Percentage Change

What you'll learn

How to find the change between an original value and a new value.
How to calculate a percentage increase or percentage decrease.
How to work out percentage profit and percentage loss.
How to round answers to 1 decimal place or 3 significant figures.

The big idea

Percentage change is about comparing how much something has changed with what it started at.

For example, an increase of £20 is quite small if the original price was £1000, but huge if the original price was £25. That is why we compare the change to the original amount, not just the new amount.

Diagram showing original amount, new amount, change, and percentage change formula

Definition

Percentage change

A percentage change tells you the change as a percentage of the original amount.

amount×100\text{percentage change}=\frac{\text{change}}{\text{original amount}}\times 100

Key Idea

Always compare with the original

The denominator is the original amount — the value before the increase, decrease, profit, or loss happened.

Prerequisite: finding the change

Before you can find a percentage change, you need the actual change.

For an increase, subtract original from new.
For a decrease, subtract new from original.

Definition

Change

The change is the difference between the original amount and the new amount.

Worked example: finding an increase

A bill was £1720 last year. This year it is £1814. Find the increase.

Example

Finding the change first

Identify the original amount and the new amount.
- Original amount: £1720
- New amount: £1814
Since the new amount is larger, this is an increase. Subtract the original from the new.
$1814 - 1720 = 94$
The increase is £94.

Common Mistake

Subtracting in the wrong order

For percentage change, it is fine to get a positive change first. If the amount went down, subtract the smaller new value from the larger original value.

Percentage increase

A percentage increase is used when the new amount is bigger than the original amount.

Definition

Percentage increase

A percentage increase is the increase written as a percentage of the original amount.

amount×100\text{percentage increase}=\frac{\text{increase}}{\text{original amount}}\times 100

Worked example: council tax increase

Last year, a council tax bill was £1840. This year, it is £1935. Work out the percentage increase to 1 decimal place.

Example

Percentage increase

Find the increase.
$1935 - 1840 = 95$
Divide the increase by the original amount, then multiply by 100.
$951840×100=5.163043478…\frac{95}{1840}\times 100=5.163043478\ldots$
Round to 1 decimal place.
$5.163043478…≈5.25.163043478\ldots \approx 5.2$
The percentage increase is 5.2%.

Tip

A quick sense check

If the change is small compared with the original amount, the percentage change should also be small. Here £95 out of £1840 is only a little over 5%, so 5.2% is sensible.

Percentage decrease

A percentage decrease is used when the new amount is smaller than the original amount.

Definition

Percentage decrease

A percentage decrease is the decrease written as a percentage of the original amount.

amount×100\text{percentage decrease}=\frac{\text{decrease}}{\text{original amount}}\times 100

Worked example: car insurance decrease

Last year, a driver paid £420 for car insurance. This year, the cost is £386. Work out the percentage decrease to 1 decimal place.

Example

Percentage decrease

Find the decrease.
$420 - 386 = 34$
Divide the decrease by the original amount, then multiply by 100.
$34420×100=8.095238095…\frac{34}{420}\times 100=8.095238095\ldots$
Round to 1 decimal place.
$8.095238095…≈8.18.095238095\ldots \approx 8.1$
The percentage decrease is 8.1%.

Common Mistake

Dividing by the new amount

Do not divide by £386 in this example. The original amount was £420, so the percentage decrease is based on £420.

Percentage change when the question does not say increase or decrease

Sometimes the question just asks for the percentage change. You need to decide whether it is an increase or a decrease.

Worked example: average price change

The average price of a flat was £256 800 in one year and £249 900 the next year. Calculate the percentage change to 1 decimal place.

Example

Deciding increase or decrease

Compare the two values.
- Original amount: £256 800
- New amount: £249 900
The new amount is smaller, so this is a decrease.
Find the decrease.
$256800 - 249900 = 6900$
Divide by the original amount and multiply by 100.
$6900256800×100=2.686915887…\frac{6900}{256800}\times 100=2.686915887\ldots$
Round to 1 decimal place.
$2.686915887…≈2.72.686915887\ldots \approx 2.7$
The percentage change is a 2.7% decrease.

