Ratio, Proportion, Partnership & Ages

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Proportion & proportional division of terms

Four numbers $a, b, c, d$ are in proportion when $\frac{a}{b} = \frac{c}{d}$ , written $a : b :: c : d$ ; then $ad = bc$ (product of extremes = product of means).

Fourth proportional to $a, b, c$ : $d = \frac{bc}{a}$ .
Third proportional to $a, b$ (a : b :: b : ?): $c = \frac{b^2}{a}$ .
Mean proportional between $a$ and $b$ : $\sqrt{ab}$ .
Continued proportion: $a, b, c$ with $\frac{a}{b} = \frac{b}{c}$ .

Componendo and dividendo: if $\frac{a}{b} = \frac{c}{d}$ then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$ — the fastest route when x and y appear as a sum and difference.

📖 Notes 🧩 5 question types ⚡ Formulas 🎯 13 practice Q

Detailed notes

The core identity

$a : b :: c : d$ means $\frac{a}{b} = \frac{c}{d}$ , and equivalently $ad = bc$ — extremes × extremes = means × means (a and d are the extremes, b and c the means). Every 'find the missing term' question is one cross-multiplication away.

The three named proportionals

Fourth proportional to $a, b, c$ : solve $\frac{a}{b} = \frac{c}{d}$ → $d = \frac{bc}{a}$ (four distinct numbers).
Third proportional to $a, b$ : solve $\frac{a}{b} = \frac{b}{x}$ → $x = \frac{b^2}{a}$ (the middle term repeats — that is what 'third' means).
Mean proportional between $a$ and $b$ : $x = \sqrt{ab}$ , from $\frac{a}{x} = \frac{x}{b}$ . It is the geometric mean; the arithmetic mean $\frac{a+b}{2}$ is the planted trap.

Continued proportion

$a, b, c$ are in continued proportion when $\frac{a}{b} = \frac{b}{c}$ , i.e. $b^2 = ac$ . So b is automatically the mean proportional of a and c. Any one member from the other two: $a = \frac{b^2}{c}$ , $c = \frac{b^2}{a}$ , $b = \sqrt{ac}$ . Four numbers $a, b, c, d$ with $\frac{b}{a} = \frac{c}{b} = \frac{d}{c}$ form a geometric progression — the same equations apply term by term.

Componendo and dividendo

If $\frac{a}{b} = \frac{c}{d} = k$ , then adding and subtracting numerator and denominator: $\frac{a+b}{a-b} = \frac{c+d}{c-d}$ . The exam use runs it backwards: given $\frac{x+y}{x-y} = \frac{p}{q}$ , jump straight to $\frac{x}{y} = \frac{p + q}{p - q}$ . Example: $\frac{x+y}{x-y} = \frac{5}{3}$ → $\frac{x}{y} = \frac{8}{2} = 4$ . With coefficient twists like $\frac{5x - 3y}{5x + 3y} = \frac{2}{7}$ , apply C&D first to get $\frac{5x}{3y} = \frac{9}{5}$ , then rearrange.

Order matters

"Fourth proportional to 4, 9, 12" and "fourth proportional to 12, 9, 4" are different numbers — the terms stay in the order written. Write the proportion exactly as the question lists the terms before cross-multiplying.

A worked chain putting it together

"Three numbers a, b, c are in continued proportion with b = 12 and a + c = 26." Then $ac = b^2 = 144$ , so a and c are the factor pair of 144 summing to 26 — that is 8 and 18. The numbers are 8, 12, 18 (check: $8 : 12 = 12 : 18 = 2 : 3$ ✓). Exam numbers are always built so the factor pair comes out clean; if yours doesn't, re-read the question, not the method.

The three means

Arithmetic mean $\frac{a+b}{2}$ , geometric (mean proportional) $\sqrt{ab}$ , harmonic mean $\frac{2ab}{a+b}$ — the exam asks for the mean proportional, i.e. the geometric mean. "The mean proportional between 3 and 12 is 6, not 7.5."

Quick revision

$ad = bc$ for $a : b :: c : d$ .
Fourth: $\frac{bc}{a}$ . Third: $\frac{b^2}{a}$ . Mean: $\sqrt{ab}$ .
Continued proportion: $b^2 = ac$ .
C&D: $\frac{x}{y} = \frac{p+q}{p-q}$ from $\frac{x+y}{x-y} = \frac{p}{q}$ .

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Fourth proportionalvery common2 practice Q

How to spot it:

'Find the fourth proportional to a, b, c' — three numbers, the fourth that completes a : b :: c : ?

\frac{a}{b} = \frac{c}{d} \Rightarrow d = \frac{bc}{a}

Write the proportion in the exact order given: a : b :: c : d.
Cross-multiply: ad = bc.
$d = \frac{bc}{a}$ — one division.

Why: the fourth term must preserve the same ratio pair-wise as the first three.

Example: The fourth proportional to 8, 12 and 18 is:

$d = \frac{12 \times 18}{8} = 27$ . (Check: $\frac{8}{12} = \frac{18}{27} = \frac{2}{3}$ ✓.)

