Simon invested some money in a bank account offering two percent compound interest per year. The interest is paid into his account at the end of every year. After the first year of interest being paid into his account, he withdraws 100 pounds. He does the same after the interest has been paid at the end of the second year. After he has withdrawn the second 100 pounds, he has 10202 pounds left in the account. Calculate how much he originally invested.
Let’s call the initial investment 𝐼. That’s what we’re looking for. What can we say about the initial investment after the first year? The initial investment, 𝐼, is being multiplied by 1.02. We can think about it like this. The whole number one represents our initial investment to which we’re adding two percent interest, which we write in decimal form as 0.02.
Combining them together, we get the initial investment plus the two percent interest, 1.02. And then we subtract 100 pounds. The end of the first year is the initial investment times 1.02 minus 100. How should we calculate the money he had after the second year? We start with the money from year one, multiply that value by 1.02, and then subtract 100.
We already know how much money he had in the account at the end of year two, 10202 pounds. But how much money did Simon have at the end of the first year? What should we plug in here? We plug in the expression that we wrote for the first year: 𝐼 times 1.02 minus 100.
We now have an equation that we can solve for 𝐼, the initial investment. 𝐼 times 1.02 minus 100 times 1.02 minus 100 equals 10202. We need to expand our multiplication over the brackets. If we multiply 𝐼 times 1.02 times 1.02, that’s the same thing as saying 𝐼 times 1.02 squared, minus 100 times 1.02, which equals 102, minus 100 equals 10202. If we combine 102 and 100, we get 202. And now we say that the initial investment times 1.02 squared minus 202 equals 10202.
To solve for the initial investment, we need to get 𝐼 by itself. We can do that by adding 202 to both sides of the equation. On the left, we have 𝐼 times 1.02 squared equals 10404. In our final step, we want to divide both sides of the equation by 1.02 squared. On the left, we’re left with 𝐼. And on the right, 10404 divided by 1.02 squared equals 10000. 𝐼 equals 10000, which makes Simon’s initial investment 10000 pounds.