Mental arithmetic

Percentages in your head – ten per cent, VAT and the common traps

Percentages are the arithmetic people use most and get wrong most. A walk-through of the techniques – ten per cent as the starting point, the trick of turning the sum around, VAT in both directions, per cent against percentage points, and why up and down by the same amount does not return you to the start.

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What this covers

Ten per cent, and how everything else is built from it
Percentages can be turned around – the most useful trick
Discounts – work out what you pay, not what you save
Value added tax, and trap number one
Per cent and percentage points
Up and down by the same amount does not return you to the start
Growth factors, and what happens when percentages are chained

What this covers (cont.)

The rule of 72 for doubling time
Working backwards

Ten per cent is the key

Almost all mental percentage work starts in the same place: move the decimal point one step and you have ten per cent. The rest is building.

Everything built from ten per cent of 640

10% = 64Move the point one step left. This is the whole foundation.
5% = 32Half of ten per cent.
20% = 128Double ten per cent.
15% = 96Ten plus five: 64 + 32.
1% = 6.4Move the point two steps. Useful for the awkward percentages.
35% = 224Three tens and a five: 64 + 64 + 64 + 32.

Note that none of the steps is a multiplication. It is moving a decimal point, halving, and a little addition. That is the whole reason percentages suit mental arithmetic: they can be done without multiplying at all, as long as you build from the ten.

Percentages can be turned around

This is the trick that saves the most time, and it is almost unknown.

x per cent of y is always the same as y per cent of x. If the sum faces the wrong way, turn it round.

8% of 50 looks like something you have to calculate. Turned round it is 50% of 8 – half of 8, which is 4.
4% of 75 → 75% of 4 = three quarters of 4 = 3.
18% of 50 → 50% of 18 = 9.
It works because both sides are the same product divided by a hundred. Order does not matter in a multiplication, and a percentage is a multiplication.

Discounts: work out what you pay

The common mistake here is not mathematical, but unnecessary work.

Two routes to the same price

Discount	The long way	The short way
30% off 450	find 135, subtract from 450	pay 70%: 315
20% off 899	find 179.80, subtract	pay 80%: 719.20
25% off 320	find 80, subtract	pay 75%: 240

Both columns give the same answer, but the right-hand one has one step fewer – and one step fewer is half the work when the work is two steps. If all you want is roughly what something costs, "pay 70 per cent" gives you the number directly.

Value added tax

This is the trap most people fall into, and it looks entirely right.

To remove 25 per cent VAT you cannot subtract 25 per cent. You have to multiply by 0.8.

Forwards: 500 before VAT becomes 500 × 1.25 = 625 with VAT.
Back, correctly: 625 × 0.8 = 500.
Back, wrongly: 625 − 25% of 625 = 625 − 156.25 = 468.75. That is 31 too little.
The reason is that the 25 per cent was calculated on the net price, while you are now subtracting it from the gross price – a bigger number. The percentage is the same, but the base has been swapped.

In a price carrying 25 per cent VAT, the tax is 20 per cent of the total – one fifth.

Of 625, 125 is VAT. That is a fifth of 625, not a quarter. If you need the tax inside a price, divide by 5. If you need the price without it, multiply by 0.8 – or subtract the fifth, which is the same calculation.

Per cent and percentage points

This is not pedantry. They are two different numbers, and the gap is often large.

A rate goes from 4 per cent to 6 per cent

Percentage points

2 percentage points

The difference between the two percentages
6 − 4 = 2
Used when comparing two percentages

Per cent

A rise of 50 per cent

The change measured against where it started
2 divided by 4 = a half
This is what your interest bill actually does

Both numbers are correct, and they describe the same event. Which is exactly why confusing them is useful to anyone wanting to overstate or understate: "only two percentage points" and "fifty per cent more expensive" are the same rate rise. When you read a figure about a change in something that is itself measured in per cent, it is worth checking which of the two is on the page.

