Module 2 · Money maths
Percentages you actually need
How to find a percent of any amount, measure a change, tell percent from percentage points and check percentages in your head.
This lesson is not translated into your language yet, so it is shown in English.
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In this lesson
- Calculate a percentage of an amount and the percentage change between two numbers
- Tell a change in percent from a change in percentage points, and read "per year" rates correctly
- Explain why a 50% rise followed by a 50% fall leaves you with less than you started with
- Estimate percentages in your head with the 10% and 1% shortcuts
Aleks bought some shares for €100. In January they rose 50%. In February they fell 50%. “Never mind,” he says, “I’m back where I started.” He is not: he has €75. By the end of this lesson you will see exactly why, and you will be able to check claims like his in your head.
A percent is a share of 100
“Per cent” means “for each hundred”. A such as 15% means 15 out of every 100, which is 0.15 as a decimal. To find a percent of an amount, turn the percent into a decimal and multiply.
- 20% of €50 = 0.20 × 50 = €10
- 3% of €1,200 = 0.03 × 1,200 = €36
- 0.5% of €400 = 0.005 × 400 = €2
You can do most of this in your head. 10% means moving the decimal point one place left: 10% of €340 is €34. 1% means moving it two places: €3.40. Build everything else from these pieces:
| You want | Build it from | 340 example |
|---|---|---|
| 5% | half of 10% | €17 |
| 15% | 10% + 5% | €51 |
| 3% | 3 × 1% | €10.20 |
| 25% | a quarter | €85 |
Estimating first is a good habit. If a calculator says 15% of €340 is €510, your quick estimate (about €50) tells you something went wrong.
Measuring a change: always divide by the start
To measure how much something changed, use this rule:
percentage change = (new − old) ÷ old × 100
A price goes from €80 to €100. The change is €20, and 20 ÷ 80 = 0.25, so the price rose 25%. Now the price goes back from €100 to €80. The change is again €20, but 20 ÷ 100 = 0.20, so it fell 20%. Same €20, different percentages, because the starting point changed.
The same rule works backwards. If a ticket costs €50 after a 25% rise, the old price was 50 ÷ 1.25 = €40. Taking 25% off €50 (which gives €37.50) is a common mistake, because it uses the wrong starting point.
A useful trick is to think in multipliers. A rise of 10% means × 1.10. A fall of 30% means × 0.70. To combine changes, multiply them. A coat with 30% off, after a later 10% price rise, costs 0.70 × 1.10 = 0.77 of its first price: that is 23% off, not 20%.
Percent or percentage points?
When a rate itself changes, there are two ways to describe it. Say a savings rises from 2% to 3%. The difference, 3 − 2 = 1, is 1 . Compared with the old rate, the change is 1 ÷ 2 = 50%. Both statements are true. A careful writer says “points” when subtracting two percentages and “percent” when comparing them.
Rates also come with a time period. Most interest rates, and (the rise in prices), are given per year, even when the label only says “p.a.” (per annum). A rate of 1.5% a month is not small: 1.5 × 12 = 18% a year, and a little more once interest is charged on interest. Before comparing two rates, put them on the same time scale.
Growth that builds on itself
Suppose €1,000 grows 10% a year. If you only ever earn 10% of the first €1,000, you get €100 each year: €1,300 after 3 years. If each year’s growth is added first, the next 10% is taken from a bigger amount: 1,000 → 1,100 → 1,210 → 1,331. The extra €31 is growth on growth. Over many years this gap becomes huge. It is called , and it is the subject of the next lesson.
Try it yourself
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