Percentage

high importance~2 Q in Tier 121 formulas⚡ 15 shortcuts5 subtopics

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Subtopic 1 of 5·Percentage increase / decrease & successive change →

Percentage meaning & conversions

Per cent means per hundred: $x\% = \frac{x}{100}$ . The single most profitable memorisation in Quant is the fraction table — every CGL percentage, profit-loss, interest and DI question gets faster.

Fraction	%	Fraction	%
$\frac{1}{2}$	50%	$\frac{1}{8}$	12.5%
$\frac{1}{3}$	$33\frac{1}{3}\%$	$\frac{3}{8}$	37.5%
$\frac{2}{3}$	$66\frac{2}{3}\%$	$\frac{5}{8}$	62.5%
$\frac{1}{4}$	25%	$\frac{1}{6}$	$16\frac{2}{3}\%$
$\frac{3}{4}$	75%	$\frac{5}{6}$	$83\frac{1}{3}\%$
$\frac{1}{5}$	20%	$\frac{1}{7}$	$14\frac{2}{7}\%$
$\frac{2}{5}$	40%	$\frac{1}{9}$	$11\frac{1}{9}\%$
$\frac{1}{12}$	$8\frac{1}{3}\%$	$\frac{1}{16}$	6.25%

Handy symmetry: $x\%$ of $y = y\%$ of $x$ (16% of 25 = 25% of 16).

$p\%$ of a quantity $N = \frac{pN}{100}$ ; to find the whole when a part is known, divide: $N = \frac{\text{part} \times 100}{p}$ .

📖 Notes 🧩 4 question types ⚡ Formulas 🎯 13 practice Q

Detailed notes

What does "per cent" mean?

Per cent means out of hundred: $x\% = \frac{x}{100}$ . So 35% of a quantity means 35 parts out of every 100 parts of it: $35\% \text{ of } N = \frac{35N}{100}$ . Percentages let us compare unlike totals: 36/60 in one test and 45/75 in another are both 60% — instantly comparable.

The fraction table (memorise cold — it pays for the whole topic)

Fraction	%	Fraction	%
1/2	50%	1/8	12.5%
1/3	$33\frac{1}{3}$ %	3/8	37.5%
2/3	$66\frac{2}{3}$ %	1/6	$16\frac{2}{3}$ %
1/4	25%	5/6	$83\frac{1}{3}$ %
3/4	75%	1/7	$14\frac{2}{7}$ %
1/5	20%	1/9	$11\frac{1}{9}$ %
2/5	40%	1/12	$8\frac{1}{3}$ %
5/8	62.5%	1/16	6.25%

Four moves that answer every basic question

x% of N → fraction × N. $12.5\%$ of 960 $= \frac{1}{8} \times 960 = 120$ .
Recover the whole from a part: if $p\%$ of N is k, then $N = \frac{100k}{p}$ . 35% of N is 189 → $N = \frac{189 \times 100}{35} = 540$ .
One quantity as a % of another: $\frac{A}{B} \times 100$ . 45 out of 360 → $\frac{45}{360} \times 100 = 12.5\%$ . The base sits right after the words "of" or "out of".
Swap trick: $x\%$ of $y = y\%$ of $x$ . $4\%$ of 25 $= 25\%$ of 4 $= 1$ .

Chained percentages on the same base

$35\%$ of N $+ 45\%$ of N $= 80\%$ of N — add the percentages first when the base is the same. The same works for subtraction: "spent 30% on rent and 25% on food → 55% gone, 45% left." But successive changes of the same quantity do not add — that is a different (and famous) trap covered in the increase/decrease subtopic.

Using the unit-percent

1% of N is N ÷ 100. Build any percentage from tens and ones: 24% of 650 → 10% is 65, so 24% = 2 × 65 + 4 × 6.5 = 130 + 26 = 156. This mental decomposition is faster than any written multiplication. The same idea runs in reverse: "what % is 78 of 650?" → 78 ÷ 6.5 = 12%.

Percentages in everyday exam language

"60% of the students are boys" → 40% are girls, and any part = total × fraction. If 35% of the class are girls and there are 21 girls, the class size = 21 × 100/35 = 60. The exam always keeps totals whole, so a fractional total is a free signal that your arithmetic slipped. Discount is a percentage of the marked price, and tax in these questions is simply added on the reduced amount afterwards.

Where the base sits

"A is 20% of B" → B is the base. "What % of B is A?" → base is B again. Exam favourite: a fall from ₹250 to ₹200 is a 20% fall (base 250), but rising back from ₹200 to ₹250 is a 25% rise (base 200) — same ₹50, different base.

