In the previous blog, I explained the difference between simple and compound interest. In this blog, I will show you how to make use of multipliers to answer compound interest questions efficiently. I will also introduce the concept of depreciation and show how you can use multipliers to answer those questions too.
What is a multiplier?
Simply, a multiplier is a number that you multiply by another number. But in the context of compound interest, I am talking about a specific use. This is best illustrated through an example.
Let’s say my annual salary is £40,000 and my boss is going to give me a 10% pay rise (lucky me). To calculate my new salary, I could find 10% of 40,000 and add it on, like this:
10% of 40,000 = 4,000
40,000 + 4,000 = 44,000
So my new salary is £44,000.
Obviously that was very easy to do that way, as the numbers were very nice to deal with. But the numbers won’t always be that kind to us. We can instead use multipliers to achieve the same result much quicker.
To do this, we need to convert 10% into a decimal (0.1). We then add this on to 1.
1 + 0.1 = 1.1
1.1 is our multiplier. If you multiply any number by 1.1, you will increase it by 10%.
So, if we apply this to our salary question, we get the same result, but much quicker:
£40,000 × 1.1 = £44,000
Applying multipliers to compound interest problems
The salary question in the previous section was very simple, so arguably there wasn’t much benefit to using a multiplier.
Where multipliers really start to earn their corn is in compound interest questions, where you need to add a percentage on a number of times. Let’s set up another example:
Let’s say I am going to invest £1,000 for 5 years with a bank which offers 5% compound interest per year. How much money will I have in my bank account at the end of the 5 years?
5% = 0.05
1 + 0.05 = 1.05
So our multiplier will be 1.05
We need to multiply our original amount by 1.05 five times, because we are investing for 5 years.
So, we can take our multiplier to the power of 5, and this will do it!
1,000 × 1.055 = 1276.2815625
So we will have £1,276.28 in our account after the 5 years.
The general rule can be written in this formula, using the same terminology as the Edexcel GCSE formula book:
where P = principal (i.e. original) amount, r = % rate of interest and n = number of years
Let’s do another example, using our new formula.
Compound interest example 1
I have £4,500 and will invest it in a bank account offering 2.4% compound interest per year for 3 years.
How much money will I have?
P = 4,500
r = 2.4
n = 3
So, using the formula:
It’s really quite easy when you know how isn’t it!
Depreciation
Depreciation is basically the opposite of interest. So instead of increasing in value, it decreases in value! Not often what we want to see but it’s a fact of life. A car, for example, will have a price when new, but depreciate in value over time because of wear and tear.
We can solve depreciation questions by using multipliers too. It’s almost exactly the same method as for compound interest, except instead of adding to 1, you subtract from 1. Let’s do an example.
Depreciation example
Fred buys a car for £800. The car depreciates at a rate of 12% per year. What will be the value of Fred’s car after 4 years?
12% as a decimal is 0.12
We subtract this from 1 to get the multiplier:
1 – 0.12 = 0.88
Then multiply our original value by the multiplier, to the power of the number of years (4):
800 × 0.884 = £479.76
Or, by tweaking the formula we used for compound interest:
where P = principal (i.e. original) value, r = % rate of depreciation and n = number of years
N.B. Depreciation is always assumed to be compound. While you could be asked to calculate both simple and compound interest, there is no such thing as simple depreciation.
Other examples unrelated to money
This method isn’t only applicable for questions involving money. We could also apply the same rule to these two examples:
Question:
A colony of ants grows by 2% every day.
Today, the population of the ant colony is 150.
What will the population of the ant colony be after 10 days?
Answer:
150 × 1.0210 = 182 ants (rounding down)
Question:
A bouncy ball is dropped from a height of 1.2 metres.
Every time the ball hits the ground, it bounces up to 75% of the height it dropped from.
How high will the ball bounce up after its third bounce?
Answer:
1.2 × 0.753 = 0.51 metres (to 2 decimal places)
I hope you have found this explanation useful. To practice GCSE-style questions involving both interest and depreciation, try our worksheet.
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