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Foundations · Numbers

Percentages

Percent of a number, percentage increase and decrease, the multiplier method, and finding the original value after a change.

First, the rules

Per cent means per hundred

25% is 25 out of every 100 — the fraction 25/100, the decimal 0.25. Once a percentage is a decimal, every percentage question is a multiplication.

★ Three forms
Example 1
35% = 35100 = 720 = 0.35
★ Percent of a number
Example 1
p% of N = N × p100 — 15% of 80 = 80 × 0.15 = 12; or 10% is 8 and 5% is 4, so 12.
What percent?
Example 1
partwhole × 100 — 18 out of 40 = 0.45 = 45%.
★ Increase / decrease: the multiplier
Example 2
+ p% → × (1 + p100) · − p% → × (1 − p100)
+ 20%: × 1.2. − 35%: × 0.65. Two changes in a row: multiply the multipliers.
★ Reverse percentage
Example 3
The new value is (1 ± p/100) of the original, so divide: after + 25% a price is 60 → original 60 ÷ 1.25 = 48.
Trap
Example 3
Taking 25% off 60 does NOT undo adding 25% to 48: 60 × 0.75 = 45, not 48.

See it

The hundred-piece bar

The bar is the number at 100%. Slide the percent to shade part of it, stretch it for an increase, cut it for a decrease — and see why a reverse question divides.

Step by step

Worked examples

1

Of, and what percent

basics

(a) Find 12.5% of 640. (b) A class of 32 has 20 girls. What percentage are boys?

Ask yourself: 12.5% is an eighth. For (b), find the part you were asked about first.

Show the solution
  1. (a) 12.5% = 18, and 640 ÷ 8 = 80 (or 640 × 0.125).
  2. (b) Boys: 32 − 20 = 12. 1232 = 0.375 = 37.5%.

Answer 80 · 37.5%

2

Multipliers, twice

percentage change

A jacket costs 120. It goes up 15%, then the new price is reduced by 15%. What is the final price, and why is it not 120?

Ask yourself: Write each change as a multiplier and apply them in order.

Show the solution
  1. Up 15%: × 1.15 → 138.
  2. Down 15% of the new price: × 0.85 → 138 × 0.85 = 117.3.
  3. Combined multiplier 1.15 × 0.85 = 0.9775, a 2.25% drop overall: the 15% off is taken from a larger number than the 15% on was added to.

Answer 117.30 — a net 2.25% decrease

3

Going backwards

reverse percentage

After a 30% discount a bike costs 371. What was the original price?

Ask yourself: The 371 is 70% of the original, not 130% and not "371 plus 30%".

Show the solution
  1. Multiplier for − 30% is 0.7, so 371 = 0.7 × original.
  2. Original = 371 ÷ 0.7 = 530.
  3. Check: 530 × 0.7 = 371 ✓. (371 × 1.3 = 482.3 would be wrong.)

Answer 530

Practice: percentages

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Percent of a number, increase and decrease with multipliers, reverse percentages and conversions. Your best score is kept in this browser only.

About this lesson

Percentages: what per cent means, finding a percentage of a quantity, percentage change with a single multiplier, reverse percentages (finding the original price), and converting between percentages, fractions and decimals. A percentage bar you can move, worked examples and practice questions with explanations. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.