Percentages

This page covers everything to do with percentages: taking a share of an amount, adding VAT or a tip, knocking off a discount, asking what percentage one value is of another, working backwards from a price, and the shopping phrases that bundle these together. Reach for it whenever a sum involves a %.

Writing a percentage

A number followed by % is a percentage. You can also write percent or per cent, and store a percentage in a name to reuse it:

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15% of £80 gives £12.00

15 per cent of £80 gives £12.00

vat = 20% gives 20%

To show an ordinary answer as a percentage, add as %:

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£18 / £120 as % gives 15%

Percentage of, on and off

The words of, on and off say what a percentage does to an amount:

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12.5% of £48 gives £6.00

£85 + 20% gives £102.00

20% off £65 gives £52.00

50% of 90 min gives 45 min

half of £37 gives £18.50

Percentages work with money, plain numbers and quantities with units alike, and the answer keeps the currency or unit. half of is a shortcut for 50% of.

A fraction written with a slash works the same way, which suits recipes and shares that are easier to think of as fractions:

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3/4 of £60 gives £45.00

1/4 off £60 gives £45.00

2/3 of 90 min gives 60 min

Write the fraction without spaces around the slash, as in 3/4.

Increases, decreases and discounts

To raise or lower an amount by a percentage, say it the way you would out loud. There’s no multiplier to work out:

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increase £80 by 15% gives £92.00

decrease £80 by 15% gives £68.00

£80 reduced by 15% gives £68.00

£80 less 15% gives £68.00

£100 less £20 gives £80.00

£10.50 with 5% off gives £9.98

£10.50 with 5% discount gives £9.98

$1,899 - 10% student discount gives $1,709.10

increased by and raised by add the percentage, and decreased by, reduced by, lowered by and discounted by take it off. less takes off a percentage or an amount. with 5% off works on a name too, as in basket with 5% off, and the discount stops at the next + or -, so food with 25% off + drinks adds the drinks at full price. A word or two describing the discount or tax, such as student discount or sales tax, keeps the line readable without changing the answer.

Chained changes

Several adjustments in a row apply from left to right, each one to the running total:

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£1,200 + 3% + 3% gives £1,273.08

£200 - 10% + 10% gives £198.00

The second line catches people out. Taking 10% off £200 gives £180, and adding 10% of £180 gives £198, not the original £200.

Percentages of named values

A percentage stored in a name behaves exactly like one written inline:

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vat = 20%

price = £250

price + vat gives £300.00

price * vat gives £50.00

price - vat gives £200.00

Names made of two or more words work the same way after of or off:

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house price = £285k

10% of house price gives £28,500.00

house price * 10% gives £28,500.00

Arithmetic between percentages

You can add, subtract and scale percentages themselves:

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vat = 20%

service = 12.5%

vat + service gives 32.5%

100% - 15% gives 85%

20% / 2 gives 10%

20% of 20% gives 4%

The type of the answer depends on the operation:

A number plus a percentage works like amount + p%, which gives compound growth a tidy form:

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£2,000 * (1 + 4%)^2 gives £2,163.20

Percentage questions

What percentage is it?

Ask what share one value is of another with is what % of, or the shorter as a % of:

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£45 is what % of £180 gives 25%

£45 as a % of £180 gives 25%

30 min is what % of 2 hours gives 25%

750 ml is what % of 2 litres gives 37.5%

Quantities are converted first, so you can mix minutes and hours or millilitres and litres.

If the first value is itself a calculation, put it in brackets:

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(£110.25 - £98) is what % of £98 gives 12.5%

Odds and survey results work the way you’d say them, with in:

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1 in 5 as % gives 20%

Percentage change

a to b as % gives the change from the first value to the second, and as x gives it as a multiplier:

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£1,500 to £1,650 as % gives 10%

£1,200 to £900 as % gives -25%

£1,500 to £1,650 as x gives 1.1x

A rise of 10% is a multiplier of 1.1x, and a fall shows as a negative percentage.

