Ratio and Proportion

Introduction

In mathematics, ratio and proportion are fundamental concepts that help us compare quantities and find unknown values. Ratios show the relationship between two quantities, while proportions involve comparing two ratios to determine if they are equal.

Ratio

A ratio is a comparison of two quantities by division. It is typically written in the form $a:b$ or $\frac{a}{b}$. For example, if there are 5 boys and 3 girls in a class, the ratio of boys to girls is $5:3$.

Example 1:

If a recipe calls for 2 cups of flour and 1 cup of sugar, what is the ratio of flour to sugar? Ratio of flour to sugar $= 2:1$.

Proportion

A proportion is an equation that states two ratios are equal. It can be written as $\frac{a}{b} = \frac{c}{d}$. For example, if $2:3 = 4:6$, then we have a proportion.

Example 2:

If a car travels 150 km in 3 hours, how far will it go in 5 hours at the same speed? Let the distance be $d$ km. The ratio of distance to time is constant. $\frac{150}{3} = \frac{d}{5}$ $d = 250$ km.

Common Mistakes

• Confusing between ratios and proportions.
• Incorrectly setting up proportions when solving problems.
• Forgetting to simplify ratios or proportions.

Key Points

• A ratio compares two quantities by division.
• A proportion is an equation stating two ratios are equal.
• Ratios can be simplified by dividing both parts by their greatest common factor.
• Proportions can be solved by cross multiplication.

Practice Questions

1. If the ratio of boys to girls in a class is $3:5$ and there are 24 students in total, how many boys are in the class?
2. If $\frac{x}{3} = \frac{5}{9}$, find the value of $x$.