Ratios

• Ratio $a:b$ means $\dfrac{a}{b}$. Sharing $T$ in the ratio $a:b$: first share $= \dfrac{a}{a+b}\times T$, second share $= \dfrac{b}{a+b}\times T$.
• Increasing/decreasing in a ratio $a:b$: new value $= \dfrac{b}{a}\times \text{old value}$ (for $a:b$ read as old : new).

Rates

$$\text{Rate} = \frac{\text{amount}}{\text{time}} \qquad \text{Speed} = \frac{\text{distance}}{\text{time}}$$

Proportion

• Direct proportion: $\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}$.
• Inverse proportion: $x_1 y_1 = x_2 y_2$.

Rates of work

If one worker finishes a job in $a$ days and another in $b$ days, together they take $$\frac{ab}{a+b}\ \text{days}$$ (work done per day: $\dfrac{1}{a} + \dfrac{1}{b}$).

Mixtures

$$\text{Price of mixture per unit} = \frac{\text{total cost of ingredients}}{\text{total quantity}}$$ Selling price per unit $= \text{cost per unit} \times \left(1 + \dfrac{\text{profit \%}}{100}\right)$.