Percentage Increase and Decrease Using Multipliers

Q: What is the multiplier for a 15% decrease?

A 15\% decrease leaves 85\%, so the multiplier is 0.85.

Q: Why doesn't a 10% rise followed by a 10% fall get back to the start?

The fall is 10\% of a bigger number. The overall multiplier is 1.1 × 0.9 = 0.99, so you end 1\% below where you began.

How multipliers turn every percentage increase and decrease into one multiplication, how to calculate the percentage change between two amounts, why a 50% rise then a 50% fall does not get you back, and reverse percentages.

3 min read Supports 2 lessons

A shop raises a £50 price by $20\%$ . The slow way: find $20\%$ of £50 (£10) and add it on, giving £60. The fast way: notice that the new price is the old one plus a fifth of it, which is $120\%$ of the old price. So multiply by $1.2$ .

100% original

+20%

An increase of $20\%$ leaves you with $120\%$ : multiply by $1.2$ .

Multipliers

\text{increase by } p\%: \;\times\left(1 + \tfrac{p}{100}\right) \qquad \text{decrease by } p\%: \;\times\left(1 - \tfrac{p}{100}\right)

A $35\%$ discount leaves $65\%$ of the price, so the multiplier is $0.65$ . A $4\%$ pay rise has multiplier $1.04$ .

Worked example

A £60 jacket is reduced by $35\%$ . What is the sale price?

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The multiplier for a $35\%$ decrease.
$1 - 0.35 = 0.65$

Finding the percentage change

To find how much something has changed in percentage terms, divide the change by the original amount.

Percentage change

\%\text{ change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100

Worked example

A house bought for £240,000 sells for £276,000. What is the percentage increase?

1
Find the change.
$276\,000 - 240\,000 = 36\,000$

Watch out

Always divide by the original, not the new amount. A fall from £80 to £60 is a $25\%$ decrease ( $\frac{20}{80}$ ), not $33\%$ .

Changes in a row

Multipliers really earn their keep when changes stack up. Each change multiplies the result of the last one.

$100$
Start with 100.

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Key idea

Repeated percentage changes multiply, they do not add. Ten years of $5\%$ growth is $\times 1.05^{10} \approx 1.63$ , a rise of $63\%$ , not $50\%$ . This is exponential growth.

Reverse percentages

Sometimes you know the amount after a change and want the original. Divide by the multiplier rather than taking the percentage off the new amount.

Worked example

After a $20\%$ discount a coat costs £72. What was the original price?

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The multiplier for a $20\%$ decrease is $0.8$ .
$\text{original} \times 0.8 = 72$

Watch out

Adding $20\%$ of £72 back on gives £86.40, which is wrong. The $20\%$ was of the original price, not the sale price.

Common questions

What is the multiplier for a 15% decrease?: A $15\%$ decrease leaves $85\%$ , so the multiplier is $0.85$ .
Why doesn't a 10% rise followed by a 10% fall get back to the start?: The fall is $10\%$ of a bigger number. The overall multiplier is $1.1 \times 0.9 = 0.99$ , so you end $1\%$ below where you began.

Practise percentage change

The Percent Increase & Decrease lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the multiplier for a 15% decrease?

A $15\%$ decrease leaves $85\%$ , so the multiplier is $0.85$ .

Why doesn't a 10% rise followed by a 10% fall get back to the start?

The fall is $10\%$ of a bigger number. The overall multiplier is $1.1 \times 0.9 = 0.99$ , so you end $1\%$ below where you began.