Ratios, Proportions, and Percentages

Learning Objectives

Set up and simplify a ratio between two quantities.

Use proportions to solve for an unknown value.

Convert between fractions, decimals, and percentages, and solve percentage problems.

Comparing Quantities

In science and everyday life, we are constantly comparing things. Ratios, proportions, and percentages are the mathematical tools we use to make these comparisons meaningful.

Ratios

A ratio is a comparison of two quantities by division. If a class has 12 boys and 15 girls, the ratio of boys to girls can be written in three ways:

Using 'to': 12 to 15

Using a colon: 12:15

As a fraction: 12/15

Like fractions, ratios should always be simplified. Since both 12 and 15 are divisible by 3, the simplified ratio is 4 to 5, 4:5, or 4/5. This means for every 4 boys, there are 5 girls.

Proportions

A proportion is an equation stating that two ratios are equal. Proportions are incredibly useful for solving problems where you know one ratio and want to find a missing part of an equivalent ratio.

Example: A map has a scale of 1 cm : 50 km. If two cities are 3.5 cm apart on the map, how far apart are they in reality?

We can set up a proportion:

(1 cm) / (50 km) = (3.5 cm) / (x km)

To solve this, we cross-multiply:

1 x = 50 3.5

x = 175

The cities are 175 km apart.

Percentages

A percentage is a special type of ratio where a number is compared to 100. The word 'percent' literally means 'per hundred.' The symbol is %. Percentages are just another way of writing fractions and decimals.

Fraction to Percent: Convert the fraction to a decimal, then multiply by 100.

3/4 = 0.75 -> 0.75 * 100 = 75%

Decimal to Percent: Multiply by 100 (or move the decimal two places to the right).

0.25 -> 25%

Percent to Decimal: Divide by 100 (or move the decimal two places to the left).

65% -> 0.65

Solving Percentage Problems

The key is to translate the words into a simple equation. The word 'of' usually means multiply, and 'is' usually means equals.

Example: What is 30% of 200?

Translate: x = 0.30 * 200

Solve: x = 60

Example: 15 is what percent of 60?

Translate: 15 = x * 60

Solve: x = 15 / 60 = 0.25

Convert to percent: 0.25 = 25%

Key Terms

**Ratio: A comparison of two numbers or quantities by division.~|~Proportion: An equation that states that two ratios are equal.~|~Cross-Multiplication: A method for solving a proportion by multiplying the numerator of one ratio by the denominator of the other, and setting the products equal.~|~Percent: A ratio that compares a number to 100. The symbol is %.