Percentages: out of a hundred

Percentages are everywhere — sale prices, test scores, phone batteries, statistics in the news. The good news is that a percentage is just a fraction with a fixed bottom number: 100. Once you see 'per cent' as 'per hundred', finding percentages, discounts and increases becomes straightforward arithmetic you can do in your head or on paper.

Per cent means per hundred

The word per cent literally means 'per hundred'. So 45% means 45 out of every 100, which is the fraction 45/100 and the decimal 0.45. These three — percentage, fraction, decimal — are just different outfits for the same number.

To turn a percentage into a decimal, divide by 100 (move the point two places left): 45% becomes 0.45. To go back, multiply by 100: 0.7 becomes 70%. Being able to switch between them quickly is half the skill.

45% means 45 squares out of a hundred are shaded.

Finding a percentage of an amount

To find a percentage of a number, turn the percentage into a decimal and multiply. '25% of 80' becomes 0.25 × 80. Some percentages have handy shortcuts: 10% is just divide by 10, 50% is half, 25% is a quarter, and 1% is divide by 100. You can build others from these — 30% is three lots of 10%.

This is exactly what you're doing when you work out a tip, a tax, or how many questions you got right on a test.

Worked example

What is 15% of 60?

10% of 60 is 6 (divide by 10).
5% is half of 10%, so 5% of 60 is 3.
15% = 10% + 5% = 6 + 3.

Answer: 15% of 60 = 9

Discounts and increases

A discount takes a percentage off. A quick way: if something is 25% off, you pay 75% of the price (because 100% − 25% = 75%). So multiply the price by 0.75. Working with what you *keep* is faster than finding the discount and subtracting.

An increase works the same way in reverse. A 20% increase means you end with 120% of the original, so multiply by 1.2. This 'multiplier' trick handles tax, interest and price rises in one step.

Worked example

A jacket costs $40 and is reduced by 30%. What is the sale price?

A 30% discount means you pay 100% − 30% = 70% of the price.
70% as a decimal is 0.70.
Multiply: 0.70 × 40 = 28.

Answer: The sale price is $28

A common trap

Percentages of *different* amounts don't cancel out. If a price goes up 10% and then down 10%, you do not get back to where you started — the second 10% is taken from a bigger number, so you end up slightly lower. Always ask 'a percentage of what?' before combining them.

Whenever a question mixes two percentage changes, work them out one step at a time on the actual amounts, rather than adding or subtracting the percentages.

Check yourself

Try these. Pick an answer to see whether it's right and why.

What is 60% written as a decimal?

Per cent means per hundred, so divide by 100: 60 ÷ 100 = 0.6.

What is 10% of 250?

Finding 10% is dividing by 10: 250 ÷ 10 = 25.

A $50 item is 20% off. What do you pay?

20% off means you pay 80%: 0.80 × 50 = 40.

What is 75% of 40?

75% is three quarters; a quarter of 40 is 10, so three quarters is 30. (Or 0.75 × 40 = 30.)

A price rises 10%, then falls 10%. Compared with the start, the final price is:

The second 10% is taken from a larger amount than the first was added to, so you end up slightly below the original price.

In a nutshell

Per cent means per hundred: 45% = 45/100 = 0.45 — the same number three ways.
To find a percentage of an amount, turn it into a decimal and multiply; use 10%, 50%, 25% and 1% as shortcuts.
For a discount, multiply by what you keep (25% off → ×0.75); for an increase, multiply by more than 1 (20% up → ×1.2).
Percentage changes of different amounts don't cancel — always ask 'a percentage of what?'

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