Calculating simple & compound interest
About this video
Starting with $4,000 at 2.5% a year for 4 years, the video compares simple interest, which pays 2.5% of the original amount each year ($400 total), with compound interest, where each year's interest is earned on the new, larger amount. It builds the expression 4000(1 + 0.025)^4, links it to the formula P(1 + r)^t, and finds about $415.25 in compound interest, a little over $15 more.
You'll be able to
- calculate simple interest for several years.
- explain how compound interest differs from simple interest.
- use P(1 + r)^t to find a compound total and subtract the principal to find the interest earned.
Key words
- principal
- The amount of money you start with (written P); the video spells it 'principle'.
- interest rate
- The percent of money added each year, here 2.5% or 0.025.
- simple interest
- Interest worked out only on the original principal, so it is the same every year.
- compound interest
- Interest worked out on the current amount, including interest already earned, so it grows each year.
- exponent
- A small number that shows how many times to multiply something by itself, like (1.025)^4.
Transcript
- [Instructor] So let's do some examples, calculating simple and compound interest. So let's say we are starting with principle and I'll use P for Principle, of $4,000. And let's say that we are going to invest it over a time period of four years. And let's say that we are looking at a rate, and we're gonna think about whether it's simple or compound, we're looking at a rate of 2.5%.
So how much money are we going to make in interest after four years? And we're assuming that we are getting 2.5% a year. I want you to pause this video and think about how much money have we earned from the interest in a simple interest situation versus a compound interest situation.
All right, now let's do this together. Let's start with the simple scenario and it is literally simpler. So here we just have to think about what is 2.5% of 4,000? So we could take 4,000 and then we can multiply that by 2.5% and we could also think about that as 0.025.
And then when you multiply it, you could verify this if you like, this is going to be equal to $100. Now this is how much you're going to get in the simple interest scenario each year. So if you're doing it over four years, you're going to get four times 100, which is equal to $400 in interest, is what you could get.
Another way you could think about it from a simple interest point of view is if I'm getting 2.5% of my original $4,000 per year, well then over four years, it's just four times 2.5%. I'm gonna get 10% of the original $4,000, which once again is $400. Now let's do the compound interest scenario.
And this one is a little bit more involved. And just as a reminder of what's going on with compound interest, we're starting with $4,000. How much will we have after the first year? Well, the first year is gonna look pretty similar, because it's gonna be 4,000 plus 4,000 times 2.5%.
So I'll say times 0.025, that's after year one. Let me write this here. After year one, would be in that situation. Now the difference between compound and simple is what we start seeing after year two, because then whatever amount that we have here, we're going to multiply that times 2.5% to figure out the new amount of interest that we're getting.
Or put another way I could simplify this expression up here. I essentially did 4,000 times one plus. Let me scroll over a little bit so I don't get too crammed. Times one plus. And I will now write 0.025 right over here. And what we're gonna do in year two is we're gonna take this amount, let me write that down.
We have 4,000 times one plus 0.025, 2.5% right over there. So that's the amount that we had after year one. And they we're gonna multiply this amount times one plus 0.025 again. So it's gonna be times one plus 0.025 again. And so you can imagine if we go all the way to year four, it's going to look like this, it's going to be 4,000.
And we're essentially going to be multiplying this thing by itself four times. So it's gonna be times one plus 0.025 to the fourth power. We're going to be doing this over four years. Now this might look very similar, it might look very familiar, I should say, you might've seen the formula of the total amount that you're going to get is going to be your principle times one plus your rate to the time...
However many times you're compounding it to that power. And so that's exactly what we had here. I didn't want to just directly plug into the formula, which we easily could have done. We could have put the 4,000 here, we could have put the rate right over here and then we could have put this time right over here to immediately get to this thing right over here.
But I wanted you to see where this is coming from. But let's calculate this. And to be clear, this is gonna be the total amount of money we have. It's not gonna be the amount that we earn. So once we figure this out, we can then subtract the original 4,000 to figure out how much did we make in interest.
So if we take, let's do this first part, I can add this part in my head. So 1.025 to the fourth power is equal to that. So essentially, whatever my principle originally was, I'm gonna get this much times that amount is gonna be the total amount that I have. So let me just multiply that times 4,000.
So this is gonna be the total amount that I have. But now let me subtract the original principle to see how much I have earned in interest. So minus 4,000 is equal to $415.25. So the amount earned is $415.25. I didn't write it here because this would be 4,400.
Actually I could write this year. This is approximately 4,415.25 . But the amount that we earned in interest is this right over here. And so you can see in this scenario, compounding got us a little bit more than an extra $15.
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