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Solving
Proportions

Master three powerful strategies for solving proportions: equivalent ratios, unit rates, and cross-multiplication equations.

Pre-Algebra

What Is a Proportion?

A proportion is an equation that says two ratios are equal. If you can write a relationship as \(\frac{a}{b} = \frac{c}{d}\), you have a proportion.

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Where Do Proportions Show Up?

Proportions appear whenever two quantities change at the same rate. You use them every day, often without realizing it.

Cooking
Doubling a recipe: if 2 cups of flour makes 12 cookies, how much for 36?
Maps
Scale: 1 inch = 50 miles. If two cities are 3.5 inches apart, how far in real life?
Shopping
Unit pricing: if 5 notebooks cost $8, how much for 12 notebooks?

Ratio vs. Proportion

A ratio compares two quantities. A proportion is a statement that two ratios are equal. Think of a ratio as a single fraction and a proportion as an equation with two fractions.

Ratio
A single comparison: \(\frac{3}{5}\) or \(3:5\). It describes one relationship.
Proportion
An equation of two ratios: \(\frac{3}{5} = \frac{6}{10}\). It says they are equal.

The Cross-Product Property

If \(\frac{a}{b} = \frac{c}{d}\), then the cross products are equal:

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Why Does Cross-Multiplying Work?

Cross-multiplication is just a shortcut for multiplying both sides of the equation by both denominators to clear the fractions.

1
Start with \(\frac{a}{b} = \frac{c}{d}\).
2
Multiply both sides by \(b\): \(a = \frac{b \cdot c}{d}\).
3
Multiply both sides by \(d\): \(a \cdot d = b \cdot c\).

Key Distinction: Three Strategies

You can solve a proportion using any of these three methods. Each is valid: pick the one that fits the numbers.

Equivalent Ratios
Multiply or divide both parts of one ratio by the same number to match the other.
Unit Rate
Find the value for "one unit" first, then scale up or down.
Cross-Multiply
Set cross products equal and solve the resulting equation for the unknown.

How to Solve a Proportion

1
Set up the proportion. Write \(\frac{\text{known ratio}}{\phantom{x}} = \frac{\text{unknown ratio}}{\phantom{x}}\), keeping matching units in the same position.
2
Choose your strategy. If you spot a simple multiplier, use equivalent ratios. If the numbers are friendly, find the unit rate. Otherwise, cross-multiply.
3
Solve for the variable. Perform the arithmetic or solve the equation to isolate the unknown.
4
Check your answer. Substitute back in and verify both ratios simplify to the same value or that cross products are equal.

Example 1: Equivalent Ratios

Solve: \(\frac{3}{5} = \frac{x}{20}\)

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× 4
× 4

Example 2: Unit Rate

A car travels 150 miles on 6 gallons of gas. How far can it go on 10 gallons?

1
Find the unit rate: \(\frac{150 \text{ miles}}{6 \text{ gal}} = 25 \text{ miles per gallon}\)
2
Scale up: \(25 \times 10 = 250 \text{ miles}\)
3
Check: \(\frac{150}{6} = 25\) and \(\frac{250}{10} = 25\) ✓

Example 3: Cross-Multiplication

Solve: \(\frac{7}{x} = \frac{4}{10}\)

1
Cross-multiply: \(7 \times 10 = 4 \times x\)
2
Simplify: \(70 = 4x\)
3
Divide both sides by 4: \(x = \frac{70}{4} = 17.5\)
4
Check: \(\frac{7}{17.5} = 0.4\) and \(\frac{4}{10} = 0.4\) ✓

Example 4: Map Scale Problem

On a map, 2 cm represents 35 km. Two cities are 9 cm apart on the map. What is the actual distance?

1
Set up: \(\frac{2 \text{ cm}}{35 \text{ km}} = \frac{9 \text{ cm}}{x \text{ km}}\)
2
Cross-multiply: \(2x = 35 \times 9 = 315\)
3
Solve: \(x = \frac{315}{2} = 157.5 \text{ km}\)

Common Mistakes

✕ Mismatched Units
Writing \(\frac{\text{miles}}{\text{hours}} = \frac{\text{hours}}{\text{miles}}\). Units in the same position must match on both sides.
✕ Cross-Adding
Adding diagonally instead of multiplying. Cross products are found by multiplying, never adding.
✕ Forgetting to Solve
Writing \(4x = 70\) and stopping. You must divide to isolate the variable: \(x = 17.5\).

Strategy Comparison

Equivalent Ratios Unit Rate Cross-Multiply
Best When Scale factor is obvious One quantity divides evenly Numbers are messy
Speed Very fast Fast Medium
Works With Decimals? Awkward Sometimes Always
Difficulty Low Low Medium

Key Takeaways

1
A proportion is an equation of two equal ratios: \(\frac{a}{b} = \frac{c}{d}\).
2
Cross products of a true proportion are always equal: \(a \cdot d = b \cdot c\).
3
You have three strategies: equivalent ratios, unit rate, and cross-multiplication.
4
Always match units in the same position and check your answer.

Question 1 of 5

Solve: \(\frac{4}{9} = \frac{x}{27}\)

Question 2 of 5

A recipe uses 3 cups of sugar for every 8 cups of flour. How many cups of sugar are needed for 20 cups of flour?

Question 3 of 5

Which of the following is NOT a proportion?

Question 4 of 5

Solve: \(\frac{5}{x} = \frac{8}{12}\)

Question 5 of 5

On a map, 3 inches represent 75 miles. Two cities are 7 inches apart on the map. What is the actual distance?