In our experience at Hamburg Maths Tutors, questions about compound interest seem to cause some of the most confusion. However, when these questions are broken down they are really simple, as long as one understands the relationship between decimals and percentages.
Let’s start with how a simple percentage increase works:
If I have £1000 and I put it into a bank, that bank will offer me ‘interest’ on my initial deposit. This interest is simply the amount of money that the bank is willing to pay me so that I will give them my money in the first place. Usually this will be a percentage of the original sum. Let’s say, for example, HMT Bank offers me 3% interest to save £1000 over the course of a year. How much will I have at the end of the year?
There are several ways to approach this problem. For example, because I know that 100% of my money is £1000, if I divide £1000 by 100 and multiply the answer by 3, I can determine that 3% of £1000 is £30.
£1000 ÷ 100 = £10
£10 × 3 = £30
This is a great, simple way to calculate percentages, but it doesn’t really help us with compound interest calculations, as we will see.
For compound interest, we will use a quicker method to increase £1000 by 3%. Let’s first convert 3% into a decimal. Well, we know that to convert a percentage to a decimal you simply divide by 100. Therefore, 3% as a decimal would be 0.03.
3 ÷ 100 = 0.03
If we now take that 0.03 and multiply it by our original amount of £1000, we see that we again get £30.
0.03 × £1000 = £30
Now, here’s the trick. What we did then was to find 3% of £1000, but we really want to know what our final amount will be after the interest is added. Of course we could just add £30 to £1000, but again, that doesn’t help us too much when we come to perform compound interest calculations.
Instead, we need to realise that if we start with 100% and we want to increase by 3%, then we need to work out what 103% of our original amount is. To do that, we again need to convert the percentage to the decimal.
103 ÷ 100 = 1.03
After we have done that, we simply multiply the £1000 by the decimal representation of 103%.
£1000 × 1.03 = £1030
Now you can see that using the decimal step makes it as simple as doing one sum in the calculator. Great.
Now let’s assume we will keep our money in the same HMT Bank for 5 years. How much money (after interest) will be in the bank in 5 years time? Let’s take each year at a time.
Year 1: £1000 × 1.03 = £1030
Year 2: £1030 × 1.03 = £1060.90
Year 3: £1060.90 ×1.03 = £1092.73
Year 4: £1092.73 × 1.03 = £1125.51
Year 5: £1125.51 × 1.03 = £1159.27
Can you see that, if we combine the 5 years, all we are really doing is multiplying by 1.03 five times?
£1000 × 1.03 × 1.03 × 1.03 × 1.03 × 1.03 = £1159.27
Ok, if we know that we are going to multiply by a number 5 times, then all we need to do is use our understanding of indices to make the task simpler. Instead of multiplying by 1.03 on 5 separate occasions, let’s multiply £1000 by 1.03 to the power of 5.
£1000 × 1.035 = £1159.27
Amazing, we get the same answer for both methods but the latter method was MUCH faster. This method is particularly helpful if we are calculating for 10s, 100s or even 1000s of years. No one wants to use a calculator 1000 times to get an answer!
initial amount × (% as a decimal)^number of years = final amount
This general principle can be applied to all compound interest questions, even if the value is depreciating, which we will discuss in a following post.