SSC CGL Maths · Ratio, Proportion and Variation

Fourth, Third and Mean Proportionals

Four numbers are in proportion when the first is to the second as the third is to the fourth, and every proportional question is solved by one rule: the product of the outer terms equals the product of the inner terms.

All Ratio, Proportion and Variation notes

Why this matters

Twenty-seven questions in the bank and twenty-five are EASY: seven on the fourth proportional, six on the third and fourteen on the mean proportional. Each is one line once you know which term is missing, so these are the fastest marks in the chapter.

Concept 1 of 3: Proportion and the fourth proportional

Two ratios are equal when one fraction is a scaled copy of the other. Cross-multiplying turns that into one equation, so if any one of the four numbers is missing, it can be found from the other three.

Definition

a, b, c, d are in proportion, written a : b :: c : d, when $\frac{a}{b} = \frac{c}{d}$ . The outer terms a and d are the extremes; b and c are the means.
The test and the solver: product of extremes = product of means, $ad = bc$ .
Fourth proportional to a, b and c, taken in that order: $d = \frac{bc}{a}$ .
Fast route: find the multiplier from the first pair and use it on the second. For 15, 24 and 35: $\frac{24}{15} = 1.6$ , so $d = 35 \times 1.6 = 56$ .
Adding or subtracting x to make four numbers proportional: solve $(a \pm x)(d \pm x) = (b \pm x)(c \pm x)$ . The $x^2$ terms cancel, so it is a one-step equation. Testing the options is just as quick.

Proportion and fourth proportional

a : b :: c : d \Leftrightarrow ad = bc \qquad d = \frac{bc}{a}

a, dthe extremes (outer terms)
b, cthe means (inner terms)

Worked example

Find the fourth proportional to 15, 24 and 35.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 2 · Q53Easy

Example 1 · Ratio, Proportion and Variation · Fourth, Third and Mean Proportionals

What should be subtracted from 10, 12, 13, 16 so that the remaining numbers may be proportional?

Taking the numbers in the wrong order

The fourth proportional to 4, 10 and 6 comes from 4 : 10 = 6 : d, which gives 15. Starting from 10 : 4 = 6 : d gives 2.4, a different number. Keep the order printed in the stem.

Drill 6 more on proportion and the fourth proportional

Concept 2 of 3: Third proportional

Three numbers are in continued proportion when each is the same multiple of the one before, like 4, 6, 9 (each is 1.5 times the last). The third proportional is simply the next number in that chain.

Definition

a, b, c are in continued proportion when a : b = b : c, so $b^2 = ac$ .
The third proportional to a and b is $c = \frac{b^2}{a}$ .
Fast route: find the step $r = \frac{b}{a}$ , then $c = b \times r$ . For 16 and 24 the step is 1.5, so the third proportional is 36.
With algebra, square the second term and divide by the first, cancelling powers as you go.

Third proportional

a : b = b : c \Leftrightarrow b^2 = ac \qquad c = \frac{b^2}{a}

Worked example

Find the third proportional to 16 and 24.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 21 Jul 2023, Shift 4 · Q52Easy

Example 2 · Ratio, Proportion and Variation · Fourth, Third and Mean Proportionals

If 28 : 98 :: 98 : y, find the value of y.

Squaring the wrong term

The third proportional to a and b is

\frac{b^2}{a}

, the SECOND term squared. For 9 and 12 it is 16; squaring the first term gives

\frac{81}{12} = 6.75

, which is wrong.

Drill 5 more on third proportional

Concept 3 of 3: Mean proportional

The mean proportional is the number that sits in the middle of a continued proportion, a : m = m : b. It is the middle of a multiplying chain, not of an adding one, so it is a square root of a product, not half of a sum.

Definition

The mean proportional of a and b is $m = \sqrt{ab}$ , because a : m = m : b gives $m^2 = ab$ .
Decimals: split each number into a perfect square: $\sqrt{0.49 \times 2.25} = 0.7 \times 1.5 = 1.05$ .
Algebra: multiply, then halve every power: $\sqrt{4x^2 \cdot 9y^4} = 6xy^2$ .
Missing number: if m is the mean proportional of a and an unknown b, then $b = \frac{m^2}{a}$ .
Which pair has mean proportional m? The pair whose PRODUCT is $m^2$ . Multiply each option; do not add them.

Mean proportional

a : m = m : b \Leftrightarrow m = \sqrt{ab} \qquad b = \frac{m^2}{a}

Worked example

Find the mean proportional of 0.49 and 2.25.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 11 Sep 2024, Shift 3 · Q54Easy

Example 3 · Ratio, Proportion and Variation · Fourth, Third and Mean Proportionals

What is the ratio of the mean proportional of 1.21 and 0.09 with the mean proportional of 0.16 and 0.36?

Halving the sum

The mean proportional of 4 and 49 is

\sqrt{196} = 14

, not

\frac{4 + 49}{2} = 26.5

. That is the ordinary average, which belongs to adding chains, not to proportion.

Misplacing the decimal in a square root

\sqrt{0.09} = 0.3

, because

0.3 \times 0.3 = 0.09

. Writing 0.03 is a common slip; count decimal places: the root has half as many as the square.

Drill 13 more on mean proportional

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 (3)

Proportion and the fourth proportional
Proportion and fourth proportional
$a : b :: c : d \Leftrightarrow ad = bc \qquad d = \frac{bc}{a}$
Third proportional
Third proportional
$a : b = b : c \Leftrightarrow b^2 = ac \qquad c = \frac{b^2}{a}$
Mean proportional
Mean proportional
$a : m = m : b \Leftrightarrow m = \sqrt{ab} \qquad b = \frac{m^2}{a}$

Watch out for (4)

Taking the numbers in the wrong order
→ Proportion and the fourth proportional
Squaring the wrong term
→ Third proportional
Halving the sum
→ Mean proportional
Misplacing the decimal in a square root
→ Mean proportional

Test yourself on Ratio, Proportion and Variation

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Ratio, Proportion and Variation chapter test Drill all 27 questions