The same question reads well in words, especially with names:

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old price = £80

new price = £92

percentage change from old price to new price gives 15%

new price is what % more than old price gives 15%

is what % less than measures a fall from the second value in the same way, so £1,200 is what % less than £1,500 is 20%.

Working backwards

When you know the result and the percentage, ask for the starting value with what:

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£72 is 20% on what gives £60.00

£48 is 20% off what gives £60.00

£300 is 20% of what gives £1,500.00

The first line is a price including 20% VAT, so the price before VAT was £60. The second is a sale price after 20% off, so the original price was £60.

more than what and less than what ask the same question in everyday words:

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£92 is 15% more than what gives £80.00

£68 is 15% less than what gives £80.00

Discount and markup rates

To find the rate that turns one amount into another, use is what % off or is what % on:

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£40 is what % off £50 gives 20%

£60 is what % on £50 gives 20%

is what % with gives the first value’s share of the two added together. For example, a £25 tip on a £225 bill is 10% of the £250 you paid:

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£25 is what % with £225 gives 10%

Proportions

For the rule of three, write a is to b as what is to c. what can stand for either value in the second pair:

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200 g is to 4 as what is to 6 gives 300 g

3 kg is to £12 as 5 kg is to what gives £20.00

5 km is to 30 min as 8 km is to what gives 48 min

The first line scales a recipe for four people up to six.

All question forms

£20 off £100, 20 off 100 and 20 on 100 without what give the ratio as a percentage (20%), not the reduced amount. To take £20 off £100, write £100 - £20 or £100 less £20.

Everyday purchase phrases

These phrases read the way you would describe a purchase:

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3 tickets at £12.50 each gives £37.50

3 coffees @ £2.80 gives £8.40

2 adults at £15 + 3 children at £8 gives £54.00

£120 after 25% discount gives £90.00

£120 with 25% off gives £90.00

£85 with 20% VAT gives £102.00

£48 with 10% tip gives £52.80

split £86 among 4 people gives £21.50

£150 split 4 ways gives £37.50

share £90 between 3 gives £30.00

£5 less than £20 gives £15.00

£5 more than £20 gives £25.00

The adjustments can be chained, and each applies to the amount immediately before it:

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4 tickets at £25 each after 10% discount with 20% VAT gives £108.00

Four tickets at £25 make £100, the discount brings that to £90, and 20% VAT on £90 makes £108.

Some details to keep in mind:

• In place of items, use any word that describes what you’re buying, such as tickets or coffees. The count may be a name, with or without the word (qty tickets at £4.50 each, or tins at £28 each with tins = 3), and must be a whole number.
• After the number of people you can add who they are, as in split £100 among 3 friends.
• Several groups at different prices add up in one line: 2 adults at £15 + 3 children at £8.
• plus works in place of with, so £85 plus 20% VAT is also £102.00.
• £10.50 plus VAT without a rate asks you for one. Set your sales tax name and rate in Settings › Calculations, or add VAT = 20% to your shared definitions, and the same line adds it for you. With VAT = 20%, £10.50 with VAT is £12.60 too, and a price marked including VAT, inc VAT, excluding VAT or ex VAT stays as written, so you can label prices the way the receipt does.
• To split a total that includes a tip, put the tipped amount in brackets: split (£64 with 12.5% tip) among 4. Storing the total in a name works too.

Putting it together

Dinner for three, with a voucher that only covers food:

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# Dinner for three

food = £88

drinks = £32

voucher = 25%

bill = food - voucher + drinks gives £98.00

total = bill with 12.5% tip gives £110.25

split total among 3 people gives £36.75

Because adjustments apply left to right, food - voucher takes 25% off the food before the drinks are added, so the voucher doesn’t touch the drinks. The tip phrase then adds 12.5% to the whole bill, and split … among 3 gives each person’s share.