Video Transcript
In this video, we’ll be exploring a particular area of exponential growth that we call compound interest. This is where for example you invest some money and then you get some interest added to it, then the original money and the interest and extra interest the next year.
This process of getting interest on interest is called compound interest. Work in this area is very similar to exponential growth but with a bit more emphasis on thinking about the percentages represented by the multipliers. For now, we’ll just consider the situation where interest is compounded once a year.
Let’s think about this question, a savings account offers interest compounded at a rate of three percent per annum, and that means three percent per year. If you invest five thousand dollars and leave the account untouched for seven years, how much money would be in it? Well first, we need to think about how we can add three percent to a number, and there are two main ways.
First, we could find three percent of the amount, so for example three percent of five thousand would be three over a hundred times five thousand, which is one hundred and fifty. And then we could add this interest to the original amount, so that’d be one hundred fifty plus the original five thousand gives us five thousand one hundred and fifty. So that’s a two-step process.
Now we were starting with a hundred percent of our original number and then we were adding another three percent on top of that, so we could just find a hundred plus three, that’s a hundred and three, percent of that number. So for example, the calculation of a hundred and three percent of five thousand would be a hundred and three over a hundred, a hundred and three percent, times five thousand, and that gives us the same answer of five thousand one hundred and fifty. Now that was just one-step process, so that was a bit quicker. Now this number here a hundred and three over a hundred, which is one point o three, that’s an important factor. We call that a multiplier.
Now if we use that multiplier, we can look at the seven-year problem. Now it’s worth pointing out at this stage that we don’t just work out three percent of five thousand and then add that seven times for the seven years. Because of the start of the second year, we’ve got five thousand one hundred and fifty dollars in our account, so we’re gonna be working out three percent of a bigger amount. And each year, we’ll have more and more money as the interest that we had last year was added and then the interest on that interest is added. We’re working out interest on larger and larger amounts.
Okay, so at the start we’ve got five thousand dollars, but by the end of year one we’ll have a hundred and three percent of our original amount. So we multiply by the multiplier one point o three.
Then in year two, we take the amount that we had at the end of year one, five thousand times one point o three, and we multiply that by one point o three so we add another three percent of that whole amount. Then in year three, we take the amount that we had at the end of year two, so that’s five thousand times one point o three times one point o three, and we multiply that by one point o three to add three percent to that amount.
And so on for year four, five, six, and seven. That means that in year one we’ve taken our investment and we’ve multiplied by the multiplier once or the multiplier to the first power. In the second year, we’ve multiplied by the multiplier twice or the multiplier to the second power. And in the third and fourth years, we’ve multiplied by the multiplier three and four times.
Now we can also say that at the start, let’s call that year zero, we had taken the investment and we’d multiply it by the multiplier no times. But the thing that we need to notice is whatever year we’re talking about, that’s how many times we’ve multiplied by the multiplier. In the second year, we’ve multiplied by the multiplier twice. In the fourth year, we’ve multiplied by the multiplier four times.
Now that means in this particular case, the amount in our account at the end of 𝑛 year, so the value of our investment after 𝑛 years, is the initial investment five thousand times the multiplier one point o three, adding three percent to a number, to the power of 𝑛.
And the question wants to know after seven years, so 𝑛 is seven. So the value after seven years is five thousand times one point o three to the power of seven, which my calculator tells me is six thousand one hundred and forty-nine point three six nine three two seven and so on. Well this is money so I’m gonna round it to two decimal places, which gives me six thousand one hundred and forty-nine dollars and thirty-seven cents.
Okay without all the explanation, let’s have a go at this question. An investment account pays one point six percent in interest per year. If you invest eleven thousand dollars, how much will you have after twelve years? Well the general format of our calculation is that the final amount we’ll end up with is the initial amount times the multiplier to the power of how many years we’re investing it for.
Well the initial amount is eleven thousand dollars and the number of years is twelve, so we just need to work out what the multiplier is going to be. Well we’re adding one point six percent to our account each year. So think about it; we’re gonna start up with a hundred percent; we’re going to add one point six percent.
That means we’re going to end up with a hundred and one point six percent of the amount that we started off with. And to work out a hundred and one point six percent of something, we do a hundred and one point six divided by a hundred times that amount.
This means our multiplier is one point o one six. So we can plug that into a calculation and then just have the numbers into our calculator. And we’re gonna round our answer to two decimal places because it’s money and that gives us thirteen thousand three hundred and eight dollars and thirteen cents.
Now compound interest questions also sometimes require you to think about the multipliers and the interest that they represent. So for example an investment account offers six percent compound interest compounded annually. If I invest some money for five years, what overall percentage will my investment increase by? Give your answer to two decimal places.
So I know that we’ve got six percent interest and we’re doing investment for five years. I’m just gonna define two variables then: 𝑎 is the initial investment amount and 𝑣 is the value of investment after five years. So 𝑣, the value of the investment is gonna be worth the initial amount times the multiplier to the power of five.