Tip

Say increase or decrease

If a question asks for percentage change, include the word increase or decrease in your final answer unless it is completely obvious from the context.

Profit and loss

Percentage change also appears in buying and selling questions.

Definition

Profit and loss

Profit happens when the selling price is more than the cost price. Loss happens when the selling price is less than the cost price.

For these questions:

The original amount is usually the cost price.
The new amount is usually the selling price.
Profit means increase.
Loss means decrease.

Worked example: percentage profit on a house

A house is bought for £240 000 and later sold for £253 200. Calculate the percentage profit.

Example

Percentage profit

Find the profit.
$253200 - 240000 = 13200$
Divide the profit by the original cost price, then multiply by 100.
$13200240000×100=5.5\frac{13200}{240000}\times 100=5.5$
The percentage profit is 5.5%.

Worked example: percentage loss on a car

A car is bought for £14 800 and sold for £10 900. Work out the percentage loss to 3 significant figures.

Example

Percentage loss

Find the loss.
$14800 - 10900 = 3900$
Divide the loss by the original cost price, then multiply by 100.
$390014800×100=26.35135135…\frac{3900}{14800}\times 100=26.35135135\ldots$
Round to 3 significant figures.
$26.35135135…≈26.426.35135135\ldots \approx 26.4$
The percentage loss is 26.4%.

Common Mistake

Using the selling price as the denominator

For profit and loss, always divide by the amount paid at the start. That is the original cost.

Multi-step selling questions

Some questions first ask you to work out the total money made from selling items. Then you compare this with the original cost.

A good order is:

Work out the total selling price.
Work out the profit or loss.
Use the percentage change formula.

Worked example: selling bottles

A shopkeeper buys 12 bottles of juice for £4.80. She sells each bottle for 60p. Work out her percentage profit.

Example

Percentage profit after selling items

Convert 60p to pounds.
$60p=0.6060\text{p}=0.60$
Work out the total selling price for 12 bottles.
$12×0.60=7.2012\times 0.60=7.20$
Work out the profit.
$7.20 - 4.80 = 2.40$
Divide the profit by the original cost and multiply by 100.
$2.404.80×100=50\frac{2.40}{4.80}\times 100=50$
The percentage profit is 50%.

Worked example: different selling prices

A trader buys 40 cans of drink for £15. He sells 28 cans for 55p each and the rest for 25p each. Work out the percentage profit to 3 significant figures.

Example

Two different selling prices

Find how many cans are sold at the lower price.
$40 - 28 = 12$
Convert the prices to pounds and find the money from the first 28 cans.
$28×0.55=15.4028\times 0.55=15.40$
Find the money from the remaining 12 cans.
$12×0.25=3.0012\times 0.25=3.00$
Find the total selling price.
$15.40 + 3.00 = 18.40$
Find the profit.
$18.40 - 15.00 = 3.40$
Calculate the percentage profit.
$3.4015.00×100=22.666666…\frac{3.40}{15.00}\times 100=22.666666\ldots$
Round to 3 significant figures.
$22.666666…≈22.722.666666\ldots \approx 22.7$
The percentage profit is 22.7%.

Tip

Keep all money in the same units

Do not mix pounds and pence in one calculation. Convert pence to pounds first, for example 45p becomes £0.45.

Questions involving mass

Some selling questions involve kilograms and grams. You must convert units before finding how many bags can be made.

Definition

Kilograms and grams

1 kg is equal to 1000 g.

Worked example: sweets in bags

A shopkeeper buys 1.5 kg of sweets for £3.60. He puts 125 g of sweets in each bag and sells each bag for 40p. Work out the percentage profit.

