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## Lecture 1: Illustrative Example - Simple Versus Compound Interest

Before we dive into all the formulae and technicalities of simple interest, we’re going to look at an example that illustrates simple and compound interest and how they differ from each other.So here we have two people; we have Jan, and we have Sarah. They both borrowed R10 000, so I’m going to put it at the beginning of their timeline.

So Jan borrows R10 000 and so does Sarah. Now, Jan borrows the R10 000 at a 10% simple interest rate. (We will explain what rates are later) That’s the rate that is used for hire purchase like when you buy a TV, and you pay later, or for odd period calculations, which we will look at later.

Sarah borrows R10 000 at 10% compound interest. Compound interest is usually what is used for investments and loans, and it is the bulk of this course, which we will explore in the later sections.

Now both of them borrow R10 000 that they have to pay back in three years. So, I’m going to put each year on my timeline. Notice that this is the end of year 1, year 2 and year 3. I’m going to do the same thing for Sarah.

Now, they have to pay back their money at the end, but they don’t just pay back the R10 000. They pay back some extra money. So that the person who they borrowed from is actually benefiting from this whole situation, let’s see how we work out the extra money, that’s called the interest.

So let’s look at Jan. At the end of year 1, let’s see how much he owes. He owes R10 000, but not just the R10 000; he owes a portion of R10 000. And the way we work it out using simple interest, is we say he owes 10% of R10 000. Now in maths, when we see the word ‘of’ that means times and when we write 10%, we write it as a decimal by dividing it by a 100. So it will look like this:

**10% of 10 000**

= 0,1 x 10 000

= 1 000

So, on top of his R10 000, he also owes R1 000, that’s the interest. So in total, after year 1, he owes R11 000.

Now, Sarah also owes her R10 000, and we need to add on the amount of interest. Now the way we work out her interest at this point after the first year is exactly the same.

**10% of 10 000**

= 0,1 x 10 000

= 1 000

So at this point, they both will owe R11 000.

Now as we move to the second year, things start to change a little bit. So let’s go back to Jan. Year 2, Jan owes that R11 000 from the previous year, but we add on another R1 000. So with simple interest, the interest is worked out on the original loan each time. So we are going to add the same amount each time.

We know at this point, at the end of the second year, that Jan owes R12 000 at the end of year 2.

Let’s see what Sarah owes at the end of year 2.

So Sarah owes the R11 000 from year 1. Now the way compound interest differs is the interest is not worked out on the original each time, it’s worked out on the end of the previous year. So here, to get this amount, we will need to work out 10% of the previous year.

**10% of 11 000**

= 0,1 x 11 000

= 1 100

So, at the end of year 2, she owes R11 000 plus R1 100 which gives us R12 100. So at the end of year 2, Jan owes R12 000, but Sarah owes R12 100.

Let’s see what happens at the end of year 3 when they actually have to pay their money back.

Jan will owe the previous year which is R12 000 + R1 000 = R13 000 at the end. That’s what he’s got to pay back.

Sarah, however, we’re going to look at her previous amount, which is R12 100, and then we’re going to add 10% of that previous amount. So it will be

**10% of 12 100**

= 0,1 x 12 100

= 1 210

So, at the end of year 3, Sarah will owe R13 310.

So if we have a look here, Jan has to pay back R13 000 at the end of his loan period, and Sarah has to pay R13 310.

So, you see, simple interest looks at the original each time. So we make a little note here – for simple interest, we’re going to look at the same amount, we’re adding the same amount each time. And that amount is the percentage of the original of the amount loaned.

Now with compound interest, it’s not the same amount each time. So, it’s not worked out on the original; it’s the percentage of the previous amount.

So, that’s the difference between simple and compound interest. And this section that we’re going into now, is going to explore simple interest – how do we work it out, what is our formula, then we’re going to introduce an idea which is called simple discount which is very similar, but you’ll see how the formula changes. And then we’ll look at what happens if we are counting days instead of just years. So, we’ll look at that – moving forward in time, moving back in time and all to do with simple interest. So get all your tools ready - your pen, paper - and let’s engage with simple interest.