Lesson video

Hi there, my name is Ms. Lambell.

You've made a really good decision to decide to join me today to do some maths.

Come on then, let's get started.

Welcome to today's lesson.

The title of today's lesson is Compound interest calculations with technology, and this is within the unit Percentages.

By the end of today's lesson, you'll be able to carry out compound interest calculations with a calculator.

A key word for today's lesson is rate of depreciation.

I'm just gonna give you a moment and I'm gonna pause and see if you can remember or if you know what we mean by the rate of depreciation.

Did you know? We'll take a look anyway and you can check if you knew.

If you didn't know, you can learn along with me now.

The rate of depreciation is the percentage by which an amount decreases.

So depreciate means the same as decrease and we often use this word when we're talking about things like the value of a car depreciating or the value of a house depreciating.

Other words that we'll be using in today's lesson are compound interest and rate of interest.

Interest is the money added to savings or loans.

Compound interest is the interest calculated on the original amount and the interest accumulated over the previous period.

And the rate of interest is the percentage which an amount will increase.

Today's lesson I've split into two separate learning cycles.

In the first one, we will concentrate on calculating compound interest, and in the second one, we'll look at what happens if we need to go backwards.

So if we know the final amount, we want to know the original amount.

Let's get going on that first one.

Aisha's grandma put 200 pounds into a savings account offering a compound interest rate.

She invested on a Aisha's 16th birthday and Aisha gets the money on her 21st birthday.

How much money will as Aisha receive? And we can see here that the savings account interest rate is 2% PA.

Can you remember what PA stands for? That's right, per annum, and it comes from the Latin, and we just need to remember per annum means per year.

And you may think about things like annual.

If you hear the word annuals, that means something that happens yearly.

So it's that an bit that helps us remember that.

Let's take a look at how much money Aisha is going to have on her 21st birthday.

Now Aisha says, "I'm going to get 220 pounds because there are five years between my 16th and 21st birthdays and 2% per year will be 10% over five years." Do you agree with Aisha? Aisha would be right if it was simple interest, but this is compound interest.

If we look to the question, we can see that it's say and it's offering a compound interest rate.

So Aisha would be right if it was simple interest because we are always calculating interest on the initial amount, in this case, that would be the 200 pounds, but here we are going to be using compound interest.

Aisha says, "Oh, I remember now.

We need to draw out the ratio tables.

But that will take ages as it's invested for five years." Aisha's now remembered that when we were working out before compound interest, we drew a ratio table one for each year.

And she's not sounding like she's too keen on drawing out five tables.

However, this lesson, we can use a calculator.

Can you think of an easier way rather than drawing out the five tables? What is the most efficient method for calculating a percentage with a calculator? Now I know you're really good at this.

You've done lots and lots of percentages work over the last few years, and it's using a multiplier.

Let's start by calculating the amount of money in the account the end of the first year.

What is the multiplier for an increase of 2%? It's 1.

02.

To find the multiplier, we have 100%, and remember, we are increasing because it's an interest rate, by 2%, giving us 102%, and the multiplier is a decimal equivalent, which is 1.

02.

You do that by dividing 102 by 100 to calculate then how much money is in the account at the end of the first year, we are going to take the initial amount invested, which is 200 pounds, and we're gonna multiply it by 1.

02 because that's our increase of 2%.

At the end of year one, Aisha has 204 pounds in her account.

Now let's look at year two.

Year two, she will have what she had year one, and we will multiply that by a multiplier to show the increase of 2%.

That's 204 multiplied by 1.

02, which is 208 pounds and 8 pence.

We can repeat that for year three.

Year three will be what was in the account at the end of year two, 'cause that's the beginning then of year three, multiplied by 1.

02, giving us 12, sorry, giving us 212.

2416.

Now I know at the moment, that doesn't really make a lot of sense because we are talking here about money, but we're going to leave our answer and we're going to round it in the very final step.

And year four, we repeat the process.

So year three multiplied by that increase of 2%, giving us 216.

486432.

And we're really there, we're now onto year five.

Year five, same calculation, but we're gonna use year four's end value as the beginning of year five, giving us 220.

8161606.

How many decimal places should we give our answer to? That's right.

Yes, two decimal places because we're talking about money, and that's the appropriate degree of accuracy when we're talking about money.

Aisha will receive 220 pounds and 82 pence.

Aisha's now saying, "You multiply by 1.

02 for each year.

Could you just do one long calculation?" What do you think? Let's take a look.

We did 200 multiplied by 1.

02, and then we took that answer and we multiplied that, so that's year one, by 1.

