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.
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.
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.
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.
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