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Word Problems Involving Proportions (SAT)

Mathematics SAT Reasoning Test Math section Difficulty 2/5

Beginning of the lesson

Word Problems Involving Proportions

Master every proportion problem type on the SAT — from basic ratios to compound variation. This comprehensive lesson covers 128 key concepts with worked examples, strategies, and common trap answers to avoid.

Part 1: Fundamental Concepts

1. Ratio Definition

A ratio is a comparison of two quantities by division. The ratio of $a$ to $b$ is written as:

$$a : b \quad \text{or} \quad \frac{a}{b}$$

Order matters! $a : b \neq b : a$ (unless $a = b$).

Example: If there are 3 cats and 5 dogs, the ratio of cats to dogs is $3:5$, but the ratio of dogs to cats is $5:3$.

2. Proportion Definition

A proportion is an equation stating that two ratios are equal:

$$\frac{a}{b} = \frac{c}{d}$$

Read as: "$a$ is to $b$ as $c$ is to $d$."

3. Terms of a Proportion

In $\frac{a}{b} = \frac{c}{d}$:

  • Extremes (outer terms): $a$ and $d$
  • Means (inner terms): $b$ and $c$
4. Cross Multiplication

The fundamental solving technique:

If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$

Product of extremes = Product of means.

5. Equivalent Ratios

Multiply or divide both terms by the same nonzero number:

$$\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12}$$

The ratio value remains unchanged.

⚠️ SAT Trap: Mismatched Units

Numerators must represent the same type of quantity, and denominators must represent the same type. If the first ratio is $\frac{\text{miles}}{\text{hours}}$, the second must also be $\frac{\text{miles}}{\text{hours}}$, NOT $\frac{\text{hours}}{\text{miles}}$.

Interactive: Equivalent Ratios Visualization
Interactive visualization — available in the full lesson

Part 2: Direct vs. Inverse Proportion

📈 Direct Proportion

As one quantity increases, the other increases proportionally.

$$y = kx \quad \Rightarrow \quad \frac{y}{x} = k \text{ (constant)}$$

Indicators:

  • "$y$ is directly proportional to $x$"
  • "$y$ varies directly with $x$"
  • When $x$ doubles, $y$ doubles

Examples:

  • More items → more cost
  • More hours → more distance
  • More ingredients → more servings

Solving method: Set up equal ratios:

$$\frac{y_1}{x_1} = \frac{y_2}{x_2}$$
📉 Inverse Proportion

As one quantity increases, the other decreases proportionally.

$$y = \frac{k}{x} \quad \Rightarrow \quad xy = k \text{ (constant)}$$

Indicators:

  • "$y$ is inversely proportional to $x$"
  • "$y$ varies inversely with $x$"
  • When $x$ doubles, $y$ halves

Examples:

  • More workers → less time
  • Faster speed → less travel time
  • Wider pipe → less time to fill

Solving method: Set products equal:

$$x_1 \cdot y_1 = x_2 \cdot y_2$$
Direct Proportion: $y = 3x$
Interactive visualization — available in the full lesson
Inverse Proportion: $xy = 24$
Interactive visualization — available in the full lesson
🧠 How to Recognize the Type

Ask yourself: "If I increase one quantity, does the other go UP or DOWN?"
UP → Direct proportion (use equal ratios) | DOWN → Inverse proportion (use equal products)

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In the full lesson

  • The complete explanation
  • Worked examples
  • Common mistakes and how to avoid them
  • Test questions: 173
  • AI tutor that answers questions about the topic
  • Progress tracking for students, parents and teachers