Example

Converting kilograms to grams first

Convert 1.5 kg to grams.
$1.5×1000=15001.5\times 1000=1500$
Find the number of bags.
$1500÷125=121500\div 125=12$
Convert 40p to pounds and find the total selling price.
$12×0.40=4.8012\times 0.40=4.80$
Find the profit.
$4.80 - 3.60 = 1.20$
Calculate the percentage profit.
$1.203.60×100=33.333333…\frac{1.20}{3.60}\times 100=33.333333\ldots$
The percentage profit is 33.3% to 3 significant figures, or 33.3% to 1 decimal place.

Common Mistake

Check whether all items are sold

In these questions, only use the money actually made from selling. If some items are not sold, they should not be included in the selling price.

Rounding your answer

The calculation often gives a long decimal. The question may ask for:

1 decimal place, meaning one digit after the decimal point.
3 significant figures, meaning the first three important digits.

Examples:

7.264... to 1 decimal place is 7.3.
7.264... to 3 significant figures is 7.26.
0.05678 to 3 significant figures is 0.0568.

Tip

Do not round too early

Keep the full calculator answer until the final step. Rounding early can make your final answer slightly wrong.

Exam technique

In the exam

Identify the original amount first — it is usually the earlier price, the cost price, or last year’s value.
Find the actual change: increase, decrease, profit, or loss.
Use $changeoriginal×100\frac{\text{change}}{\text{original}}\times 100$ , then round exactly as the question asks.

Self review

Check yourself

Can you explain why the original amount goes on the bottom of the fraction?
If a value goes from £80 to £92, is this an increase or a decrease, and what is the change?
In a profit question, which amount is the original: the buying price or the selling price?

Percentage Change

What you'll learn

How to find the change between an original value and a new value.
How to calculate a percentage increase or percentage decrease.
How to work out percentage profit and percentage loss.
How to round answers to 1 decimal place or 3 significant figures.

The big idea

Percentage change is about comparing how much something has changed with what it started at.

Diagram showing original amount, new amount, change, and percentage change formula

Definition

Percentage change

A percentage change tells you the change as a percentage of the original amount.

amount×100\text{percentage change}=\frac{\text{change}}{\text{original amount}}\times 100

Key Idea

Always compare with the original

The denominator is the original amount — the value before the increase, decrease, profit, or loss happened.

Prerequisite: finding the change

Before you can find a percentage change, you need the actual change.

For an increase, subtract original from new.
For a decrease, subtract new from original.

Definition

Change

The change is the difference between the original amount and the new amount.

Worked example: finding an increase

A bill was £1720 last year. This year it is £1814. Find the increase.

Example

Finding the change first

Identify the original amount and the new amount.
- Original amount: £1720
- New amount: £1814
Since the new amount is larger, this is an increase. Subtract the original from the new.
$1814 - 1720 = 94$
The increase is £94.

Common Mistake

Subtracting in the wrong order

For percentage change, it is fine to get a positive change first. If the amount went down, subtract the smaller new value from the larger original value.

Percentage increase

A percentage increase is used when the new amount is bigger than the original amount.

Definition

Percentage increase

A percentage increase is the increase written as a percentage of the original amount.

amount×100\text{percentage increase}=\frac{\text{increase}}{\text{original amount}}\times 100

Worked example: council tax increase

Last year, a council tax bill was £1840. This year, it is £1935. Work out the percentage increase to 1 decimal place.

Example

Percentage increase

Find the increase.
$1935 - 1840 = 95$
Divide the increase by the original amount, then multiply by 100.
$951840×100=5.163043478…\frac{95}{1840}\times 100=5.163043478\ldots$
Round to 1 decimal place.
$5.163043478…≈5.25.163043478\ldots \approx 5.2$
The percentage increase is 5.2%.

Tip

A quick sense check

If the change is small compared with the original amount, the percentage change should also be small. Here £95 out of £1840 is only a little over 5%, so 5.2% is sensible.

Percentage decrease

A percentage decrease is used when the new amount is smaller than the original amount.

