U8 Ratio and Proportion: Essential Formulas

This section covers the fundamental concepts of ratio, proportion, and variation, which are crucial for solving problems involving direct and inverse relationships between quantities.

1 Ratio and Simplification

Definition of a Ratio

A ratio $a : b$ compares two quantities. It can be simplified by dividing both terms by their greatest common divisor (H.C.F.). For example, the ratio $15 : 20$ simplifies to $3 : 4$.

$$ a : b = \frac{a}{b} \quad \text{for} \quad b \neq 0 $$
ab:

2 Proportion and Properties

Direct Proportion

Two quantities $x$ and $y$ are directly proportional if their ratio is constant. This is denoted as $y \propto x$.

$$ y = kx \quad \text{where } k \text{ is the constant of proportionality.} $$

Inverse Proportion

Two quantities $x$ and $y$ are inversely proportional if their product is constant. This is denoted as $y \propto \frac{1}{x}$.

$$ y = \frac{k}{x} \quad \text{or} \quad xy = k $$

3 Joint and Partial Variation

Joint Variation

A quantity varies jointly as two or more other quantities. For example, $z$ varies jointly as $x$ and $y$.

$$ z \propto xy \quad \Rightarrow \quad z = kxy $$

Partial Variation

A quantity is the sum of two parts, where one part is constant and the other part varies directly with another variable.

$$ y = k_1 + k_2 x $$

4 Applications and Problem Solving

Sharing a Quantity in a Given Ratio

To divide a quantity $Q$ in the ratio $a : b : c$, the shares are proportional to the terms of the ratio.

$$ \text{First share} = \frac{a}{a+b+c} \times Q $$

Similarly, $\text{Second share} = \frac{b}{a+b+c} \times Q$ and $\text{Third share} = \frac{c}{a+b+c} \times Q$.

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