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%

£1,250 + 20pc VAT gives £1,500.00

Beside an amount of money, pc against a number is percent as well, so a quote that writes “20pc VAT” adds up as it stands.

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 percentage and a fixed amount charged together work the same way, which is how card processing and booking fees are quoted. Write both, then on and the amount they apply to:

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1.4% + 20p on £2,255 gives £31.77

2.9% + 30c on $120 gives $3.78

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 up 20% gives £96.00

£80 cut by 20% gives £64.00

£18 marked up by 150% gives £45.00

£80 less 15% gives £68.00

£100 less £20 gives £80.00

£5 off £29 gives £24.00

£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, raised by, up and marked up by add the percentage, and decreased by, reduced by, lowered by, cut by and discounted by take it off. less takes off a percentage or an amount, and A off B takes the first amount off the second, the way a voucher is written. 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.

Sale signs written as fractions work the same way:

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£110.40 with a third off gives £73.60

£60 with half off gives £30.00

£110.40 less a third gives £73.60

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%

30% + 0.4 gives 70%

100% + 2 + 30% gives 330%

20% / 2 gives 10%

20% of 20% gives 4%

A plain number in the sum counts as a share of one, so 0.4 is 40% and 2 is 200%. Several terms apply from left to right, which is a quick way to add up a multiplier written in parts.

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%

of between two amounts of one kind gives the share directly, so you can read a portion off a label or a progress line without writing the question out:

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5 kg of 20 kg gives 25%

20 min of 1 hour gives 33.3333333333%

3 of 4 kg gives 75%

Two measurements of different kinds have no share, so of does the sum that does make sense: a depth over an area is a volume, and a mass in a volume is a concentration.

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25 mm of 57.6 m² gives 1.44 m³

30 g of 1 L gives 30 g/L

Anything else, such as 5 kg of 2 m, shows nothing rather than a bare percentage that would mean nothing either.

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

A fraction asks the same question, which suits a share you think of in quarters or fifths:

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50 is 1/5 of what gives 250

£50 is 1/5 of what gives £250.00

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.

When one side of the pair is a percentage, say it that way instead. Give the pair you know, a comma, then the value you’re asking about:

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20% is 500, what is 750 gives 30%

if 20 is 30%, what is 60% gives 40

The first line reads a progress bar: if 500 counts as 20%, then 750 is 30%. The second goes the other way, from a percentage to an amount: when 20 is 30% of the target, 60% of it is 40. The comma is what separates the pair you know from the question, so keep it in.

All question forms

Between two plain numbers, 20 off 100 and 20 on 100 give the ratio as a percentage (20%) rather than the reduced amount. Between two amounts of money, £20 off £100 is the reduced amount, £80.00, so a voucher line reads the way it’s written.

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

£97 between 6 of us gives £16.17

£3,603.91 in 9 instalments gives £400.43

12 payments of £50 gives £600.00

£45.99 x2 gives £91.98

£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. items is an ordinary plural noun like the rest, so it needs no each: 6 items at £9.90 is £59.40. 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 it can be a fraction for anything sold by weight.
• What each one holds can be a measurement instead of a price, which turns a packing list into a total: 14 boxes at 12 kg each is 168 kg and 9 at 1h 45m each is 15 hours 45 min.
• The number of people in a split must be a whole one, and split £86 among 0 reports a divide by zero.
• 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.