Definition

Percentage decrease

A percentage decrease is the decrease written as a percentage of the original amount.

amount×100\text{percentage decrease}=\frac{\text{decrease}}{\text{original amount}}\times 100

Worked example: car insurance decrease

Last year, a driver paid £420 for car insurance. This year, the cost is £386. Work out the percentage decrease to 1 decimal place.

Example

Percentage decrease

Find the decrease.
$420 - 386 = 34$
Divide the decrease by the original amount, then multiply by 100.
$34420×100=8.095238095…\frac{34}{420}\times 100=8.095238095\ldots$
Round to 1 decimal place.
$8.095238095…≈8.18.095238095\ldots \approx 8.1$
The percentage decrease is 8.1%.

Common Mistake

Dividing by the new amount

Do not divide by £386 in this example. The original amount was £420, so the percentage decrease is based on £420.

Percentage change when the question does not say increase or decrease

Sometimes the question just asks for the percentage change. You need to decide whether it is an increase or a decrease.

Worked example: average price change

The average price of a flat was £256 800 in one year and £249 900 the next year. Calculate the percentage change to 1 decimal place.

Example

Deciding increase or decrease

Compare the two values.
- Original amount: £256 800
- New amount: £249 900
The new amount is smaller, so this is a decrease.
Find the decrease.
$256800 - 249900 = 6900$
Divide by the original amount and multiply by 100.
$6900256800×100=2.686915887…\frac{6900}{256800}\times 100=2.686915887\ldots$
Round to 1 decimal place.
$2.686915887…≈2.72.686915887\ldots \approx 2.7$
The percentage change is a 2.7% decrease.

Tip

Say increase or decrease

If a question asks for percentage change, include the word increase or decrease in your final answer unless it is completely obvious from the context.

Profit and loss

Percentage change also appears in buying and selling questions.

Definition

Profit and loss

Profit happens when the selling price is more than the cost price. Loss happens when the selling price is less than the cost price.

For these questions:

The original amount is usually the cost price.
The new amount is usually the selling price.
Profit means increase.
Loss means decrease.

Worked example: percentage profit on a house

A house is bought for £240 000 and later sold for £253 200. Calculate the percentage profit.

Example

Percentage profit

Find the profit.
$253200 - 240000 = 13200$
Divide the profit by the original cost price, then multiply by 100.
$13200240000×100=5.5\frac{13200}{240000}\times 100=5.5$
The percentage profit is 5.5%.

Worked example: percentage loss on a car

A car is bought for £14 800 and sold for £10 900. Work out the percentage loss to 3 significant figures.

Example

Percentage loss

Find the loss.
$14800 - 10900 = 3900$
Divide the loss by the original cost price, then multiply by 100.
$390014800×100=26.35135135…\frac{3900}{14800}\times 100=26.35135135\ldots$
Round to 3 significant figures.
$26.35135135…≈26.426.35135135\ldots \approx 26.4$
The percentage loss is 26.4%.

Common Mistake

Using the selling price as the denominator

For profit and loss, always divide by the amount paid at the start. That is the original cost.

Multi-step selling questions

Some questions first ask you to work out the total money made from selling items. Then you compare this with the original cost.

A good order is:

Work out the total selling price.
Work out the profit or loss.
Use the percentage change formula.

Worked example: selling bottles

A shopkeeper buys 12 bottles of juice for £4.80. She sells each bottle for 60p. Work out her percentage profit.

Example

Percentage profit after selling items

Convert 60p to pounds.
$60p=0.6060\text{p}=0.60$
Work out the total selling price for 12 bottles.
$12×0.60=7.2012\times 0.60=7.20$
Work out the profit.
$7.20 - 4.80 = 2.40$
Divide the profit by the original cost and multiply by 100.
$2.404.80×100=50\frac{2.40}{4.80}\times 100=50$
The percentage profit is 50%.

Worked example: different selling prices

A trader buys 40 cans of drink for £15. He sells 28 cans for 55p each and the rest for 25p each. Work out the percentage profit to 3 significant figures.

