Ratios and Proportions: A Complete Guide

A ratio compares two quantities (e.g., 3 cups flour to 2 cups sugar). A proportion says two ratios are equal. When one value is unknown, we can solve for it using cross-multiplication.

Understanding ratios

Ratios can be written as 3 : 2, 3 to 2, or 3/2. They all mean the same: for every 3 of one thing, there are 2 of another. Ratios should be simplified like fractions — 6 : 4 becomes 3 : 2.

What is a proportion?

A proportion is an equation of two equal ratios: a/b = c/d. If 2/3 = 4/6, the ratios are proportional. We use proportions when we know three values and need to find the fourth.

Cross-multiplication

If a/b = c/d, then a × d = b × c. Multiply diagonally across the equals sign. This is the key to solving proportion problems.

Example: 3/4 = x/12. Find x.

Answer: x = 9. Check: 3/4 = 9/12 ✓

Real-world example

A recipe uses 2 eggs for every 3 cups of flour. How many eggs for 9 cups of flour?

Set up: eggs/flour = eggs/flour → 2/3 = x/9

Answer: 6 eggs

Unit rates

A unit rate compares a quantity to 1 unit (e.g., miles per gallon, price per pound). To find it: divide the first quantity by the second. 120 miles in 3 hours → 120 ÷ 3 = 40 miles per hour.

Common mistakes

Mixing up the order: If the ratio of boys to girls is 3:5, don't write it as 5:3. The order matters and must match the context.

Cross-multiplying wrong: In a/b = c/d, cross-multiply as a × d = b × c. Don't multiply the same side.

More examples

Example: If 3 pencils cost $1.50, how much do 7 pencils cost?

Example: A recipe calls for 2 cups flour for 3 cups milk. How much flour for 9 cups milk?

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