Percentage increase and decrease with a multiplier

Increasing a number by a percentage means adding that percentage of the number on top of it, and it turns out to be a single multiplication. The usual hurdle is understanding why we multiply by 1.2 rather than 0.2, so that is what we look at first.

When do I use this?

When a price gets VAT added, when something goes up or down in a sale, when a bank adds interest, or when a question asks how many per cent something has risen or fallen. The background, that per cent means hundredths and that the amount you start with is the whole, is in what is a percentage?

Adding a percentage: VAT

When we buy an item, there is a tax on it that we call VAT. In the UK, VAT is 20%. Sometimes a price is shown "excluding VAT", which means the price shown is without the tax. (my Danish notes use 25%, because Danish VAT is 25%. The method is exactly the same, only the number changes.)

Let's say we are out shopping and want to buy something that costs £100 without VAT. When we get to the till we have to pay the VAT on top of it, so 20% is added. How much does the item cost with VAT?

We start by finding 20% of 100:

$$\frac{100}{100} \times 20 = 20$$

With 100 it is pretty straightforward, but what we do is to find 1% by dividing by 100 and then multiply by 20 to find 20%. It turns out that 20% of 100 is 20. Now we add these 20 on top of the price without VAT, and we have the price with VAT:

$$100 + 20 = 120$$

So the item costs £120 with VAT. When we have a price without VAT, we add 20% to get the price with VAT. More generally we can write

$$\text{price without VAT} \times 1.2 = \text{price with VAT}$$

We multiply by 1.2 because 20% is 0.2 as a decimal. So we can add 20% to a price by multiplying by 1.2: the 1 keeps the whole price, and the 0.2 adds the 20% on top. If you cannot quite see the multiplication by 1.2, you can also do it like this:

$$\text{price} + \left( \frac{\text{price}}{100} \times 20 \right) = \text{price with VAT}$$

Here we first find 20% of the price (that is what is inside the brackets) and then add it to the original price. It probably makes a bit more sense to do it this way, but it is a bit more of a bother than just multiplying by 1.2.

The number we multiply by, 1.2, is what the interest chapter calls the growth factor: 1 plus the percentage as a decimal. For a 3% rise the growth factor is 1.03, for a 50% rise it is 1.5.

Taking a percentage off

Taking a percentage off works the same way, just downwards. Let's say a scooter costs £2000 and we take 20% off. We can find 20% of the price and subtract it:

$$2000 - \left( \frac{2000}{100} \times 20 \right) = 1600$$

or just

$$2000 \times 0.8 = 1600$$

Taking 20% off leaves 80%, and 80% is 0.8 as a decimal. So the multiplier for a decrease is 1 minus the percentage as a decimal: 0.8 for 20% off, 0.9 for 10% off, 0.75 for 25% off.

Finding a percentage increase or decrease

The other way round, we sometimes know the two values and want the change in per cent. When we talk about a percentage increase or decrease, we look at the difference between two numbers. Let's say 500 has gone up to 700 and we want the increase in per cent. We first find the rise, and then we look at how much it has risen compared with the starting value. The difference here is

$$700 - 500 = 200$$

Now we look at how many per cent 200 is of 500, that is

$$\frac{200}{500} \times 100 = 40$$

So there has been a rise of 40%. More generally we can say that a percentage change from $x$ to $y$ can be found by

$$\frac{y - x}{x} \times 100$$

Notice that we always compare the change with the starting value $x$, not with the new value. The second step is the same move as in the guide about writing one number as a percentage of another.

Common mistakes

• Multiplying by 0.2 instead of 1.2. If we just multiplied the price by 0.2, we would find 20% of the price. Multiplying by 1.2 adds the 20% on top of the price, which is what an increase is.
• "Adding 20% and then taking 20% off are opposites." They are not. When we take 20% off the bigger price, we take off 20% of a bigger number than the 20% we added. To get back to the original price you divide by 1.2 instead, which is the guide about reverse percentages.
• Comparing the change with the wrong number. A percentage rise is the rise as a percentage of the value you started from. From 500 to 700 the rise is 200, and it is 200 out of 500, not 200 out of 700.

Related

• What is a percentage?
• Reverse percentages: finding the original value
• How to find a percentage of an amount
• Compound interest: the interest formula