02, that's year two, and then we repeated that again for each of the years.

Because we're multiplying the previous amount, yes, we could do it as one long calculation.

Aisha now says, "There is a simpler way of writing repeated multiplication." Can you remember what Aisha is referring to? Is writing in exponent form.

Remember, a repeated multiplication we can write using its exponent form.

We've got a repeated multiplication here of 1.

02 five times.

What could we replace that with? We could replace it with 1.

02 to the power of 5.

Remember that exponent is showing the number of repeated multiplication.

So here, years one to five, we've multiplied by 1.

025 times, and we're going to represent that as multiplied by 1.

02 to the power of 5, and we get exactly the same answer.

And you might decide now that you want to check that using your calculator.

From now on, we are going to use the exponent form to save us some time.

1,800 pounds is invested in this account for four years.

How much interest is earned over four years? What is the multiplier for an increase of 3.

11%? Well, it's 100% the original amount invested plus the 3.

11%, and the sum of those is 103.

11%.

We want it remember as a multiplier, and that's a decal equivalent, which is 1.

0311.

And here all we've done is divided by 100.

We have our original amount invested, 1,800, multiplied by 1.

0311, that represents our increase of 3.

11%, to the power of 4 because that's the number of years that we are going to invest that money for.

This gives us 2,034.

58413.

But remember, we need to give it to two decimal places.

That's the appropriate degree of accuracy for money.

So the answer is 2,034 pounds and 58 pence.

Is that the answer though? Well, that's the amount of money that is in the account at the end of four years.

Let's go back to the question.

The question actually said, how much interest is earned over the four years? So we need to subtract the original investment of 1,800 pounds from the 2,034 pounds and 58 pence, meaning that the account is earned 234 pounds and 58 pence in interest.

Let's take a look at another one.

24,000 pounds is invested in this account for six years.

How much is in the account at the end of six years? What is my multiplier for an increase of 4.

02%? It's 100 plus the 4.

02, which is 104.

02%, but we want that as a multiplier, the decimal equivalent.

So, therefore, it's 1.

0402.

Let's take our amount invested, multiply it by our multiplier, and we're going to do that to the power of the number of years, which in this case is six, which gives us 30,402.

7129.

Again, we need to give that to two decimal places.

So our answer here is 30,402.

71.

Is that my final answer this time? Let's go back to the question, says how much is in the account at the end of the six years? So yes, this is our final answer.

Now I'd like you to do this check for understanding.

I'd like you to match each of the interest rates on the left-hand side to its correct multiplier.

Pause the video.

And when you've correctly matched, 'cause I know you'll do it correctly, matched up those five, come back and we'll check those for you.

How did you get on? Let's check.

2.

3% would be 1.

023, 2.

03% would be 1.

0203, 0.

3, sorry, 0.

23% would be 1.

0023, 23% is 1.

23, and 0.

023% is 1.

00023.

We now know to find the account balance, the final account balance, we take the amount of money invested, we multiply it by the percentage increase as a multiplier to the power of n, where n is the number of time periods.

From now on, we are going to use this to help us work out the answers to these questions.

A bank is paying an interest rate of 0.

5% per month.

How much interest is earned on 600 pounds after six months? The final account balance = amount invested, which is 600 pounds, and then we want the percentage increase as a multiplier, which is 1.

005.

Remember, if you are not sure, if you can't do that quickly in your head, you're going to 100, add 0.

5, and then divide by 100.

We need to do this to the power of 6.

This time, we were working in months, but the interest rate was in months, so it's okay.

We're still going to be writing it as the power of 6.

This gives us 618.

23.

But remember, the question ask for interest so we must work out just that extra bit of money.

So we subtract the original starting balance, which was 600 pounds, giving us 18 pounds, 23 pence.

Let's just do one more together and then I know you'll be ready to have a go at one independently.

3,500 pounds is invested in an account offering an interest rate of 2.

84% PA for four years.

How much is in the account at the end of four years? We know the final balance is equal to the amount invested, or the original amount, multiplied by the interest rate as an increase to the power of the number of time periods.

So this is my calculation, 3,500 multiplied by 1.

0284 to the power 4, which gives me a final balance of this, but I know I need to give it to two decimal places.

So 3,914 pounds and 86 pence.

Now it's over to you.

Pause the video, have a go at this one.

When you're ready, come back.

Make sure you write down all steps you were working just in case our answers don't match, you'll be able to see where you've gone wrong.

Good luck and I'll be here waiting when you get back.

How did you get on? Well, let's check.

The final balance is 5,800 multiplied by 1.

