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

A rectangular garden is 84 m long and 56 m wide. What is the ratio of its length to its width?

Solution: The ratio is $84 : 56$ . The HCF of 84 and 56 is 28, so dividing both sides by 28:

84 : 56 = 3 : 2

Worked example: different units

Find the ratio of 120 cm to 2 m.

Solution: Convert both to centimetres first, since 2 m = 200 cm:

120 : 200 = 3 : 5

Writing $120 : 2 = 60 : 1$ 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 ₹2,400 between P and Q in the ratio $7 : 5$ .

Solution: There are $7 + 5 = 12$ parts, so each part is $2400 \div 12 = 200$ :

\text{P} = 7 \times 200 = 1400, \qquad \text{Q} = 5 \times 200 = 1000

Check: $1400 + 1000 = 2400$ and $1400 : 1000 = 7 : 5$ .

Combining two ratios into one

If $a : b = 3 : 4$ and $b : c = 6 : 7$ , find $a : b : c$ .

Solution: The two ratios share $b$ , but it's 4 in one and 6 in the other. Scale both so $b$ becomes the LCM of 4 and 6, which is 12:

a : b = 3 : 4 = 9 : 12, \qquad b : c = 6 : 7 = 12 : 14

a : b : c = 9 : 12 : 14

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 14, 21, 18, 24 in proportion?

Solution: Check whether $14 \times 24 = 21 \times 18$ :

14 \times 24 = 336, \qquad 21 \times 18 = 378

336 ≠ 378, so 14, 21, 18, 24 are not in proportion.

Compare that with 12, 15, 28, 35:

12 \times 35 = 420, \qquad 15 \times 28 = 420

These match, so 12, 15, 28, 35 are in proportion.

Worked example: solving for a missing term

If $24 : 42 :: x : 56$ , find $x$ .

Solution: Cross-multiplying, $24 \times 56 = 42 \times x$ :

x = \dfrac{24 \times 56}{42} = \dfrac{1344}{42} = 32

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 $x^2 = ab$	9 and 16	$x = \sqrt{144} = 12$
Third proportional to $a$ and $b$	$a : b :: b : x$ , so $x = \dfrac{b^2}{a}$	5 and 10	$x = \dfrac{100}{5} = 20$
Fourth proportional to $a, b, c$	$a : b :: c : x$ , so $x = \dfrac{bc}{a}$	3, 5 and 6	$x = \dfrac{30}{3} = 10$

Direct and inverse proportion

Direct proportion: both quantities grow together, so their ratio stays fixed. If 4 notebooks cost ₹92, then 7 notebooks cost $x$ where $4 : 7 :: 92 : x$ :

x = \dfrac{7 \times 92}{4} = 161

Inverse proportion: one quantity grows as the other shrinks, so their product stays fixed instead. If 8 pumps empty a tank in 15 hours, 12 pumps take $x$ hours where:

8 \times 15 = 12 \times x \;\Rightarrow\; x = 10 \text{ hours}

Common mistakes

Comparing different units. Convert first: 120 cm to 2 m is $3 : 5$ , not $60 : 1$ .
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 pumps means fewer hours. 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 $3 : 4$ and $6 : 7$ 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 2027 paper includes.

Checked against GATE 2027 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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