Percentage

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Subtopic 3 of 5·← Percentage increase / decrease & successive change·Population growth, depreciation & elections →

'x% more than' ↔ 'x% less than' & chains

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The 'more than / less than' pair has different bases — this asymmetry is tested constantly.

If A is $x\%$ more than B: $A = B\left(1 + \frac{x}{100}\right)$ , and B is $\frac{100x}{100 + x}\%$ less than A.
If A is $x\%$ less than B: $A = B\left(1 - \frac{x}{100}\right)$ , and B is $\frac{100x}{100 - x}\%$ more than A.

Chains: if A = a% of B and B = b% of C, then A $= \frac{ab}{100}\%$ of C.

The 'less than' percentage is always LARGER than the 'more than' percentage — use that to sanity-check options.

📖 Notes 🧩 4 question types ⚡ Formulas 🎯 14 practice Q

Detailed notes

The base decides everything

"A is x% more than B" uses B as the base; "B is ?% less than A" uses A as the base. Different denominators → different percentages. From 100 to 125 is $+25\%$ , but from 125 back to 100 is $-20\%$ — a 25% rise is undone by a 20% fall, never by a 25% fall.

The flip formulas

$\text{More} \to \text{less}: \frac{100x}{100 + x}\%, \qquad \text{Less} \to \text{more}: \frac{100x}{100 - x}\%$ Standard pairs to memorise: 25% more ↔ 20% less; 20% less ↔ 25% more; 50% more ↔ $33\frac{1}{3}$ % less; 20% more ↔ $16\frac{2}{3}$ % less. Sanity check that kills wrong options: the "less" answer is always smaller than x, the "more" answer is always bigger than x.

Assume the base = 100

The bullet-proof method: set the unmentioned quantity to 100, compute both values, then compute the asked percentage on its own base. Example: A is 60% more than B → B = 100, A = 160 → B is $\frac{60}{160} \times 100 = 37.5\%$ less than A. This also handles the recovery version: production falls from 1,500 to 1,200 → to regain 1,500 it must now rise by $\frac{300}{1200} \times 100 = 25\%$ , not 20% — the base shrank, so the recovery percentage is always bigger than the fall.

Chains of percentages

A = 30% of B and B = 40% of C → $A = \frac{30}{100} \times \frac{40}{100} \times C = 12\%$ of C. Multiply the fractions — never add. Workplace version: 40% of the employees are graduates and 25% of the graduates are post-graduates → post-graduates $= \frac{40}{100} \times \frac{25}{100} = 10\%$ of all employees. The two percentages sit on different totals, so they chain.

How the examiner flips the wording

Same arithmetic, four costumes:

"A's salary is 25% more than B's" ↔ "B's salary is 20% less than A's".
"A is heavier than B by 25%..." — weight of B is the base.
"If A's income is 25% more than B's, B's income is less than A's by what %?" — the classic flip.
"The marks of A are 20% less than B" → A/B = 4/5, so B is $\frac{1}{4}$ more = 25% more. Convert to the ratio A : B = 80 : 100 whenever the wording tangles you.

Speed pairs worth memorising

"more than"	equals "less than" (the other way)
25% more	20% less
$33\frac{1}{3}$ % more	25% less
50% more	$33\frac{1}{3}$ % less
$66\frac{2}{3}$ % more	40% less
100% more	50% less
Reading the table right to left covers the "less → more" questions too.

Quick revision

More→less: $\frac{100x}{100 + x}$ . Less→more: $\frac{100x}{100 - x}$ .
Base = 100 solves every variant, including chained ones.
Chains: multiply fractions of successive bases.
Recovery % > fall %; "less" answer < x < "more" answer.

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: 'More than' flipped to 'less than'very common3 practice Q

How to spot it:

'A's income is x% more than B's — B's income is what % less than A's?' The base changes mid-question.

\text{less}\% = \frac{100x}{100 + x}

Flip formula: $\frac{100x}{100 + x}\%$ less.
Or set B = 100 → A = 100 + x; the gap x sits on base (100 + x).
Ratio route: A : B = (100 + x) : 100; less % = excess/base × 100.

Why: the same gap is x% of B but a smaller % of the bigger A.

Example: A's income is 60% more than B's. B's income is what per cent less than A's?

