Video Transcript
Mason saves 20 dollars every month in an account that pays an annual interest rate of four percent compounded monthly. How much will be in Mason’s account after four years of regular saving? Give your answer to the nearest cent. If the interest was compounded quarterly, how much would be in the account after four years?
There are two parts to this question. Firstly, when the annual interest is compounded monthly and secondly when it is compounded quarterly. We will begin by clearing some space and considering the first scenario. We begin by writing the annual interest rate of four percent as a decimal, 0.04. As this is compounded monthly, the monthly interest rate is 0.04 divided by 12. This is equal to one over 300. As Mason saves 20 dollars every month, we can begin to model this as a geometric sequence with first term 20. We know that any geometric sequence has a common ratio, or multiplier, between consecutive terms. In this case, it will be equal to one plus one over 300. This is equal to 301 over 300.
Since we have a common ratio, or multiplier, we can model this as a geometric sequence with first term 20 and common ratio 301 over 300. The second term is equal to 20.06 and so on. We are asked to calculate the amount in Mason’s account after four years. And since the interest is compounded monthly, there will be four multiplied by 12, or 48, monthly payments. Our geometric sequence has first term 20, common ratio 301 over 300, and 48 terms. The sum of the first 𝑛 terms of a geometric sequence can be calculated using the formula 𝑎 sub one multiplied by 𝑟 to the 𝑛th power minus one all divided by 𝑟 minus one. In this question, we want to find the sum of the 48 terms.
Substituting our values into the formula, we have 20 multiplied by 301 over 300 to the power of 48 minus one all divided by 301 over 300 minus one. Typing this into our calculator gives us 1039.1920 and so on. We are asked to give the answer to the nearest cent. This is equal to 1039 dollars and 19 cents. After four years of regular saving, the amount of money in Mason’s account is 1039 dollars 19 cents.
Let’s now consider the second part of the question. This states “If the interest was compounded quarterly, how much would be in the account after four years?” We still have an annual interest rate of four percent. This time, as the interest is compounded quarterly, the quarterly interest rate is equal to 0.04 divided by four. This is equal to one over 100, or one hundredth. As a decimal, we can write this as 0.01. The multiplier, or common ratio, here will be equal to one plus 0.01, which is equal to 1.01. Across the four years, there will be four multiplied by four, or 16, quarterly payments. And at this stage, we might think we have all the information to substitute into our formula. We know that 𝑟 is equal to 1.01, and we know that 𝑛 is 16. We might think that we simply substitute 𝑎 as 20 dollars once again.
However, this was the amount that Mason saves every month. There are three months in a quarter, which means that the first term of our sequence is 20 dollars multiplied by three. This is the amount of money that Mason will save between each quarterly interest payment. The first term 𝑎 sub one in our sequence is equal to 60 dollars. Substituting in our values of 𝑎 sub one, 𝑟, and 𝑛, we have 60 multiplied by 1.01 to the 16th power minus one all divided by 1.01 minus one. This is equal to 1035.4718 and so on. Once again, rounding to the nearest cent, we get 1035 dollars and 47 cents. This is the amount of interest in Mason’s account if the interest is compounded quarterly.
We notice this is slightly less than the answer in the first part of our question. And we can therefore conclude that if the interest is compounded monthly rather than quarterly, then Mason will earn more interest across the four-year term.
