Ratio and Proportion — chapter notes

A ratio $a : b$ ($b \ne 0$) compares two quantities of the same kind; it is unchanged when both terms are multiplied or divided by the same non-zero number, so always give it in lowest terms.

From $a : b$ (with $a, b > 0$):

• duplicate ratio $a^2 : b^2$; triplicate ratio $a^3 : b^3$;
• sub-duplicate ratio $\sqrt{a} : \sqrt{b}$; sub-triplicate ratio $\sqrt[3]{a} : \sqrt[3]{b}$;
• reciprocal (inverse) ratio $b : a$;
• compound ratio of $a : b$ and $c : d$: $ac : bd$.

Example: the compound ratio of the duplicate ratio of $3 : 4$ and the sub-duplicate ratio of $16 : 25$ is $(9 \times 4) : (16 \times 5) = 9 : 20$.

Four numbers $a, b, c, d$ are in proportion, written $a : b = c : d$ or $a : b :: c : d$, when $\dfrac{a}{b} = \dfrac{c}{d}$.

• $a$ and $d$ are the extremes; $b$ and $c$ are the means.
• Test: product of extremes $=$ product of means, $ad = bc$.
• $d$ is the fourth proportional to $a, b, c$: $d = \dfrac{bc}{a}$.

Example: $1.2, 2, 4.8, 8$ are in proportion since $1.2 \times 8 = 9.6 = 2 \times 4.8$; the fourth proportional to $5, 7, 15$ is $\dfrac{7 \times 15}{5} = 21$.

$a, b, c$ are in continued proportion when $a : b = b : c$, so $b^2 = ac$.

• $b = \sqrt{ac}$ is the mean proportional between $a$ and $c$.
• $c = \dfrac{b^2}{a}$ is the third proportional to $a$ and $b$.
• $a, b, c, d$ are in continued proportion when $a : b = b : c = c : d$; then $a : d = a^3 : b^3$ (the triplicate ratio of $a : b$).

Example: the mean proportional between $0.32$ and $0.5$ is $\sqrt{0.16} = 0.4$; the third proportional to $5$ and $7.5$ is $\dfrac{56.25}{5} = 11.25$.

If $a : b = c : d$, then

• invertendo: $b : a = d : c$ (invert both ratios);
• alternendo: $a : c = b : d$ (interchange the means).

Each new proportion still has equal cross-products, so it is a true proportion.

If $a : b = c : d$, then

• componendo: $\dfrac{a + b}{b} = \dfrac{c + d}{d}$ (add $1$ to both sides);
• dividendo: $\dfrac{a - b}{b} = \dfrac{c - d}{d}$ (subtract $1$ from both sides);
• componendo and dividendo: $\dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$ (divide the two results).

The steps are reversible: if $\dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$, then $a : b = c : d$.

• Equations: when one side is (sum) $\div$ (difference), apply componendo and dividendo to both sides. Example: $\dfrac{2x + 9}{2x - 9} = \dfrac{5}{3}$ gives $\dfrac{4x}{18} = 4$, so $x = 18$.
• Square roots: $\dfrac{\sqrt{x + 4} + \sqrt{x - 4}}{\sqrt{x + 4} - \sqrt{x - 4}} = 2$ gives $\dfrac{\sqrt{x + 4}}{\sqrt{x - 4}} = 3$; square and solve: $x = 5$.
• Finding a ratio: $\dfrac{5x + 3y}{5x - 3y} = \dfrac{7}{3}$ gives $\dfrac{10x}{6y} = \dfrac{10}{4}$, so $x : y = 3 : 2$.
• Proofs (k-method): put $\dfrac{a}{b} = \dfrac{c}{d} = k$, so $a = bk$, $c = dk$ (or $b = ak$, $c = ak^2$ for continued proportion), and simplify each side separately.

Write the proportion, use product of extremes $=$ product of means, solve, and check in the original ratios.

• Scales and recipes: $3$ cm for $20$ km means $7.5$ cm stands for $\dfrac{20 \times 7.5}{3} = 50$ km.
• Changing ratios: numbers $2k$ and $5k$ with $3$ subtracted from each in the ratio $1 : 4$ give $k = 3$.
• Making numbers proportional: adding $3$ to each of $3, 6, 7, 12$ gives $6, 9, 10, 15$.
• Areas and volumes: areas of similar figures are in the duplicate ratio, and volumes of cubes in the triplicate ratio, of their lengths.

Not in the ICSE 2027 syllabus: applications of proportion beyond direct simple applications; this chapter keeps to direct ones.