Up and down are not symmetrical

This surprises most people the first time, and it has practical consequences.

100, up 25 per cent and down 25 per cent

Start: 100The starting point.
Up 25% → 125The 25 per cent is taken of 100, and comes to 25.
Down 25% → 93.75Now the 25 per cent is taken of 125, and comes to 31.25. The fall is bigger than the rise was.

You do not lose money because of the mathematics, but because the second percentage has a bigger base than the first. To get back to 100 from 125 you need to fall 20 per cent, not 25. The rule is that a rise of p per cent is undone by a fall of p divided by 100 plus p – for 25 that is 25/125, or 20 per cent.

Growth factors

The way out of this mess is to stop adding and subtracting, and start multiplying.

The change as a single number

Change	Multiply by	Note
Up 25%	1.25	adds a quarter
Down 20%	0.8	the counterpart to 1.25
Up 10%	1.1
Up 10% twice	1.21	that is 21 per cent, not 20
Down 20%, then up 25%	1.0	back where you started
Up 50%, then down 50%	0.75	25 per cent below the start

The advantage is that several changes in a row simply multiply together, in any order. The fourth row is why "we grew ten per cent two years running" is not twenty per cent: the second year you grow off a bigger number. The bottom row is worth remembering the next time something falls fifty per cent and rises fifty per cent again.

The rule of 72

An estimate, not a formula – but a good estimate, and the only thing you need to judge a rate of return in your head.

72 divided by the rate gives the years to double

Rate	The rule says	Actual factor after those years
3%	24 years	2.03
4%	18 years	2.03
6%	12 years	2.01
8%	9 years	2.00
10%	7.2 years	1.99
12%	6 years	1.97

The right-hand column is what the money is actually multiplied by after the number of years the rule gives. Between 3 and 12 per cent it lands within a few per cent, and it is most precise around 8. Outside that range it gets worse, and it says nothing about what happens if the rate changes along the way.

Working backwards

The last case: you know the percentage and the result, but not what you started with.

Three backwards questions

35 is 70% of what?Divide by the growth factor: 35 ÷ 0.7 = 50. Or reason that if 70% is 35, then 10% is 5, and 100% is 50.
What percentage is 18 of 24?18 divided by 24 is three quarters, so 75%.
What percentage is 45 of 360?45 divided by 360 is an eighth, so 12.5%.

The bottom two are why it pays to know the fractions as percentages – then the answer is recognition rather than calculation. The first is the reason to think in growth factors: going forwards you multiply, going back you divide by the same number.

What actually needs practising

Ten per cent, and the four steps from it. Half, double, ten plus five, and the point two steps over. With those, you have covered most of what you meet.

Key points

Find ten per cent by moving the decimal point, and build everything else from there.
Percentages can be turned around: 8 per cent of 50 is the same as 50 per cent of 8, which is 4.
For a discount it is quicker to work out what you pay than what you save.
To remove 25 per cent VAT you multiply by 0.8 – you do not subtract 25 per cent.

Key points (cont.)

In a price carrying 25 per cent VAT, the tax is 20 per cent of the total – one fifth.
From 4 to 6 per cent is 2 percentage points, but a rise of 50 per cent.
Up 25 per cent and down 25 per cent leaves you at 93.75 – not 100.
A growth factor is multiplied. Ten per cent twice is 21 per cent, not 20.

Key points (cont.)

The rule of 72 estimates doubling time, and holds well for rates between 3 and 12 per cent.

Questions this answers

How do you work out percentages in your head?
Why is ten per cent the starting point for everything?
Is 8 per cent of 50 the same as 50 per cent of 8?
What is the fastest way to work out a discount?
How do you find the price before VAT?
What is the difference between per cent and percentage points?

Questions this answers (cont.)

Why do you not end up back where you started when something rises and falls by the same percentage?
What is a growth factor?
What is the rule of 72, and how accurate is it?
How do you work backwards from a percentage?

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