# How to Calculate Compound Percentage Changes

## A Soroban for some quick counting

## What is a compound percentage change?

We're all aware of percentage changes. Whether it's 25% off the cost of a new television in the Black Friday sales or a 5% rise in train fares (again), changing an amount by a percentage is an everyday skill. But what about compound percentage changes?

Imagine you put £100 into the bank in a savings account with a fixed 4% interest rate paid out annually. At the end of the year (assuming you haven't touched the original deposit) your money will have increased by 4%, giving you an extra £4 and a total of £104 in the account.

If you leave all of that money in the account for another year, what happens then? Do you get another £4 and a total of £108 in the bank? No. For the second year, not only do you get 4% on your original £100, which is still in the bank, but you also get 4% on the extra £4 that you earned through interest the previous year. 4% of £104 is £4.16 meaning at the end of the second year you will have £104 + £4.16 = £108.16 in your account. Assuming you don't touch the money at an point and that the 4% interest rate remains constant, you will earn more money each year as the amount in your account rises. This is compound interest.

Note: If you just received the £4 every year, this would be known as **simple** interest.

## How to calculate compound percentage growth

Let's look at how to calculate compound percentage growth (also know as compound interest when dealing with examples like ours).

As before, you start off with £100 in the bank account and a fixed interest rate of 4%. We could find 4% by dividing the £100 by 100 to get 1% and then multiplying this by 4. This is great for one year, but if we wanted to work out how much we're going to have in the account 5 or 10 years down the line, it's going to take a long time.

Instead, we are going to use something called the multiplier method. If we call our original deposit 100%, then after a 4% increase, we are going to end up with 104%. To calculate 104% of an amount we first convert the percentage into a decimal by dividing by 100, giving us 104 / 100 = 1.04. Multiplying by this 1.04 will increase an amount by 4% in one go.

For our example, we have £100 to start with so after one year we have £100 x 1.04 = £104. After another year we have £104 x 1.04 = £108.16, then £108.16 x 1.04 = £112.49 and so on. However, we can speed it up even more.

We are multiplying by the same multiplier, 1.04, once for every year that passes, so if we want to find the total several years further on, we can multiply by 1.04 that many times by using powers.

For example after 5 years, we will have £100 x 1.04 x 1.04 x 1.04 x 1.04 x 1.04 which is the same as £100 x 1.04^{5} = £121.67.

After 25 years we would have £100 x 1.04^{25} = £266.58. Imagine how long that would have taken if we worked out 4% for each year separately!

## Another example of compound percentage growth

Let's try another example of compound percentage growth.

A town's population is increasing by 12% every year. If it starts at 30 000 people, and assuming this increase remains constant, what will the population be in 6 years time? What about in 20 years time?

So, we are starting with 100% and want a 12% increase, hence we will end up with 112% which is 1.12 as a decimal.

Therefore after 6 years the population will be 30 000 x 1.12^{6} = 59 215.

After 20 years it will be 30 000 x 1.12^{20} = 289 389.

## What about compound percentage decreases?

A compound percentage decrease (also known as compound decay) is when an amount decreases by the same percentage multiple times. The method for finding this is very similar to finding an increase.

Suppose you bought a car for £20 000 and each year, the car's value drops by 15%. We want to find out how much the car will be worth in five years time.

We could find 15% of £20 000, subtract this, then find 15% of the new amount and so on, but again, this is going to to take a while. Instead, let's look at using multipliers as we did above.

If we start at 100%, a 15% reduction will leave us with 85%. So instead of thinking of this as finding a 15% decrease every year, we can instead think of it as finding 85%. 85% as a decimal is 85 / 100 = 0.85, so to find 85% we multiply by 0.85. To do this multiple times we use powers as we did above.

So, going back to our car example, after 5 years the value will be £20 000 x 0.85^{5} = £8 874.11.

After 10 years the value will be £20 000 x 0.85^{10} = £3 937.49.

Check out the video below for further examples.

## Compound interest on the DoingMaths YouTube channel

**© 2020 David**

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