frac{80 \text{ st}}{x \text{ tchr}}$	$\frac{20}{1} = \frac{x}{80}$

Ratios with three or more parts. Boys to girls to teachers is $5:3:2$ , and there are 200 students total. Add the parts: $5 + 3 + 2 = 10$ . Each part = $\frac{200}{10} = 20$ people.

Boys: $5 \times 20 = 100$ .
Girls: $3 \times 20 = 60$ .
Teachers: $2 \times 20 = 40$ .

Note: "total" here was 200 — but if the problem said 200 students, you'd add only the student parts ( $5 + 3 = 8$ ).

Direct vs. inverse proportionality:

Direct: $y = kx$ . As $x$ doubles, $y$ doubles. (Distance and time at constant speed.)
Inverse: $y = k/x$ . As $x$ doubles, $y$ halves. (Number of workers and time to finish a job.)

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Direct or inverse proportion?

When one quantity increases, does the other increase (same direction)?

Yes ↓

→ DIRECT — set up $\frac{y_1}{x_1} = \frac{y_2}{x_2}$

No ↓

Does the other quantity decrease in proportion?

Yes ↓

→ INVERSE — set up $x_1 \cdot y_1 = x_2 \cdot y_2$

No ↓

→ Not a simple proportion — re-read the problem

SAT-specific tip: when two quantities are proportional, the ratio between them stays constant. Set up the proportion using the constant, then solve.

Quick check

Quick check. Set up the proportion with matching units (same thing on top of both fractions), then cross-multiply.

A map uses a scale of 1 inch = 25 miles. Two cities are 7.5 inches apart on the map. What is the actual distance between the cities in miles?

187.5

150

175

200

Worked examples

Example 1

A recipe calls for 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar do you need?

Example 2

A tank empties in 12 hours when 5 valves are open. If the rate is constant per valve, how long does it take to empty if only 3 valves are open?

Common pitfalls

Setting up proportions with mismatched units

$\frac{2 \text{ inches}}{5 \text{ miles}} = \frac{x \text{ miles}}{8 \text{ inches}}$ is wrong — the units flipped. Whatever's on top in one fraction must be on top in the other.

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Confusing direct and inverse proportion

More workers → less time is INVERSE. More hours worked → more pay is DIRECT. Read the problem carefully: as one increases, does the other increase (direct) or decrease (inverse)?

Forgetting to add ratio parts to find total

Ratio $5:3:2$ has $5 + 3 + 2 = 10$ parts total. If the total quantity is given, divide by 10, then multiply each part. Don't divide by individual ratio numbers.

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Cross-multiplying without two fractions

Cross-multiplication is for $\frac{a}{b} = \frac{c}{d}$ . If you have $\frac{x}{3} = 5$ , just multiply both sides by 3 — no cross-anything needed.

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Key takeaways

A proportion $\frac{a}{b} = \frac{c}{d}$ solves via cross-multiplication: $a \cdot d = b \cdot c$ .
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Set up proportions so the same unit is on the same side of the bar.
Direct proportion $y = kx$ : more of one means more of the other.
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Inverse proportion $y = k/x$ : more of one means less of the other.
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For three-part ratios: sum the parts, divide the total by the sum, multiply each ratio number by that quotient.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Ratios and Proportions, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Linear Equations→

Ratios and Proportions

You are here

→Percentages →Unit Conversion