Introduction

In mathematics, ratios, proportions, and rates are fundamental concepts that help us compare quantities and understand relationships between different values. Understanding these concepts is crucial for solving various real-life problems involving comparisons and scaling.

Ratio

A ratio is a comparison of two quantities. It is written in the form $a:b$ or $\frac{a}{b}$, where $a$ and $b$ are non-negative numbers.

Example: If the ratio of boys to girls in a class is $3:2$, and there are 25 girls, how many boys are there? $$ \text{Number of boys} = 3 \times \frac{25}{2} = 37.5 $$ So, there are 37.5 boys in the class.

Proportion

A proportion is an equation stating that two ratios are equal. It can be written as $\frac{a}{b} = \frac{c}{d}$.

Example: If 4 pens cost Ksh 80, how much will 7 pens cost? Let the cost of 7 pens be x. We can set up a proportion: $$ \frac{4}{80} = \frac{7}{x} $$ Cross multiply to solve for x: $$ 4x = 80 \times 7 \ x = \frac{80 \times 7}{4} = 140 $$ Therefore, 7 pens will cost Ksh 140.

Rates

A rate is a special ratio that compares two quantities with different units. It is written as $\frac{a}{b}$ where $a$ is a quantity measured in one unit and $b$ is a quantity measured in a different unit.

Example: If a car travels 240 km in 4 hours, what is its speed in km/h? The rate of speed is given by: $$ \text{Speed} = \frac{240 \text{ km}}{4 \text{ hours}} = 60 \text{ km/h} $$ Therefore, the car's speed is 60 km/h.

Common Mistakes

• Confusing ratios with fractions.
• Misinterpreting the order of numbers in a ratio.
• Forgetting to simplify ratios and proportions.

Key Points

• Ratios compare two quantities, proportions state that two ratios are equal, and rates compare quantities with different units.
• Ratios can be written as $a:b$ or $\frac{a}{b}$.
• Proportions are equations that equate two ratios.
• Rates are special ratios that compare quantities with different units.

Practice Questions

1. The ratio of mangoes to apples in a basket is 3:4. If there are 24 apples, how many mangoes are there?
2. If 5 books cost Ksh 150, how much will 12 books cost?