Per mille

When a hundredth is too rough, a thousandth steps in

You already know a percent: a hundredth of the whole. Sometimes a hundredth is too big a piece. When you want to talk about really small parts, per mille is handy: a thousandth of the whole.

The sign is ‰ — it looks like a percent with an extra zero. And that is how you remember it: 1 ‰ = 1/1000 = 0.001. Ten per mille make one percent.

Where will you meet per mille? On railway signs and roads: a gradient of 25 ‰ means the track climbs 25 metres for every 1 000 metres of length. In chemistry with weak solutions. And adults know it from alcohol in blood — for drivers it is 0 ‰.

You calculate it just like percentages, only instead of a hundred you divide by a thousand. Who handled the last lessons gets per mille for free — it is just a finer metre on the same ruler.

Jonáš and Filip go by train to the mountains and Filip points out of the window at a white sign by the track: “25 ‰ — what is that?” Jonáš: “Gradient. In a kilometre the track climbs 25 metres.” Filip counts: “We will go uphill about 8 kilometres… so 8 × 25 = 200 metres of climb?” Jonáš nods: “Exactly. That is why it hums here.” Filip thinks: “And why don’t they write it in percent?” — “25 ‰ is 2.5%. On gentle climbs you would keep getting decimals. Thousandths give nicer numbers.”

Calculating with per mille

Per mille of a number — the same ladder as with percents, just with a thousand:

Step 1: divide the base by 1 000 → you get 1 ‰. Step 2: multiply by the number of per mille.

How much is 6 ‰ of 5 000? One per mille: 5 000 : 1000 = 5. Six per mille: 6 × 5 = 30.

The path via a decimal: 6 ‰ = 0.006, so 0.006 × 5 000 = 30. Watch the zeros — per mille has one extra place after the point than the same percent.

The trio of questions works too: you look for the part (base × per mille), the base (ladder: divide by per mille, times 1000) and the number of per mille (part : whole × 1000).

Track gradient: the figure in ‰ = how many metres of height per 1 000 metres of length. A gradient of 12 ‰ on 3 km: 12 × 3 = 36 metres of climb.

A conversion bridge to write down: 1 ‰ = 0.1% = 0.001 of the whole. Ten per mille = one percent. A thousand per mille = the whole. If you are not sure, convert per mille to percent and count the old way.

How much is 8 ‰ of 2 500?

Path via 1 ‰: one thousandth of 2 500 is 2 500 : 1000 = 2.5. Eight per mille: 8 × 2.5 = 20.

Path via a decimal: 8 ‰ = 0.008. I count 0.008 × 2 500 = 20.

Result: 20. Check via percent: 8 ‰ = 0.8%, so a bit less than 1% (which is 25). And 20 really is a bit less than 25. That checks out.

A track climbs at 18 ‰. How many metres does it climb in 4.5 km?

Step 1: what the figure says: 18 metres of height for every 1 000 metres of length.

Step 2: the length is 4.5 km = 4 500 m, so 4.5 thousand.

Step 3: climb = 18 × 4.5 = 81 metres.

Result: 81 metres. Estimate for a check: on 5 km it would be 90 metres, on 4 km 72 — the result sits between them. That checks out. Notice how per mille simplifies the work: you just multiply by the number of kilometres.

In 500 g of solution there are 2 g of salt. How many per mille is that?

Step 1: the part is 2 g, the whole 500 g.

Step 2: the share: 2 : 500 = 0.004.

Step 3: I convert to per mille by multiplying by a thousand: 0.004 × 1000 = 4 ‰.

Result: 4 ‰. In percent that would be 0.4% — you see why per mille is better here: a whole number is easier to remember and say than a decimal. Fine concentrations are exactly its pitch.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Calculate per mille of a number — via one thousandth. Write only the number.

Practise the questions

On paper: thousandths in action

For each question write the conversion to percent too — so you see both units side by side.

1. Calculate 3 ‰ of 12 000 and 15 ‰ of 400. Both via one per mille.

2. A road climbs at 60 ‰ (a steep hill). How many metres does it climb in 2.5 km? And how many percent is that?

3. In 2 000 g of sea water there are 70 g of salt. How many per mille of salt does the water contain? (Real sea salinity is around 35 ‰ — compare.)

4. Fill the conversion table: 1% = ? ‰, 25 ‰ = ? %, 0.5% = ? ‰, 300 ‰ = ? %.

Now you 💪

1. I can say what per mille is: a thousandth of the whole, a tenth of a percent.
2. I calculate per mille of a number via 1 ‰ and via a decimal.
3. I understand a gradient in per mille: metres of height per kilometre of length.

What to take from this lesson

• Per mille = a thousandth of the whole. 1 ‰ = 0.1% = 0.001.
• The ladder works the same: base : 1000 = one per mille.
• A gradient of x ‰ = x metres of height per 1 km of length.
• Per mille is handy where percents would give small decimals.
• In conversions watch the zeros: a percent drops two places, per mille three.

© 2026 Ing. Martin Polak / AlgoRhino · 内容使用条款