Percents

Per cent means per hundred, so every percent is a fraction with a hundred underneath.

Shade sixty-three squares of a hundred-square grid.

How much is shaded? Sixty-three out of a hundred — which is what "sixty-three per cent" says out loud.

a hundred squares, sixty-three shaded 63 out of 100 — which is what "per cent" says the same grid works for a fraction and a decimal

The same picture is also 63100, and also 0.63. One grid, three names.

Nothing has been converted. They were always the same amount, written differently.

A percent is not a new kind of number. It is a fraction whose bottom is always a hundred, which is exactly what makes percents easy to compare.

A hundred-square grid with 64 squares shaded 64% = 64/100 = 0.64 one whole = 100 squares

Three ways to write one amount

63%, 63100 and 0.63 are the same amount in three notations.

Percent to decimal: divide by 100, which moves the point two places left.

Decimal to percent: multiply by 100, moving it two places right.

Two places, either way

The only question is which direction. Percents are the bigger-looking numbers, so going to a decimal always makes the number smaller.

Finding a percent of something

Turn the percent into a decimal and multiply.

25% of 80 is 0.25 × 80 = 20.

For friendly percents, mental shortcuts are faster: 10% is a tenth, 50% is a half, 25% is a quarter.

Build from 10%

Find 10% by moving the point one place, then scale. 30% is three of those; 5% is half of one.

Discounts and increases in one step

A 20% discount leaves 80%, so multiply by 0.8 — one step, not two.

A 15% increase leaves 115%, so multiply by 1.15.

Working out the change and then subtracting gives the same answer with more chances to slip.

The multiplier carries the whole story

The 1 in 1.15 keeps what you already had; the .15 adds the growth. Multiplying by 0.15 alone throws the original away.

Percent change

Percent change compares the change with the original, not with the new value.

A price rising from $40 to $52 has changed by $12, and 1240 = 0.3, so 30%.

Dividing by the new price instead gives a different and wrong answer.

Always over the original

Up 30% then down 30% does not return you to the start, because the second percent is taken of a larger number.

Working backwards

If a sale price is $36 after 25% off, then $36 is 75% of the original.

So divide: 36 ÷ 0.75 = 48.

Adding 25% back onto $36 would give $45, which is not the price it started at.

Undo by dividing

Multiplying got you there, so dividing gets you back. Adding the percent back on is the commonest error in this unit.

Worked examples

A percent of a number

Find 35% of 60.

  1. 10% of 60 is 6. Move the point one place.
  2. So 30% is 18, and 5% is 3. Scale from the friendly one.
  3. 18 + 3 = 21. Or directly: 0.35 × 60 = 21.

21

A discount

A $60 coat is reduced by 20%. What is the sale price?

  1. A 20% discount leaves 80%. What you keep, not what comes off.
  2. Multiply by 0.8: 60 × 0.8. One step.
  3. = $48. Check: the saving is $12, which is a fifth of $60.

$48

Working backwards

After 25% off, a game costs $36. What was the original price?

  1. 25% off leaves 75%, so $36 is 0.75 of the original. Identify what the $36 represents.
  2. Divide: 36 ÷ 0.75. Undo the multiplication.
  3. = $48. Check: 25% of 48 is 12, and 48 - 12 = 36.

$48

Words to know

Percent
A number out of a hundred, written with %.
Percent of a number
A part found by multiplying, as in 25% of 80.
Percent increase
How much something grew, as a percentage of what it was.
Discount
An amount taken off a price, usually given as a percent.
Multiplier
The single number you multiply by, such as 0.8 for 20% off.

Mistakes to watch for

Multiplying by the discount instead of what is left. A 20% discount on $60 means multiplying by 0.8 to get $48 — multiplying by 0.2 gives $12, which is the saving, not the price.

Adding the percent back to reverse a discount. $36 after 25% off came from $48, found by dividing by 0.75. Adding 25% of $36 gives $45, which is simply a different number.

Dividing by the new value for percent change. The change is always compared with the original: $40 to $52 is 1240 = 30%, not 1252.

Moving the decimal point the wrong way. 63% is 0.63, not 6.3. Percents look bigger, so the decimal must come out smaller.

Assuming up then down by the same percent returns to the start. A $100 item up 20% is $120; down 20% from there is $96, because the second percent is taken of a larger amount.

"Per cent" is Latin for "by the hundred". Roman emperors levied a tax of 1100 on goods sold, and the idea stuck.

The % symbol evolved from a scribbled abbreviation of "per 100" — you can still see the two zeros in it.

Percentages above 100 are perfectly sensible: a price that triples has risen by 200%, because the increase is twice the original.

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