¶Proportions and Ratios

Ratios and proportions are fundamental mathematical concepts used to describe relationships between quantities. They form the basis for understanding percentages, probability, trigonometry, and advanced algebra.

¶Understanding Ratios

A ratio compares the size of two or more quantities. It tells us how much of one thing there is compared to another. Ratios can be written in three main ways:

1. Using a colon: $a : b$
2. Using the word "to": $a \text{ to } b$
3. As a fraction: $\frac{a}{b}$

For example, if a bowl contains 3 red marbles and 5 blue marbles, the ratio of red to blue marbles is $3:5$. The ratio of red marbles to total marbles is $3:8$.

¶Equivalent Ratios and Proportions

Two ratios that represent the same relationship are called equivalent ratios. A proportion is an equation stating that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$

You can generate equivalent ratios by multiplying or dividing both terms of the ratio by the same non-zero number, similar to finding equivalent fractions.

¶Cross-Multiplication

The most reliable method for solving proportional equations involving an unknown value is cross-multiplication. In any true proportion $\frac{a}{b} = \frac{c}{d}$, the product of the extremes equals the product of the means: $a \times d = b \times c$

Example: If $\frac{4}{7} = \frac{x}{21}$, then: $4 \times 21 = 7 \times x$ $84 = 7x$ $x = 12$

¶References

• Khan Academy: Ratios, rates, & percentages
• Common Core State Standards for Mathematics: CCSS.6.RP.A.1, CCSS.6.RP.A.3