How to calculate percentages

What a percent is

Per cent means per hundred. A percent is a ratio that compares a part to a whole scaled to 100, which makes different amounts easy to compare. Scoring 18 out of 24 and 30 out of 40 sound different, but both are 75%.

Every percent can be written three ways. 25% is the fraction 25/100, which simplifies to 1/4, and the decimal 0.25. To turn a percent into a decimal, divide by 100, which moves the decimal point two places left: 7% is 0.07, 150% is 1.5, and 0.5% is 0.005. To go back, multiply by 100.

Percents can be more than 100. A price that is 150% of the old price is one and a half times as much.

The three basic percent questions

Almost every percent problem is one of three questions about the same relationship, part = percent × whole:

• Find the part: what is 35% of 80? Multiply: 0.35 × 80 = 28.
• Find the percent: what percent of 24 is 18? Divide the part by the whole: 18 ÷ 24 = 0.75, which is 75%.
• Find the whole: 12 is 40% of what number? Divide the part by the percent: 12 ÷ 0.40 = 30.

Worked example: a percent of a number

For mental math, break the percent into easy pieces. 10% of a number is that number divided by 10, and 5% is half of 10%.

Worked example: percent increase

Percent change always compares the change with the original value, the amount you started from. Dividing by the new value is the most common mistake.

Discounts, tax, tips and multipliers

A faster way to handle increases and decreases is a single multiplier. Increasing by r% means multiplying by 1 + r/100; decreasing by r% means multiplying by 1 − r/100.

So a 20% discount multiplies the price by 0.80, an 8.25% sales tax multiplies it by 1.0825, and an 18% tip multiplies the bill by 1.18. A $60 jacket at 20% off costs 60 × 0.80 = $48. With 8.25% tax on that, the total is 48 × 1.0825 = $51.96.

Worked example: finding the original amount

Reverse percent problems give you the amount after a change and ask for the amount before it. Dividing by the multiplier undoes the change.

Common percent mistakes

Percents are full of traps that look reasonable at first:

• Dividing by the new value instead of the original when finding percent change.
• Undoing a percent change with the same percent. A 20% cut followed by a 20% rise gives 0.80 × 1.20 = 0.96 of the start, a 4% loss overall.
• Adding successive percents. Two 10% increases make 1.10 × 1.10 = 1.21, a 21% increase, not 20%.
• Writing 5% as 0.5 instead of 0.05. Always move the decimal point two places.
• Confusing percent change with percentage points. A rate going from 4% to 6% rose by 2 percentage points, which is a 50% increase.
• Thinking a percent increase cannot go past 100%. A positive amount cannot fall by more than 100%, but it can rise by any percent: doubling is a 100% increase, and tripling is a 200% increase.

How to practise and check your work

Estimate before you calculate. 35% of 80 should be a bit more than a third of 80, so around 27 or 28; an answer of 2.8 or 280 means the decimal point slipped. Benchmarks help: 50% is half, 25% is a quarter, 10% is a tenth, 1% is a hundredth.

Practise with real receipts, sale signs and nutrition labels. For any discount, work out the sale price two ways, by subtracting the discount and by using the multiplier, and check that the answers agree.