Percent Increase and Decrease

Quick answer

To find a percent change, divide the amount of change by the ORIGINAL value and multiply by 100. Going from 40 to 50 is a change of 10, and 10 divided by 40 is 25%, so a 25% increase. To apply a change instead, multiply by 1 plus or minus the decimal: a 20% rise means multiplying by 1.2, and a 20% fall means multiplying by 0.8.

What you'll learn

Calculate a percent increase or decrease from two values
Apply a percent change in a single multiplication
Explain why a rise and fall of the same percent do not cancel

The formula

\text{percent change} = \frac{\text{change}}{\text{original}} \times 100

Going from $40$ to $50$ : the change is $10$ , and

\frac{10}{40} = 0.25 \quad\Rightarrow\quad 25\% \text{ increase}

A decrease works identically. Going from $50$ to $40$ : the change is $10$ , but the original is now $50$ :

\frac{10}{50} = 0.2 \quad\Rightarrow\quad 20\% \text{ decrease}

Same two numbers, different answers — because they are measured against different starting points.

Why it is always the original

This is the one idea in the lesson worth understanding rather than memorising.

A percent change describes how big the change was compared with where you started. “Prices rose 25%” means the rise was a quarter of the old price. The old price is what the sentence is about, so the old price is the whole.

Divide by the new value instead and you have answered a question nobody asked. The number will look plausible, which is what makes the error dangerous.

Applying a change in one step

Finding the change and adding it works, but there is a faster route that also makes later work possible.

To increase by $20\%$ : you keep $100\%$ and add $20\%$ , which is $120\%$ in total, or $1.2$ .

\text{new} = \text{original} \times 1.2

To decrease by $20\%$ : you keep $80\%$ .

\text{new} = \text{original} \times 0.8

Change	Multiplier	Reasoning
up $5\%$	$1.05$	keep all of it, add $5\%$
up $50\%$	$1.5$	keep all of it, add half
down $10\%$	$0.9$	$100\% - 10\% = 90\%$
down $35\%$	$0.65$	$100\% - 35\% = 65\%$

One multiplication instead of two steps, and no chance of adding when you meant to subtract.

Why a rise then an equal fall does not return you

Put $100 up by $20\%$ , then down by $20\%$ .

100 \times 1.2 = 120 \qquad 120 \times 0.8 = 96

Not $100$ . The rise was $20\%$ of $100$ , but the fall was $20\%$ of $120$ — a bigger amount, because it was taken from a bigger number.

The multipliers show it at once: $1.2 \times 0.8 = 0.96$ , so the pair together is a $4\%$ decrease no matter what you start with.

Worked examples

Common mistakes

Practice problems

A price goes from $20$ to $25$ . What is the percent increase?

Answer

$25\%$

Full solution

Change is $5$ , and $\tfrac{5}{20} = 0.25$ , so a $25\%$ increase.
A price goes from $80$ to $60$ . What is the percent decrease?

Answer

$25\%$

Full solution

Change is $20$ , and $\tfrac{20}{80} = 0.25$ , so a $25\%$ decrease.
Increase $340$ by $10\%$ .

Hint

One multiplication will do it.

Answer

$374$

Full solution

$340 \times 1.1 = 374$ .
Decrease $90$ by $30\%$ .

Answer

$63$

Full solution

Keep $70\%$ : $90 \times 0.7 = 63$ .
A town grows from $12{,}000$ to $13{,}800$ . What is the percent increase?

Answer

$15\%$

Full solution

Change is $1800$ , and $\tfrac{1800}{12000} = 0.15$ , so a $15\%$ increase.
A $40 item rises by $25\%$ and then falls by $25\%$ . What is the final price?

Hint

Do not expect $40.

Answer

$37.50

Full solution

$40 \times 1.25 = 50$ , then $50 \times 0.75 = 37.50$ .

The combined multiplier is $1.25 \times 0.75 = 0.9375$ , a $6.25\%$ overall decrease.
A salary rises $4\%$ one year and $5\%$ the next. What is the total percent increase?

Hint

Not $9\%$ .

Answer

$9.2\%$

Full solution

$1.04 \times 1.05 = 1.092$ , which is a $9.2\%$ increase.

The extra $0.2\%$ is the second year’s rise applied to the first year’s rise.
After a $20\%$ discount, a coat costs $64. What was the original price?

Hint

$64 is $80\%$ of the original.

Answer

$80

Full solution

$\tfrac{64}{0.8} = 80$ .

Checking: $80 \times 0.8 = 64$ ✓

Adding $20\%$ to $64 would give $76.80, which is wrong — that adds a fifth of the new price rather than removing a fifth of the old.
A bag of rice was $500$ g and is now advertised as ” $20\%$ extra free”. How much is in the bag?

Answer

$600$ g

Full solution

$500 \times 1.2 = 600$ g.
A shop raises a $50 price by $10\%$ , then advertises ” $10\%$ off”. A customer says the price is back to $50. Are they right?

Hint

What is each $10\%$ taken from?

Answer

No. The price is $49.50.

Full solution

$50 \times 1.1 = 55$ , then $55 \times 0.9 = 49.50$ .

The rise was $10\%$ of $50, which is $5. The discount was $10\%$ of $55, which is $5.50 — more came off than went on.

So the customer is wrong, though not in the direction they feared: the final price is fifty cents below where it started.

Frequently asked questions

How do I calculate a percent increase?

Subtract to find the change, divide by the ORIGINAL value, then multiply by 100. Going from 40 to 50 is a change of 10, and 10 divided by 40 gives 25%.

Why divide by the original and not the new value?

Because a change is described relative to where it started. Dividing by the new value answers a different question and gives a different number — from 40 to 50 that would be 20%, not 25%.

What is the quickest way to add 20% to a price?

Multiply by 1.2. The 1 keeps the original amount and the 0.2 adds the extra, so one multiplication does both. To take 20% off, multiply by 0.8.

If a price rises 20% then falls 20%, is it back to the start?

No, it ends lower. The rise is 20% of the original and the fall is 20% of the larger new figure, so more comes off than went on. 100 becomes 120, then 96.

Can a decrease be more than 100%?

Not for a quantity that cannot go below zero. Losing 100% of something leaves nothing, so there is nothing further to lose. Increases have no such ceiling.

What to learn next

Discounts, Sales Tax and TipsHow to work out a sale price, add sales tax, and calculate a tip, using one multiplication instead of two steps — and why the order does not matter.

Formulas on this page

Percent change

Key terms in this lesson

Percentage point: A percentage point is the plain difference between two percents. A rise from 4% to 6% is 2 percentage points, but a 50% increase — the two describe the same change and give different numbers.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.7.RP.A.3Ratios and Proportional RelationshipsUse proportional relationships to solve multistep ratio and percent problems.
CCSS.MATH.CONTENT.6.RP.A.3cRatios and Proportional RelationshipsFind a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

The formula

Why it is always the original

Applying a change in one step

Why a rise then an equal fall does not return you

Worked examples

Common mistakes

Practice problems

Frequently asked questions

How do I calculate a percent increase?

Why divide by the original and not the new value?

What is the quickest way to add 20% to a price?

If a price rises 20% then falls 20%, is it back to the start?

Can a decrease be more than 100%?

What to learn next

Formulas on this page

Key terms in this lesson

Standards alignment

More Percents lessons