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Ratios &
Proportions

Master the art of comparing quantities and solving for unknown values using cross-multiplication, scaling, and equivalent fractions.

SAT Math

What Is a Ratio?

A ratio is a comparison of two quantities by division. If a class has 12 boys and 18 girls, the ratio of boys to girls is \(\frac{12}{18}\), which simplifies to \(\frac{2}{3}\).

Boys 12 : Girls 18 Simplified: 2 : 3

What Is a Proportion?

A proportion is an equation stating that two ratios are equal. If \(\frac{a}{b} = \frac{c}{d}\), then \(a\), \(b\), \(c\), and \(d\) are in proportion.

2 3 = 8 12 Both are equivalent to 2 : 3

Why Ratios Matter on the SAT

Ratio and proportion questions appear across multiple SAT sections. They test your ability to set up relationships, scale quantities, and solve efficiently.

Word Problems
Mixing solutions, splitting costs, recipe scaling
Geometry
Similar figures, scale factors, map distances
Data & Graphs
Rates, unit conversion, percentages as ratios

The Cross-Multiplication Rule

If \(\frac{a}{b} = \frac{c}{d}\), then \(a \cdot d = b \cdot c\). This is the fundamental tool for solving any proportion.

=
\(a\)
\(b\)
\(c\)
\(d\)
\( a \cdot d = b \cdot c \)

Equivalent Ratios & Scaling

Two ratios are equivalent if you can multiply (or divide) both parts by the same nonzero number. This is the same as simplifying fractions.

2 : 5 × 2 4 : 10 × 3 12 : 30 All equivalent: they all simplify to 2 : 5

Ratio vs. Proportion vs. Rate

Ratio Proportion Rate
Definition Comparison of two quantities Equation: two ratios are equal Ratio with different units
Example \(3 : 4\) \(\frac{3}{4} = \frac{6}{8}\) 60 miles per hour
Key Skill Simplify Cross-multiply Unit conversion
SAT Focus Part-to-part or part-to-whole Solving for unknowns Speed, density, cost

Part-to-Part vs. Part-to-Whole

This is the single most important distinction the SAT tests. A ratio of \(3 : 5\) can mean very different things depending on context.

Part-to-Part
Compares one group to another: 3 boys : 5 girls. Total = \(3 + 5 = 8\) parts.
Part-to-Whole
Compares one group to the total: 3 boys out of 8 students, or \(\frac{3}{8}\).

How to Solve Proportions

1
Identify the ratio. Write the known relationship as a fraction \(\frac{a}{b}\).
2
Set up the proportion. Write the second ratio with the unknown: \(\frac{a}{b} = \frac{c}{x}\).
3
Cross-multiply. Compute \(a \cdot x = b \cdot c\).
4
Solve for the unknown. Divide both sides by the coefficient of \(x\).
5
Check units and context. Does the answer make sense? Are your units consistent?

Example 1: Basic Proportion

A recipe calls for 3 cups of flour for every 5 cups of sugar. How much flour is needed for 20 cups of sugar?

1
Set up: \(\frac{3}{5} = \frac{x}{20}\)
2
Cross-multiply: \(3 \times 20 = 5 \times x\), so \(60 = 5x\)
3
Solve: \(x = \frac{60}{5} = 12\)

Example 2: Part-to-Whole

In a bag of marbles, the ratio of red to yellow marbles is \(2 : 7\). If there are 63 marbles total, how many are red?

1
Total parts: \(2 + 7 = 9\) parts
2
Value per part: \(\frac{63}{9} = 7\) marbles per part
3
Red marbles: \(2 \times 7 = 14\)

Example 3: Rate Problem

A car travels 240 miles in 4 hours. At the same rate, how far will it travel in 7 hours?

240 mi / 4 hr = x mi / 7 hr Unit rate: 60 mph
\(x = 60 \times 7 = 420 \text{ mi}\)

Example 4: Multi-Part Ratio

A concrete mix uses cement, sand, and gravel in the ratio \(1 : 2 : 3\). If 180 kg of the mix is needed, how much sand is required?

1
Total parts: \(1 + 2 + 3 = 6\)
2
Value per part: \(\frac{180}{6} = 30\) kg
3
Sand = 2 parts: \(2 \times 30 = 60\) kg

Common Mistakes

✕ Flipping the Ratio
Writing \(\frac{5}{3}\) when the problem says "3 to 5." Always match the order given in the problem.
✕ Part vs. Whole Confusion
Using a part-to-part ratio as if it were part-to-whole. If the ratio is \(2 : 3\), the total is \(5\) parts, not \(3\).
✕ Mismatched Units
Setting up a proportion with mismatched units, like \(\frac{\text{miles}}{\text{hours}} = \frac{\text{hours}}{\text{miles}}\). Always keep units consistent.

Strategy Summary

Problem Type Method Formula
Solve for unknown Cross-multiply \(ad = bc\)
Part-to-whole Add ratio parts \(\frac{\text{part}}{a + b} \times \text{total}\)
Scaling / equivalent Multiply both parts \(a : b = ka : kb\)
Unit rate Divide quantity by units \(\frac{\text{quantity}}{\text{units}}\)
Multi-part ratio Sum all parts, find value per part \(\frac{\text{total}}{a + b + c}\)

Question 1 of 5

If \(\frac{x}{12} = \frac{5}{4}\), what is the value of \(x\)?

Question 2 of 5

The ratio of cats to dogs at a shelter is \(3 : 5\). If there are 40 animals total, how many cats are there?

Question 3 of 5

A map uses a scale of \(1\) inch \(= 15\) miles. Two cities are \(3.5\) inches apart on the map. What is the actual distance?

Question 4 of 5

A paint mixture uses red, blue, and white in the ratio \(2 : 3 : 7\). If you need 36 liters of the mixture, how many liters of blue paint are required?

Question 5 of 5

If \(8\) workers can build a wall in \(10\) days, how many days will it take \(5\) workers to build the same wall (at the same individual rate)?