B = 100, A = 160 → gap 60 on base 160 → $\frac{60}{160} \times 100 = 37.5\%$ less.

Type 2: 'Less than' flipped to 'more than'very common4 practice Q

How to spot it:

'B is x% less than A — A is what % more than B?' Also appears as 'by what % must it rise to get back?'

\text{more}\% = \frac{100x}{100 - x}

Flip formula: $\frac{100x}{100 - x}\%$ more.
Or set A = 100 → B = 100 − x; the gap x sits on the smaller base (100 − x).
Recovery version: fall from F to N → rise needed $= \frac{F - N}{N} \times 100\%$ — base is the reduced value.

Why: after a fall the base is smaller, so the matching rise is always the bigger number.

Example: The production of a factory falls from 1,500 units to 1,200 units. By what per cent must it now increase to reach 1,500 units again?

Gap $= 300$ on the reduced base 1,200 → $\frac{300}{1200} \times 100 = 25\%$ . (Not 20% — that would undo it on the old base.)

Type 3: Chained percentages (A is a% of B, B is b% of C)common3 practice Q

How to spot it:

'40% of the employees are graduates, 25% of the graduates are post-graduates' — a percentage of a percentage.

A = \frac{a}{100} \times \frac{b}{100} \times C \ \Rightarrow\ A = \frac{ab}{100}\% \text{ of } C

Write each statement as a fraction of its own base.
Multiply the fractions: $\frac{a}{100} \times \frac{b}{100} \times C$ .
Read A as $\frac{ab}{100}\%$ of the outermost base C.

Why: each percentage sits on a different total, so they compose multiplicatively — the same logic as successive discounts.

Example: In an office, 40% of the employees are graduates and 25% of the graduates are post-graduates. Post-graduates form what per cent of all employees?

$\frac{40}{100} \times \frac{25}{100} = \frac{10}{100}$ → 10% of all employees. (Adding 40 + 25 = 65% is the trap.)

Type 4: Two quantities compared through a common basecommon2 practice Q

How to spot it:

'A is x% more than C and B is y% more than C — A is what % of B?' Both percentages sit on the same hidden base.

A : B = (100 + x) : (100 + y) \ \Rightarrow\ \frac{A}{B} = \frac{100 + x}{100 + y} \times 100\%

Set the common base C = 100.
A = 100 + x and B = 100 + y.
The asked ratio is A/B × 100 — one division.

Why: a shared base turns two independent percentages into one ratio.

Example: A's height is 50% more than C's, and B's height is 20% more than C's. B's height is what per cent of A's height?

C = 100 → A = 150, B = 120 → $\frac{120}{150} \times 100 = 80\%$ .

Formulas

More → less

A = B\left(1 + \frac{x}{100}\right) \Rightarrow B = A\left(\frac{100}{100+x}\right),\ \text{less by } \frac{100x}{100+x}\%

Less → more

A = B\left(1 - \frac{x}{100}\right) \Rightarrow \text{B is } \frac{100x}{100-x}\% \text{ more than A}

Chain of percentages

A = \frac{a}{100}B,\ B = \frac{b}{100}C \Rightarrow A = \frac{ab}{100}\% \text{ of } C

Shortcut tricks

⚡ Flip with 100x/(100±x)

More→less uses 100+x in the denominator; less→more uses 100−x. The answer must be bigger than x for the less→more direction.

Example: A's salary is 25% more than B's. B's salary is what per cent less than A's?

$\frac{100 \times 25}{125} = 20\%$ .

⚡ Assume the base = 100

Set the unmentioned quantity to 100 and read off both values.

Example: A's salary is 20% more than B's. By what per cent is B's salary less than A's?

B = 100, A = 120. Gap 20, base A = 120 ⇒ 20/120 = 16⅔%.

⚡ Chain percentages by multiplying fractions

Convert each link to a fraction of the next and multiply.

Example: A's income is 40% of B's, and B's is 25% of C's. A's income is what % of C's?

$\frac{40}{100} \times \frac{25}{100} = \frac{10}{100}$ ⇒ 10%.

Where students lose marks

Answering 'x% less' with the same x (the correct value $\frac{100x}{100+x}$ is always smaller than x).
Using 100 − x in the denominator for the more→less direction.
Chaining by adding percentages instead of multiplying fractions.
Ignoring which quantity the question calls the base.

Practice sets — 14 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

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