Ratio and Proportion

What is a ratio?

A ratio compares two quantities of the same kind, showing how many times one contains the other. The ratio of $a$ to $b$ is written $a : b$ , and it's simplified the same way a fraction is: by dividing both sides by their HCF.

Both quantities must be in the same unit before you compare them, and the order matters: $a : b$ and $b : a$ are different ratios unless $a = b$ .

Worked example: simplifying a ratio

The length and breadth of a rectangular park are 75 m and 60 m respectively. What is the ratio of the length to the breadth?

Solution: The ratio is $75 : 60$ . The HCF of 75 and 60 is 15, so dividing both sides by 15:

75 : 60 = 5 : 4

Worked example: different units

Find the ratio of 50 paise to ₹2.

Solution: Convert both to paise first, since ₹2 = 200 paise:

50 : 200 = 1 : 4

Writing $50 : 2 = 25 : 1$ without converting would be wrong by a factor of 100.

Dividing a quantity in a given ratio

To split a total in the ratio $m : n$ , think of it as $m + n$ equal parts. The first share is $\dfrac{m}{m+n}$ of the total and the second is $\dfrac{n}{m+n}$ .

Example: Divide ₹1,200 between A and B in the ratio $5 : 3$ .

Solution: There are $5 + 3 = 8$ parts, so each part is $1200 \div 8 = 150$ :

\text{A} = 5 \times 150 = 750, \qquad \text{B} = 3 \times 150 = 450

Check: $750 + 450 = 1200$ and $750 : 450 = 5 : 3$ .

Combining two ratios into one

If $a : b = 2 : 3$ and $b : c = 4 : 5$ , find $a : b : c$ .

Solution: The two ratios share $b$ , but it's 3 in one and 4 in the other. Scale both so $b$ becomes the LCM of 3 and 4, which is 12:

a : b = 2 : 3 = 8 : 12, \qquad b : c = 4 : 5 = 12 : 15

a : b : c = 8 : 12 : 15

What is a proportion?

Four numbers $a, b, c, d$ are said to be in proportion, written $a : b :: c : d$ , when the ratio of the first pair equals the ratio of the second pair:

\dfrac{a}{b} = \dfrac{c}{d} \iff a \times d = b \times c

That cross-multiplied form (the product of the outer (extreme) terms equals the product of the inner (middle) terms) is the fastest way to test or solve a proportion.

The name of each term in a : b :: c : d

Term	Name	Meaning
$a$ and $c$	Antecedents	The first term of each ratio
$b$ and $d$	Consequents	The second term of each ratio
$a$ and $d$	Extremes	The outer two terms
$b$ and $c$	Means	The inner two terms
$d$	Fourth proportional	The term found from $a$ , $b$ and $c$

Worked example: testing a proportion

Are 20, 18, 5, 6 in proportion?

Solution: Check whether $20 \times 6 = 18 \times 5$ :

20 \times 6 = 120, \qquad 18 \times 5 = 90

120 ≠ 90, so 20, 18, 5, 6 are not in proportion.

Compare that with 30, 40, 45, 60:

30 \times 60 = 1800, \qquad 40 \times 45 = 1800

These match, so 30, 40, 45, 60 are in proportion.

Worked example: solving for a missing term

If $36 : 81 :: x : 63$ , find $x$ .

Solution: Cross-multiplying, $36 \times 63 = 81 \times x$ :

x = \dfrac{36 \times 63}{81} = \dfrac{2268}{81} = 28

Mean, third and fourth proportionals

Each of these is the same cross-multiplication with a different term missing.

Find	Set up	Example	Answer
Mean proportional of $a$ and $b$	$a : x :: x : b$ , so $x^2 = ab$	4 and 9	$x = \sqrt{36} = 6$
Third proportional to $a$ and $b$	$a : b :: b : x$ , so $x = \dfrac{b^2}{a}$	4 and 6	$x = \dfrac{36}{4} = 9$
Fourth proportional to $a, b, c$	$a : b :: c : x$ , so $x = \dfrac{bc}{a}$	2, 3 and 4	$x = \dfrac{12}{2} = 6$

Direct and inverse proportion

Direct proportion: both quantities grow together, so their ratio stays fixed. If 5 pens cost ₹60, then 8 pens cost $x$ where $5 : 8 :: 60 : x$ :

x = \dfrac{8 \times 60}{5} = 96

Inverse proportion: one quantity grows as the other shrinks, so their product stays fixed instead. If 6 workers finish a job in 10 days, 4 workers take $x$ days where:

6 \times 10 = 4 \times x \;\Rightarrow\; x = 15 \text{ days}

Common mistakes

Comparing different units. Convert first: 50 paise to ₹2 is $1 : 4$ , not $25 : 1$ .
Swapping the order. "Boys to girls" and "girls to boys" are reciprocal ratios. Keep the order the question uses.
Treating an inverse relationship as direct. More workers means fewer days. Ask whether the quantities move together before setting up $a : b :: c : d$ .
Adding ratios like fractions. To combine $a : b$ and $b : c$ , make the shared term equal, as in the example above. Adding $2 : 3$ and $4 : 5$ term by term gives nothing meaningful.

Where this comes up in exams

Ratios and percentages are listed under numerical computation and estimation in the General Aptitude syllabus, a section every GATE 2026 paper includes.

Checked against GATE 2026 information brochure (General Aptitude syllabus). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

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