Quick revision

$x\% = \frac{x}{100}$ ; the fraction table is the speed key.
Part → whole: $N = \frac{100k}{p}$ .
"What % of": divide by the base, × 100.
$x\%$ of $y = y\%$ of $x$ .
Same base → add/subtract the percentages directly.

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: Find x% of a number (fraction table)very common3 practice Q

How to spot it:

'What is 24% of 650?' or 'find $12\frac{1}{2}$ % of 960' — a direct percentage of a given number.

x\% \text{ of } N = \frac{x}{100} \times N

Replace x% by its table fraction (12.5% → 1/8, 16⅔% → 1/6 …).
Divide N by the denominator, multiply by the numerator.
For ugly percentages, build them from 10% and 1% of N.

Why: percentages are fractions with a fixed denominator — the table just gives friendlier denominators.

Example: Find $12\frac{1}{2}$ % of 960.

$12\frac{1}{2}\% = \frac{1}{8}$ → $\frac{960}{8} = 120$ . No long multiplication anywhere.

Type 2: Recover the base, then apply another percentagevery common2 practice Q

How to spot it:

'If 35% of a number is 189, find 80% of it' — a part is given, a different part is asked.

N = \frac{100k}{p}, \quad \text{then } q\% \text{ of } N = \frac{qk}{p}

Recover the whole: $N = \frac{\text{part} \times 100}{p}$ .
Take the asked percentage of N.
Shortcut: find 1% of N (k ÷ p) and scale — 1% × 80 gives 80% directly.

Why: both parts are fixed fractions of the same whole, so the whole unlocks every other part.

Example: If 35% of a number is 189, what is 80% of that number?

$N = \frac{189 \times 100}{35} = 540$ ; $80\%$ of 540 $= 432$ .

Type 3: Fraction ↔ percentage conversionvery common2 practice Q

How to spot it:

'Express 5/16 as a percentage' or 'write 87.5% as a fraction' — pure conversion, both directions.

\frac{p}{q} \times 100\%, \qquad x\% = \frac{x}{100} = \text{reduce the fraction}

Fraction → %: multiply by 100 (equivalently, scale the table value: 1/8 = 12.5% → 5/8 = 62.5%).
% → fraction: put x over 100, reduce. Use the table in reverse for standard values.
Decimals shift two places: 0.375 = 37.5%.

Why: 'per cent' is just a denominator of 100; conversion moves the number across that fixed denominator.

Example: Express $\frac{5}{16}$ as a percentage.

$\frac{5}{16} \times 100 = \frac{125}{4} = 31.25\%$ (1/16 is 6.25%, so 5 of them are 31.25%).

Type 4: One quantity as a percentage of anothervery common3 practice Q

How to spot it:

'45 is what per cent of 360?' / 'A's income is what % of B's?' — two quantities, the base named after 'of'.

\frac{\text{part}}{\text{base}} \times 100\%

Identify the base (the value after 'of').
Divide the part by the base, multiply by 100.
Comparative version: 'A is x% and B is y% more than C' → A is $\frac{100 + x}{100 + y} \times 100\%$ of B.

Why: a percentage always needs a base; the question names it with 'of'.

Example: Two numbers are respectively 20% and 50% more than a third number. The first number is what per cent of the second?

Let the third number be 100 → numbers 120 and 150. $\frac{120}{150} \times 100 = 80\%$ .

Formulas

Per cent of a number

x\% \text{ of } N = \frac{xN}{100}

Fraction to per cent

\frac{p}{q} \times 100\%

Per cent to fraction

x\% = \frac{x}{100}

Commutativity

x\% \text{ of } y = y\% \text{ of } x

Recovering the base

N = \frac{\text{part} \times 100}{\text{percentage}}

Shortcut tricks

⚡ The fraction table IS the shortcut

Convert percentages to fractions before multiplying: 12.5% of 640 = 1/8 × 640 = 80. One line, no calculator.

Example: Find $16\frac{2}{3}\%$ of 720.

$16\frac{2}{3}\% = \frac{1}{6}$ ⇒ $\frac{720}{6} = 120$ .

⚡ Swap the percentages

Use $x\%$ of $y = y\%$ of $x$ when the swapped pair is friendlier.

Example: Find 4% of 25.

$= 25\%$ of $4 = 1$ .

⚡ Part-to-whole in one division

'p% of a number is k' ⇒ number = 100k/p. Look at options: only one will make the product land on k.

Example: If 40% of a number is 256, find the number.

$\frac{256 \times 100}{40} = 640$ .

Where students lose marks

Treating $x\%$ of $y$ as $xy$ (missing the ÷100).
Converting 1/8 as 18% instead of 12.5%.
Dividing by the wrong base: % change must always use the ORIGINAL value as denominator.
Reading 'B's income is x% of A's' as 'x% more than A's'.

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 6 min · wrong answers go to your mistake notebook automatically.