Percentage

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

⏱ 4 min read🧩 5 question types🎯 14 practice Q

The idea in one minute

'A is $25\%$ more than B' compares A with B, so B is the base. Said the other way round, B is compared with A, and the answer is different: B is $20\%$ less than A.

This subtopic is about flipping such statements both ways, quickly and correctly.

The base decides the answer

Take "A is $25\%$ more than B". Put B at 100, so A is 125.

Now read it backwards: B is less than A by $25$ out of $125$ , and $\dfrac{25}{125} \times 100 = 20\%$ . So B is $20\%$ less than A.

Same pair, two different percents. The base changed.

Rule: The quantity after "than" is the base. Flip the statement and the base flips too.

The two flip formulas

You do not need to test with 100 every time.

x\% \text{ more} \Rightarrow \frac{100x}{100+x}\% \text{ less}

x\% \text{ less} \Rightarrow \frac{100x}{100-x}\% \text{ more}

Check with 25: $25\%$ more becomes $\dfrac{100 \times 25}{125} = 20\%$ less. And $20\%$ less becomes $\dfrac{100 \times 20}{80} = 25\%$ more.

Watch: $25\%$ more and $25\%$ less are not a matching pair. Their partners are $20\%$ less and $33\dfrac{1}{3}\%$ more.

Set the base to 100

When unsure, do not trust memory of formulas. Build the numbers.

"A's salary is $60\%$ more than B's." B $= 100$ , A $= 160$ . B is less by $\dfrac{60}{160}$ of A, and $\dfrac{60}{160} \times 100 = 37.5\%$ .

Example: A town's population fell from 1,500 to 1,200. The rise needed to get back: $\dfrac{300}{1200} \times 100 = 25\%$ . The fall was only $\dfrac{300}{1500} = 20\%$ .

Pairs worth memorising

The same pairs appear every year. Learn the table and skip the working.

Statement	Reverse
$25\%$ more	$20\%$ less
$20\%$ more	$16\dfrac{2}{3}\%$ less
$50\%$ more	$33\dfrac{1}{3}\%$ less
$100\%$ more	$50\%$ less
$25\%$ less	$33\dfrac{1}{3}\%$ more
$20\%$ less	$25\%$ more
$75\%$ less	$300\%$ more

Tip: Every pair multiplies back to itself. Undoing a change is the flip question in disguise.

Chains of percentages

"A $40\%$ of B" language means multiply.

" $40\%$ of the candidates are graduates, and $25\%$ of the graduates are engineers." Engineers $= \dfrac{25}{100} \times \dfrac{40}{100} = \dfrac{1}{10} = 10\%$ of all candidates.

Note: Each "of" starts from the group named just before it, not from the total.

Awkward percents through fractions

Third-sixths and sevenths flip best as fractions.

$16\dfrac{2}{3}\%$ more means $\times \dfrac{7}{6}$ . Reversed, that is less by $\dfrac{1}{7}$ , and $\dfrac{1}{7} \times 100 = 14\dfrac{2}{7}\%$ .

"X is $16\dfrac{2}{3}\%$ more than Y" and "Y is $14\dfrac{2}{7}\%$ less than X" are the same fact.

Careful: $14\dfrac{2}{7}\%$ is exact, not approximately 14.2. Keep it as the fraction $1/7$ .

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Convert 'x% more' into 'less'

How to spot it:

'A is some percent more than B. B is less than A by what percent?'

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

Method

Note x, the percent by which A beats B.
Compute 100x over 100 + x.
State it as the per cent by which B is less.

Why it works:

The same gap measured on the bigger base A gives a smaller percent.

Try this

A's salary is 60% more than B's. B's salary is less than A's by what percentage?

Show solution

$\dfrac{100 \times 60}{100 + 60} = \dfrac{6000}{160}$ .
$= 37.5$ .

Answer

37.5%

Type 2very common4 practice Q

Convert 'x% less' into 'more'

How to spot it:

'A is some percent less than B. B is more than A by what percent?'

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

Method

Note x, the percent of the fall.
Compute 100x over 100 - x.
State it as the per cent by which B is more.

Why it works:

The same gap measured on the smaller base A gives a bigger percent.

Try this

A's income is 20% less than B's. B's income is more than A's by what percentage?

Show solution

$\dfrac{100 \times 20}{100 - 20} = \dfrac{2000}{80}$ .
$= 25$ .

Answer

25%

Type 3common3 practice Q

Per cent of a per cent

How to spot it:

One group is a percent of a second group, which is itself a percent of a total.

\frac{a}{100} \times \frac{b}{100}

Method

Attach each percent to its own group.
Multiply the two percents as fractions.
Simplify to a single percent of the total.

Why it works:

Every 'of' multiplies, so a percent of a percent is a product.

Try this

In an examination centre, 40% of the candidates are graduates, and 25% of the graduates are engineers. Engineers are what percentage of all candidates?

Show solution

$\dfrac{40}{100} \times \dfrac{25}{100} = \dfrac{1}{10}$ .
$\dfrac{1}{10} = 10\%$ .

