Video Transcript
An account pays five percent interest every three months. On an initial deposit of 𝑀 dollars, what is the amount after 𝑡 years?
In this compound interest question, we’re given a very important statement that the account pays five percent interest every three months. The really important thing to note here is that five percent interest every three months is not the same as five percent annual interest compounded every three months. If we did have an interest rate compounded every three months, then we would use the slightly more complex compound interest formula. But we don’t need to use that here.
So let’s have a look at the formula on the left-hand side. Usually, when we use this formula 𝑉 equals 𝑃 times one plus 𝑟 to the power of 𝑛, the value of 𝑉 represents the value of the investment, 𝑃 represents the principal or starting amount, 𝑟 is the interest rate, and 𝑛 is the number of years. In this problem, however, we’ll have slightly different meanings for some of these letters.
We’re given that the interest rate is an interest rate every three months. And so we can’t have an exponent that is just the number of years. But we’ll also need to consider how many lots of three months there are in each of these years. And so the formula that we need to use must have an exponent of four 𝑛 because there are four lots of three months in every year.
Let’s now note down the information from the question so that we can assign the different values to each of these letters. The value of the investment is what we wish to work out. So let’s keep that as 𝑉. Next, we’re given that the initial deposit is 𝑀 dollars, so that means that 𝑃 is equal to 𝑀. The interest rate we’re given is five percent. And remember that we can write that as a fraction as five over 100 or as a decimal of 0.05.
Finally, we need to work out the amount after 𝑡 years, so that means that our value of 𝑛, the number of years, will be equal to 𝑡. We can then substitute these values into our amended formula. This gives us 𝑉 equals 𝑀 times one plus 0.05 to the power of four 𝑡. We can then simplify this to 𝑉 equals 𝑀 times 1.05 to the power of four 𝑡. And so this expression on the right-hand side would allow us to calculate the value of the investment given any starting amount of 𝑀 dollars.
