Video Transcript
A bank offers its customers an account with an interest rate of three percent compounded annually. Write an equation to represent 𝑆, the value of an investment that is left in the account for 𝑡 years. Let 𝑆 sub zero represent the initial investment. If an amount of money is saved in the account, what will the percentage increase in its value be, provided it is left in the account for five years? Give your answer to the nearest percent.
Let’s start by looking at the first question. And we’ll consider the information that we’re given, namely that there is a bank account which has an interest rate of three percent compounded annually. We should remember the difference between simple and compound interest. Compound interest is calculated on the principle or starting amount and also on the interest of previous periods. If we had instead been asked to work out the simple interest, we’d be working out the interest only on the starting amount.
In order to help us answer the question, we should recall the formula to calculate compound interest when the interest is compounded annually, 𝑣 equals 𝑝 times one plus 𝑟 over 100 to the power of 𝑦, where 𝑣 is the value of the investment, 𝑝 is the principal or starting amount, 𝑟 is the interest rate, and 𝑦 is the number of years after the initial investment. So let’s take this formula and fill in any values or variables that we’re given.
We’re told to use 𝑆 to represent the value of an investment. So our equation will start with 𝑆 equals. 𝑆 sub zero should be used to represent the initial investment. The interest rate is three percent. And the number of years is represented by the variable 𝑡. If we look at the equation, we can simplify one plus three over 100 as 1.03. We can, therefore, answer the first question that the equation can be given as 𝑆 equals 𝑆 sub zero multiplied by 1.03 to the power of 𝑡. Now that we have an equation like this, if we were required to find the value of one of these three unknowns, we could do so given the value of the other two unknowns.
We can now take a look at the second question. In this question, we’re asked to find the percentage increase in a value of money given it’s left in the account for five years. We can use the equation that we worked out in the first part of this question to help us.
We’re told that the money is in the account for five years, so that means the value of 𝑡 could be replaced with five. This will give us 𝑆 equals 𝑆 sub zero multiplied by 1.03 to the power of five. At this point, we haven’t got enough information to either work out the value of 𝑆, the value of the investment, or 𝑆 sub zero, the starting amount. However, we’re just asked to find the percentage increase. And the first step will be to actually work out the value of 1.03 to the power of five.
Using our calculator, we can evaluate this as 1.15927 and so on. So let’s have a think about what this means in the context of the question. The starting amount, 𝑆 sub zero, is multiplied by this value, 1.159 and so on, to give us the value of the investment as 𝑆. If we break down the value 1.15927 into one plus 0.15927, then this will allow to more easily see how the starting amount 𝑆 sub zero has increased.
The proportion that the starting amount is increased by is given by this value, 0.15927 and so on. The question is, what is this value as a percentage? Well, it can be written as 15.927 and so on over 100. So as a percentage, this means it’s 15.927 percent. However, we need to give our answer to the nearest percent. This means we need to check our first decimal digit to see if it’s five or more. And as it is, then the answer rounds up to 16 percent.
So we’ve found that after five years, any amount of money in this account will have increased by 16 percent. Notice that it’s different to the interest rate as the interest rate was three percent.