Type 2: Third proportionalcommon2 practice Q

How to spot it:

'Find the third proportional to a and b' — only two numbers; the middle repeats (a : b :: b : ?).

\frac{a}{b} = \frac{b}{x} \Rightarrow x = \frac{b^2}{a}

Repeat the second term: a : b :: b : x.
Cross-multiply: ax = b².
$x = \frac{b^2}{a}$ .

Why: 'third proportional to a and b' is the third member of a continued proportion starting a, b.

Example: The third proportional to 9 and 12 is:

$x = \frac{12^2}{9} = 16$ . (Check: $\frac{9}{12} = \frac{12}{16} = \frac{3}{4}$ ✓.)

Type 3: Mean proportionalvery common2 practice Q

How to spot it:

'Find the mean proportional between a and b' — the number x with a : x :: x : b.

x = \sqrt{ab}

Multiply the two numbers.
Take the square root (it must come out exact in exam numbers).
Sanity check: x² = ab. The arithmetic mean (a+b)/2 is NOT the answer.

Why: the mean proportional is the self-repeating middle term, i.e. the geometric mean.

Example: The mean proportional between 12 and 48 is:

$\sqrt{12 \times 48} = \sqrt{576} = 24$ . (Check: $\frac{12}{24} = \frac{24}{48} = \frac{1}{2}$ ✓.)

Type 4: Continued proportion (find a member)common2 practice Q

How to spot it:

'a, b, c are in continued proportion; given two of them, find the third' — sometimes disguised as a geometric progression.

b^2 = ac \ \Rightarrow\ b = \sqrt{ac},\ a = \frac{b^2}{c},\ c = \frac{b^2}{a}

Write the defining equation $\frac{a}{b} = \frac{b}{c}$ , i.e. $b^2 = ac$ .
Substitute the two known members and solve.
The middle term is the mean proportional of the outer two.

Why: 'continued proportion' fixes one exact relationship among the three members.

Example: The numbers x, 12 and 18 are in continued proportion. Find x.

$12^2 = 18x$ → $x = \frac{144}{18} = 8$ . (Check: $\frac{8}{12} = \frac{12}{18} = \frac{2}{3}$ ✓.)

Type 5: Componendo and dividendocommon2 practice Q

How to spot it:

$(x + y)/(x - y)$ given as a ratio; find x : y — sometimes with coefficients attached to x and y.

\frac{x + y}{x - y} = \frac{p}{q} \Rightarrow \frac{x}{y} = \frac{p + q}{p - q}

Apply C&D: the sum-over-difference of each side is equal.
$\frac{x}{y} = \frac{p+q}{p-q}$ — read it directly.
With coefficients ( $\frac{mx \pm ny}{mx \pm ny}$ ), C&D gives $\frac{mx}{ny}$ ; divide off m : n.

Why: one application converts the sum/difference form into the pure ratio, saving a full solve.

Example: If $\frac{a+b}{a-b} = \frac{7}{5}$ , then $a : b$ equals:

C&D: $\frac{a}{b} = \frac{7 + 5}{7 - 5} = 6$ → $a : b = 6 : 1$ .

Formulas

Basic proportion

\frac{a}{b} = \frac{c}{d} \iff ad = bc

Fourth proportional

d = \frac{bc}{a}

Third proportional

c = \frac{b^2}{a}

Mean proportional

\text{mean} = \sqrt{ab}

Componendo & dividendo

\frac{a}{b} = \frac{c}{d} \Rightarrow \frac{a+b}{a-b} = \frac{c+d}{c-d}

Invertendo / alternando

\frac{b}{a} = \frac{d}{c},\quad \frac{a}{c} = \frac{b}{d}

Shortcut tricks

⚡ Extremes × means

Set up the proportion, cross-multiply, done. For 'fourth proportional' keep the order exactly as written.

Example: Find the fourth proportional to 4, 9 and 12.

$\frac{4}{9} = \frac{12}{d}$ ⇒ $4d = 108$ ⇒ $d = 27$ .

⚡ Mean proportional = geometric mean

Multiply the two numbers and take the square root. Perfect squares make it instant.

Example: Find the mean proportional between 25 and 81.

$\sqrt{25 \times 81} = 5 \times 9 = 45$ .

⚡ Componendo-dividendo for x : y

When (x + y) and (x − y) both appear, write the ratio and jump straight to x/y.

Example: If $\frac{x+y}{x-y} = \frac{5}{3}$ , find $\frac{x}{y}$ .

$\frac{x}{y} = \frac{5+3}{5-3} = 4$ .

Where students lose marks

Mixing up third and fourth proportional (third has a repeated middle term).
Forgetting the middle term repeats when checking a continued proportion.
Applying componendo-dividendo without the base proportion being true.
Taking the arithmetic mean (a+b)/2 instead of √(ab) for the mean proportional.

Practice sets — 13 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 13 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.