But what’s the value of the multiplier going to be? Each year we’re gonna start off with a hundred percent and we’re gonna add six percent so we’re gonna work out a hundred and six percent of the previous balance. And to work out a hundred and six percent of something, you multiply by a hundred and six over a hundred.
And a hundred and six divided by a hundred is one point o six. So our multiplier is gonna be one point o six. That completes a calculation 𝑣 is equal to 𝑎 times one point o six to the power of five Well I don’t know what the final value is gonna be, and I don’t know what the initial investment amount is going to be. But I know over the course of five years, I’m gonna be multiplying that initial investment by one point three three eight two two five five seven eight and so on.
So I need to work out what percentage increase is represented by multiplying my initial amount by the multiplier one point three three eight two two five five seven eight and so on. Now remember that that multiplier would have come from multiplying something by a hundred and thirty-three point eight two two five five seven eight over a hundred, which means that our percentage was a hundred and thirty-three point eight two two five five seven eight and so on percent. And that means that I started with a hundred percent and I was adding thirty-three point eight two two five five seven eight and so on percent.
Now we need to round that figure to two decimal places, and I can see that my overall increase was thirty-three point eight two percent.
Let’s look at another question like that again. A rare vintage car gains eighteen percent per year in value. What would be the percentage increase in value after nine years? Give your answer to two decimal places. So first of all, let’s work out the multiplier for adding eighteen percent to something. Well we’ve got a hundred percent plus eighteen percent. It means we’ve got to work out one hundred and eighteen percent of that number. And a hundred and eighteen percent of a number you can calculate by doing a hundred and eighteen divided by a hundred times that number.
So our multiplier is a hundred and eighteen over a hundred, which is one point one eight. Now I’m going to increase the value of the car by eighteen percent nine times over the course of nine years. So I’m gonna multiply by one point one eight nine times, so that’s multiplying the initial value by one point one eight to the ninth power.
Now when I work out one point one eight to the ninth power on my calculator I get four point four three five and then lots of other digits I don’t know what the initial value was. But whatever it was, I’m gonna be multiplying it by four point four three five four five three eight five nine and so on. And that’ll give me the final value of the car.
So this value here is my overall multiplier for the nine-year total. And that multiplier means I’m trying to find out four hundred and forty-three point five four five three eight five nine something percent of my original initial value.
And that represents the calculation that I was starting off with a hundred percent and then adding three hundred and forty-three point five five percent if I round to two decimal places. So the percentage increase in value after nine years is three hundred and forty-three point five five percent.
So the car will end up being worth four hundred and forty-three point five five percent of what it was worth originally, but the increase therefore will be three hundred and forty-three point five five percent. Let’s have a look at one last question then. An investment account offers interest compounded annually. The following formula is used to calculate the final value 𝑣 of the initial investment 𝑐 after 𝑡 years: 𝑣 is equal to 𝑐 times one point o one three to the power of 𝑡. What annual rate of interest is the account offering?
Well if 𝑣 is equal to 𝑐 times one point o one three to the 𝑡th power, then this value here, one point o one three, is our multiplier. That means that each year we’re multiplying the balance of the previous year by a hundred and one point three over a hundred; that’s the same as one point o one three.
And multiplying by a hundred and one point three over a hundred is the same as finding a hundred and one point three percent of something. And that means we were starting off with a hundred percent, but we were adding one point three percent so our annual rate of interest is one point three percent.
Just to finish off, let’s summarise what we’ve learned then. So for annually compounded interest, this is interest which is compounded just once per year, the final value of the investment is equal to the initial value of the investment times the multiplier to the power of 𝑛, where 𝑛 is the number of years that we make our investment for.
And as with all financial advice, there are extra caveats. So for example, this formula only works if we don’t make any extra deposits or withdrawals from the account during that time. So we just make our initial investment and we just leave it there for the whole period of the investment. So how do we work out the multiplier? Well for example, if we had a three percent annual interest rate, each year will be taking a hundred percent of our amount and then adding three percent giving us a hundred and three percent.
And to work out a hundred and three percent of something, you do a hundred and three over a hundred times that thing. And a hundred and three divided by a hundred is one point o three, so our multiplier in this case would be one point o three.
Now actually, we can summarise that process in a little formula. If we say the rate of interest is 𝑟 percent — and first of all we did a hundred percent plus three percent so that’s 𝑟 percent in our case — that gives us a hundred plus three, a hundred and three, and the multiplier is simply that number divided by a hundred. So if 𝑟 is the rate of annual interest then, a hundred plus 𝑟 over a hundred is the multiplier.
Or we could even split that up into a hundred over a hundred plus 𝑟 over a hundred and of course a hundred over a hundred is just one. So if you’re good at remembering formulae, then this is the formula that will tell you what the multiplier is: one plus the annual rate of interest divided by a hundred.
So now we can update our compound interest formula. The final value is equal to the initial investment times one plus 𝑟 over a hundred all to the power of 𝑛, where 𝑛 is the number of years that we’re making the investment for and 𝑟 is the annual rate of interest in percent.