Example

Two different selling prices

Find how many cans are sold at the lower price.
$40 - 28 = 12$
Convert the prices to pounds and find the money from the first 28 cans.
$28×0.55=15.4028\times 0.55=15.40$
Find the money from the remaining 12 cans.
$12×0.25=3.0012\times 0.25=3.00$
Find the total selling price.
$15.40 + 3.00 = 18.40$
Find the profit.
$18.40 - 15.00 = 3.40$
Calculate the percentage profit.
$3.4015.00×100=22.666666…\frac{3.40}{15.00}\times 100=22.666666\ldots$
Round to 3 significant figures.
$22.666666…≈22.722.666666\ldots \approx 22.7$
The percentage profit is 22.7%.

Tip

Keep all money in the same units

Do not mix pounds and pence in one calculation. Convert pence to pounds first, for example 45p becomes £0.45.

Questions involving mass

Some selling questions involve kilograms and grams. You must convert units before finding how many bags can be made.

Definition

Kilograms and grams

1 kg is equal to 1000 g.

Worked example: sweets in bags

A shopkeeper buys 1.5 kg of sweets for £3.60. He puts 125 g of sweets in each bag and sells each bag for 40p. Work out the percentage profit.

Example

Converting kilograms to grams first

Convert 1.5 kg to grams.
$1.5×1000=15001.5\times 1000=1500$
Find the number of bags.
$1500÷125=121500\div 125=12$
Convert 40p to pounds and find the total selling price.
$12×0.40=4.8012\times 0.40=4.80$
Find the profit.
$4.80 - 3.60 = 1.20$
Calculate the percentage profit.
$1.203.60×100=33.333333…\frac{1.20}{3.60}\times 100=33.333333\ldots$
The percentage profit is 33.3% to 3 significant figures, or 33.3% to 1 decimal place.

Common Mistake

Check whether all items are sold

In these questions, only use the money actually made from selling. If some items are not sold, they should not be included in the selling price.

Rounding your answer

The calculation often gives a long decimal. The question may ask for:

1 decimal place, meaning one digit after the decimal point.
3 significant figures, meaning the first three important digits.

Examples:

7.264... to 1 decimal place is 7.3.
7.264... to 3 significant figures is 7.26.
0.05678 to 3 significant figures is 0.0568.

Tip

Do not round too early

Keep the full calculator answer until the final step. Rounding early can make your final answer slightly wrong.

Exam technique

In the exam

Identify the original amount first — it is usually the earlier price, the cost price, or last year’s value.
Find the actual change: increase, decrease, profit, or loss.
Use $changeoriginal×100\frac{\text{change}}{\text{original}}\times 100$ , then round exactly as the question asks.

Self review

Check yourself

Can you explain why the original amount goes on the bottom of the fraction?
If a value goes from £80 to £92, is this an increase or a decrease, and what is the change?
In a profit question, which amount is the original: the buying price or the selling price?

Percentage Change

What you'll learn

The big idea

Prerequisite: finding the change

Worked example: finding an increase

Percentage increase

Worked example: council tax increase

Percentage decrease

Worked example: car insurance decrease

Percentage change when the question does not say increase or decrease

Worked example: average price change

Profit and loss

Worked example: percentage profit on a house

Worked example: percentage loss on a car

Multi-step selling questions

Worked example: selling bottles

Worked example: different selling prices

Questions involving mass

Worked example: sweets in bags

Rounding your answer

Recap questions

You've reached the end

Percentage Change Revision Guide

Percentage Change

What you'll learn

The big idea

Prerequisite: finding the change

Worked example: finding an increase

Percentage increase

Worked example: council tax increase

Percentage decrease

Worked example: car insurance decrease

Percentage change when the question does not say increase or decrease

Worked example: average price change

Profit and loss

Worked example: percentage profit on a house

Worked example: percentage loss on a car

Multi-step selling questions

Worked example: selling bottles

Worked example: different selling prices

Questions involving mass

Worked example: sweets in bags

Rounding your answer

Recap questions

You've reached the end

Percentage Change Revision Guide