Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 3 • Question 5

An investment account offers 2% compound interest. The following formula can be used to calculate the interest earned on the investment after 𝑛 years: 𝐼 = 𝐴₀ × ((1.02)^𝑛 − 1). 𝐼 is the amount of interest. 𝐴₀ is the amount initially invested. Sandy invests some money for 6 years and receives £2535.14 interest. Calculate the amount of money Sandy initially invested, giving your answer to the nearest hundred pounds.

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Video Transcript

An investment account offers two-percent compound interest. The following formula can be used to calculate the interest earned on the investment after 𝑛 years. 𝐼 equals 𝐴 naught multiplied by 1.02 to the power of 𝑛 minus one. 𝐼 is the amount of interest. 𝐴 naught is the amount initially invested. Sandy invests some money for six years and receives 2535 pounds and 14 pence interest. Calculate the amount of money Sandy initially invested, giving your answer to the nearest 100 pounds.

We’ve been told that Sandy invests her money in an account that offers two-percent compound interest. It might be tempting to try and recall the formula that we generally use when working with compound interest. However, we’ve actually been given a slightly different formula. It looks somewhat similar to the formula that we normally use, but this time it only calculates the interest earned. So rather than being a compound interest question, this is actually a substitution and solving question. Let’s see what else we know.

We’re told that Sandy invests her money for six years. So in this equation, our value of 𝑛 is going to be six. The amount of interest she receives at the end of that time is 2535 pounds and 14 pence. So in our equation, 𝐼 is going to be 2535.14.

Let’s substitute what we currently have into this formula then. If we replace 𝐼 with 2535.14 and 𝑛 with six, our equation becomes 2535.14 equals 𝐴 naught multiplied by 1.02 to the power of six minus one. This question is asking us to calculate the amount of money Sandy initially invested. That’s 𝐴 naught. So we’re going to need to solve this equation by performing a series of inverse operations.

To figure out what inverse operations we need to apply, let’s look at what’s happening to 𝐴 naught. It’s being multiplied by 1.02 to the power of six minus one. So to solve this equation for 𝐴 naught, we need to divide both sides by 1.02 to the power of six minus one. And of course, we could evaluate that by typing 1.02 to the power of six minus one into our calculator.

Alternatively, we can leave it as it is and work this out as one big sum at the end. Let’s do the latter. We get that 𝐴 naught is equal to 2535.14 all over 1.02 to the power of six minus one. Now at this point, you’ve probably noticed we don’t need the extra set of brackets around the 1.02 to the power of six, though it doesn’t actually make a difference whether we include them or not at this stage.

Our final step is to work out the value of 𝐴 naught by typing 2535.14 all over 1.02 to the power of six minus one into our calculator. This should be fairly straightforward on a scientific calculator. If you’re not working with one of those then, remember, this fraction line means divide. So you’ll need to work out 1.02 to the power of six minus one and then do 2535.14 divided by that number. If that’s the case, you’ll be performing the calculation 2535.14 divided by 0.126162 and so on. Either way, we should get a value for 𝐴 naught as 20094.256 and so on.

We need to round our answer to the nearest 100 pounds. This is the one hundreds column. So we look to the digit immediately to the right of this. That’s called the deciding digit. Remember, if that deciding digit is five or above, we round the number up. If it’s less than five, we round the number down. This time, the deciding digit is greater than five. So we’re going to round our number up. And this means the amount of money that Sandy initially invested was 20100 pounds, correct to the nearest 100 pounds.

And we could of course check our working by substituting this back into the original formula. That’s 20100 multiplied by 1.02 to the power of six minus one. That gives us an answer of 2535.86. Remember, we rounded, so it would make sense that our answer is just a little bit out from the original interest we were given. That was 2535.14. So again, we can see it’s fairly close, so we’ve probably done this calculation correct. The original amount of money she initially invested is 20100 pounds.

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