Understanding Ratios and Proportions
Ratios and proportions appear everywhere — from cooking recipes and map scales to unit prices and mixing paint colors. This guide covers how to write, simplify, and solve ratios and proportions step by step.
What Is a Ratio?
A ratio compares two quantities by showing how much of one thing there is relative to another. You can write a ratio in three ways: using a colon (3:5), as a fraction (3/5), or with the word “to” (3 to 5).
Example: A bag has 3 red marbles and 5 blue marbles.
The ratio of red to blue is 3:5. The ratio of blue to total marbles is 5:8.
Simplifying Ratios
To simplify a ratio, divide both parts by their greatest common divisor (GCD). A simplified ratio uses the smallest whole numbers possible.
Example: Simplify 12:8
The GCD of 12 and 8 is 4.12 ÷ 4 = 3 and 8 ÷ 4 = 2
So 12:8 = 3:2.
Example: Simplify 45:30
The GCD of 45 and 30 is 15.45 ÷ 15 = 3 and 30 ÷ 15 = 2
So 45:30 = 3:2.
What Is a Proportion?
A proportion is an equation that says two ratios are equal. For example, 2/3 = 4/6 is a proportion. You can check whether two ratios form a proportion by cross-multiplying: if the cross products are equal, the ratios are proportional.
Example: Are 3/4 and 9/12 proportional?
Cross-multiply: 3 × 12 = 36 and 4 × 9 = 36.
The cross products are equal, so yes, they are proportional.
Solving Proportions with Cross-Multiplication
When one value in a proportion is unknown, you can find it by cross-multiplying and then solving for the variable.
Steps: Write the proportion → cross-multiply → divide to isolate the variable.
Example: Solve 3/4 = x/20
Cross-multiply: 3 × 20 = 4 × x60 = 4xx = 60 ÷ 4 = 15
Example: Solve 5/8 = 15/x
Cross-multiply: 5 × x = 8 × 155x = 120x = 120 ÷ 5 = 24
Example: Solve x/6 = 10/15
Cross-multiply: x × 15 = 6 × 1015x = 60x = 60 ÷ 15 = 4
Real-World Applications
Ratios and proportions are used constantly in everyday life. Here are a few common scenarios.
Scaling a Recipe
A recipe serves 4 and calls for 2 cups of flour. How much flour for 6 servings?2/4 = x/6 → x = 12 ÷ 4 = 3 cups of flour.
Reading a Map
A map scale says 1 cm = 50 km. Two cities are 3.5 cm apart on the map.1/50 = 3.5/x → x = 50 × 3.5 = 175 km apart.
Comparing Unit Prices
Brand A: 12 oz for $3.60. Brand B: 16 oz for $4.00.
Brand A: $3.60 ÷ 12 = $0.30/oz. Brand B: $4.00 ÷ 16 = $0.25/oz.
Brand B is the better deal.
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