AFCAT Numerical Ability · Ratio and Proportion

Proportion: Fourth, Third and Mean Proportionals

Four numbers are in proportion when the first two make the same ratio as the last two; from that one equation come the fourth, third and mean proportionals.

All Ratio and Proportion notes

Why this matters

The largest page in the chapter's practice bank, and nearly all of it is one of three formulas: the fourth proportional bc/a, the third proportional b²/a, or the mean proportional √(ab). Learn which is which and most of these questions take one line.

Concept 1 of 4: Four numbers in proportion, and the fourth proportional

a, b, c, d

are in proportion when

a : b = c : d

. Cross-multiplying turns that into one equation: the product of the outer two equals the product of the inner two. Every question on this concept is that equation.

Definition

$a : b :: c : d$ means $\dfrac ab = \dfrac cd$ , so $ad = bc$ . $a$ and $d$ are the extremes, $b$ and $c$ the means.
The fourth proportional to $a, b, c$ is $d = \dfrac{bc}{a}$ .
'What must be added to each number to make them proportional?': add $x$ to all four and solve $(a + x)(d + x) = (b + x)(c + x)$ . The $x^2$ terms cancel.
Letters work the same way: the fourth proportional to $p, q, r$ is $\dfrac{qr}{p}$ .

Product of extremes equals product of means

a : b :: c : d \;\text{means}\; ad = bc, \qquad d = \dfrac{bc}{a}

Worked example

Find the fourth proportional to 5, 12 and 20.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 1 · Ratio and Proportion · Proportionals

Find the fourth proportional to

a^2

ab

, and

bc

Extremes times extremes

a : b :: c : d

the test is

a \times d = b \times c

. Multiplying

a \times c

, the first terms of each ratio, tests nothing.

Drill 19 more on four numbers in proportion, and the fourth proportional

Concept 2 of 4: Third proportional

The third proportional to

a

and

b

is the number

c

that continues the pattern

a : b = b : c

. The middle number is used twice.

Definition

$c$ is the third proportional to $a$ and $b$ when $a : b = b : c$ , so $c = \dfrac{b^2}{a}$ .
Order matters: the third proportional to 2 and 6 is 18, but to 6 and 2 it is $\tfrac 23$ .
Given $c$ and $a$ , find $b$ from $b^2 = ac$ .

Third proportional

a : b = b : c \;\Rightarrow\; c = \dfrac{b^2}{a}

Worked example

Find the third proportional to 16 and 20.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Ratio and Proportion · Proportionals

The third proportional to (x² - y²) and (x - y) is:

Square the second number, not the first

The third proportional to 25 and 30 is

\dfrac{30^2}{25} = 36

\dfrac{25^2}{30}

is the third proportional to 30 and 25, the other way round.

Drill 10 more on third proportional

Concept 3 of 4: Mean proportional

The mean proportional between

a

and

b

is the number in the middle of

a : x = x : b

. It is the square root of their product.

Definition

$x$ is the mean proportional between $a$ and $b$ when $a : x = x : b$ , so $x = \sqrt{ab}$ .
For decimals and fractions, multiply first, then take the root: $\sqrt{0.02 \times 0.32} = \sqrt{0.0064} = 0.08$ .
Surds often multiply out to a whole number: $(3 + \sqrt 5)(3 - \sqrt 5) = 4$ .
Given the mean proportional $m$ and one number $a$ , the other is $\dfrac{m^2}{a}$ .

Mean proportional

a : x = x : b \;\Rightarrow\; x = \sqrt{ab}

Worked example

Find the mean proportional between 12 and 75.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Ratio and Proportion · Proportionals

The mean proportional between

(2 + \sqrt{3})

and

(8 - \sqrt{48})

A square root, not an average

The mean proportional between 4 and 16 is

\sqrt{4 \times 16} = 8

. Their ordinary average, 10, is a different number.

Drill 11 more on mean proportional

Concept 4 of 4: Continued proportion

Three numbers

a, b, c

are in continued proportion when

a : b = b : c

. Each is the one before times the same factor, so they form a geometric progression, and the middle one is the mean proportional of the outer two.

Definition

$a, b, c$ in continued proportion means $\dfrac ab = \dfrac bc$ , so $b^2 = ac$ .
They form a G.P.: $a, ar, ar^2$ .
Given $a : c$ , the middle term is the square root of the product: if $a : c = 1 : 9$ , then $a : b : c = 1 : 3 : 9$ .
In an expression in $a, b, c$ , replace $b^2$ by $ac$ . To check an answer quickly, try a G.P. such as 1, 2, 4.

Continued proportion

a : b = b : c \;\text{means}\; b^2 = ac

Worked example

x, y, z

are in continued proportion and

x : z = 9 : 49

. Find

x : y : z

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Ratio and Proportion · Proportionals

q

is the mean proportional between

p

and

r

, then

\frac{p^2 - q^2 + r^2}{p^{-2} - q^{-2} + r^{-2}}

is equal to

Continued proportion is a G.P., not an A.P.

In 4, 6, 9 the middle term passes

6^2 = 4 \times 9

. The test

p + r = 2q

is for an arithmetic progression and is a different condition.

Drill 5 more on continued proportion

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Four numbers in proportion, and the fourth proportional
Product of extremes equals product of means
$a : b :: c : d \;\text{means}\; ad = bc, \qquad d = \dfrac{bc}{a}$
Third proportional
Third proportional
$a : b = b : c \;\Rightarrow\; c = \dfrac{b^2}{a}$
Mean proportional
Mean proportional
$a : x = x : b \;\Rightarrow\; x = \sqrt{ab}$
Continued proportion
Continued proportion
$a : b = b : c \;\text{means}\; b^2 = ac$

Watch out for (4)

Extremes times extremes
→ Four numbers in proportion, and the fourth proportional
Square the second number, not the first
→ Third proportional
A square root, not an average
→ Mean proportional
Continued proportion is a G.P., not an A.P.
→ Continued proportion

Drill the questions on this topic

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