Answer

10%

Type 4occasional2 practice Q

Two quantities on a common base

How to spot it:

Two quantities are each compared with the same third quantity, and then with each other.

Method

Put the shared base quantity at 100.
Write both quantities from the two statements.
Take the ratio of the two quantities.

Why it works:

A common base makes the two quantities comparable in one ratio.

Try this

X is 50% more than Z, and Y is 20% more than Z. Y is what percentage of X?

Show solution

Z $= 100$ , so X $= 150$ and Y $= 120$ .
$\dfrac{120}{150} \times 100 = 80$ .

Answer

80%

Type 5occasional

Awkward percents through fractions

How to spot it:

Percents like $16\dfrac{2}{3}\%$ or $14\dfrac{2}{7}\%$ appear, and a flip is asked.

Method

Write the percent as a fraction from the table.
Add or subtract that fraction from 1 for the bigger quantity.
The leftover fraction, over the bigger quantity, is the reverse percent.

Why it works:

Fraction chips reverse exactly, with no division of awkward decimals.

Try this

X is $16\dfrac{2}{3}\%$ more than Y. By what percentage is Y less than X?

Show solution

$16\dfrac{2}{3}\% = \dfrac{1}{6}$ , so X $= \dfrac{7}{6}$ Y.
Y $= \dfrac{6}{7}$ X: less by $\dfrac{1}{7}$ .
$\dfrac{1}{7} \times 100 = 14\dfrac{2}{7}$ .

Answer

14 2/7%

Formula sheet

x% more, reversed

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

x% more than becomes this much less than.

x% less, reversed

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

x% less than becomes this much more than.

Per cent of a per cent

\frac{a}{100} \times \frac{b}{100} = \frac{ab}{10000}

Each 'of' multiplies.

Shortcuts that save time

⚡ Memorise the standard pairs

Four pairs cover most papers: 25/20, 20/ $16\dfrac{2}{3}$ , 50/ $33\dfrac{1}{3}$ , and 100/50. Read the question, write the partner, done.

Example

If A's income is 20% less than B's, then B's income is more than A's by:

Show solution

$x\%$ less reverses to $\dfrac{100x}{100 - x}\%$ more.
$\dfrac{100 \times 20}{80} = 25$ .

Answer

25%

⚡ Base-100 when memory fails

Put the base quantity at 100, write the other quantity, and divide by the new base. Slow but never wrong.

Example

A's salary is 60% more than B's. B's salary is less than A's by what percentage?

Show solution

B $= 100$ , so A $= 160$ .
Gap $= 60$ over base $160$ .
$\dfrac{60}{160} \times 100 = 37.5$ .

Answer

37.5%

⚡ Awkward percents: use the fraction

$16\dfrac{2}{3}\%$ is $1/6$ , so 'more by $1/6$ ' reverses to 'less by $1/7$ ' of the bigger number.

Example

X is 16 2/3% more than Y. By what percentage is Y less than X?

Show solution

$16\dfrac{2}{3}\% = \dfrac{1}{6}$ , so X $= \dfrac{7}{6}$ Y.
Y $= \dfrac{6}{7}$ X, less by $\dfrac{1}{7}$ .
$\dfrac{1}{7} \times 100 = 14\dfrac{2}{7}$ .

Answer

14 2/7%

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Answering ' $25\%$ less' for the reverse of ' $25\%$ more'.

The reverse is $\dfrac{100 \times 25}{125} = 20\%$ less.

Mistake 02

Writing 'A is $25\%$ more than B' as $A = B - 25\%$ of B.

More means add: $A = B + \dfrac{25}{100}B = \dfrac{5}{4}B$ .

Mistake 03

Using $100 + x$ in the less-to-more formula.

Less-to-more divides by $100 - x$ ; more-to-less divides by $100 + x$ .

Mistake 04

Adding chained percents: $40\%$ and $25\%$ of them called $65\%$ .

Multiply instead: $25\%$ of $40\% = 10\%$ .

Mistake 05

Calling $75\%$ less the reverse of $75\%$ more.

$75\%$ less reverses to $\dfrac{100 \times 75}{25} = 300\%$ more.

Mistake 06

Rounding $14\dfrac{2}{7}\%$ to $14.2\%$ and losing the fraction.

Keep sevenths as fractions; the options use $14\dfrac{2}{7}$ .

Quick revision

Read this the night before the exam.

$x\%$ more $\Rightarrow \dfrac{100x}{100+x}\%$ less.
$x\%$ less $\Rightarrow \dfrac{100x}{100-x}\%$ more.
Base-100 always works: put the 'than' quantity at 100.
Pairs: 25/20, 50/ $33\dfrac{1}{3}$ , 100/50, 75/300.
Each 'of' in a chain multiplies: $a\%$ of $b\% = ab/10000$ .
$16\dfrac{2}{3}\%$ more $\Leftrightarrow$ $\dfrac{1}{7}$ less.

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

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

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