Ratio, Proportion & Variation

At a glance 11 formulas

1. 1Split in a ratio$\text{share of } a = \dfrac{a}{a + b + c} \times \text{total}$
2. 2Chain ratios$\dfrac{A}{C} = \dfrac{A}{B} \times \dfrac{B}{C}$$A:B = 2:3,\ B:C = 4:5 \Rightarrow A:B:C = 8:12:15$
3. 3One against the rest$A : \text{rest} = a : b \Rightarrow A = \dfrac{a}{a + b} \times \text{total}$
4. 4Fourth, third, mean$a : b = c : x \Rightarrow x = \dfrac{bc}{a}$$\text{third proportional} = \dfrac{b^2}{a},\ \text{mean} = \sqrt{ab}$
5. 5Transfers keep the totalWhen money only moves from one person to another, the total is fixed: put the old and new ratios over the same total and read the change in parts.
6. 6Income, spend, savingsWrite incomes as $ax, bx$ and spends as $cy, dy$. Each saving is income minus spend, which gives two equations in $x$ and $y$.
7. 7Componendo-dividendo$\dfrac{a}{b} = \dfrac{c}{d} \Rightarrow \dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$
8. 8Equal ratios$\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{a + c}{b + d}$$= \dfrac{pa + qc}{pb + qd}$
9. 9Direct and inverse$y \propto x \Rightarrow \dfrac{y_2}{y_1} = \dfrac{x_2}{x_1}$$y \propto \dfrac{1}{x} \Rightarrow \dfrac{y_2}{y_1} = \dfrac{x_1}{x_2}$
10. 10Joint variation$y \propto \dfrac{x^{2}}{z} \Rightarrow \dfrac{y_2}{y_1} = \left(\dfrac{x_2}{x_1}\right)^{2} \times \dfrac{z_1}{z_2}$$\text{multiply each factor's ratio}$
11. 11Fixed plus variable$C = a + kn$$k = \dfrac{C_2 - C_1}{n_2 - n_1},\ a = C_1 - kn_1$