0106 to the power of 5.

So the final balance would've given us this, but we needed two significant, sorry, two decimal places, I should say.

So that's 6,113 pounds and 99 pence.

How did you get on? Great work.

Now you are ready to have a go at these questions.

So I'm gonna ask you to pause the video.

And then when you've got your answers, come back.

Remember to show all steps of your working.

Good luck and I'll be here waiting when you get back.

Super work.

Let's check those answers.

And here we can see that A, the answer is 83,314 pounds and 28 pence, B is 20,614 pounds and 96 pence, C, 726 pounds, 45 pence, and D, 136,362.

60 pound.

Hopefully there you spotted the ones where you needed to give the total and the ones where you needed to just give the interest.

Now let's move on to the second learning cycle where we're going to be finding the original amount.

So if we know the final amount, what did we start with? The value of a car depreciates by 25% a year over three years.

The value of the car is now 10,125 pounds.

What was the value of the car three years ago? Depreciate, so we talked about this on that key word slide.

It just means it's decreasing.

It's losing 25% of its value every year.

What is the multiplier for a decrease of 25%? It's 100%.

The car was worth 100% of its original value and we are losing 25%.

So we're subtracting 25%, giving a 75%.

The multiplier, remember, is the decimal equivalent, and therefore, 0.

75.

75 divided by 100 is 0.

75.

So our final value is the original value multiplied by 0.

75 to the power of 3 because it was depreciating by 25% over three years.

We know the final value this time.

We know what the car is worth now its final value.

So 10,125.

We are trying to find the original value.

Original value, so if I rearranged this equation, I end up with 10,125 divided by 0.

75 cubed, which is 24,000 pounds.

It's worth stopping and thinking here, does that seem sensible? And to me, it does.

The original value of the car was 24,000 pounds.

Now I'd like you to match each percentage to the correct multiplier, and here, I've given you a mixture of increases and decreases.

So just take care to read carefully whether I'm asking you for an increase or a decrease.

Pause the video.

And then when you come back, we'll check those answers.

Super work.

3.

4% increase would be 1.

034, a 34% decrease, 0.

66, 3.

04% decrease, 0.

9696, a 34% increase, 1.

34, and a 3.

4% decrease would be 0.

966.

Now we're going to look at a question with Alex's cat.

Alex's cat lost 14% of its mass each year.

In 2024, she was a mass of 4.

8 kilogrammes.

By how many grammes has her mass changed from 2022 to 2024? What is the multiplier for a decrease of 14%? We start with 100 and we subtract 14, and we end up with 86%.

We want the decimal equivalent, which is 0.

86.

So the final mass is equal to the original mass, the multiplier, to the power, what will my power be here? It's going to be to power of 2 because there's two years between 2022 and 2024.

Now we've got two unknowns here, so we cannot solve this yet, but we do know from the question that the Alex's cat has a mass of 4.

8 in 2024.

So the final mass is 4.

8 kilogrammes.

I can rearrange this now to find the original mass.

It's 4.

8 divided by 0.

86 squared, which gives me 6.

490, or you may have 6.

49 kilogrammes.

The change in mass though, the question asked how many grammes has a mass changed from 2022 to 2024? So again, really important to go back and check what was the question really asking us to do.

We know that the change in mass will be the original mass subtract the final mass.

The original mass was 6.

490 and the final mass was 4.

8 kilogrammes.

That gives us 1.

69 kilogrammes.

Now have I answered the question now? Not quite.

I'm almost there.

But if we read carefully what the actual final part of the question was saying, by how many grammes has a mass changed? What is 1.

69 kilogrammes in grammes? Yeah, it's 1,690 grammes.

Alex's cat has lost 1,690 grammes of its mass over those two years.

Alex improves his score on a computer game by 20% a month.

At the end of April, his score is 248,832.

What was his score at the beginning of the year? Well, let's take a look how we're going to solve this one.

What's my multiplier for an increase of 20%? It's 1.

2.

And you can see there where I've got that from if you need to stop and pause the video and have take a look.

The final score, so we're talking about scores here, Alex's scores on the computer game is the original multiplied by 1.

2 to the power of 4 because we were saying we wanted to know what his, we knew his score at the end of April, which is the fourth month.

His final score is 248,832.

We can rearrange this equation to find this total, sorry, to find the original score, and we do 248,832 divided by 1.

2 to the power of 4, which gives us 120,000.

Again, look and think, could that be right? One thing to really double check is we know that his score was improving each month, so, therefore, our answer has to be lower.

So if you've got an answer that's higher, you know need to go back and check your working.

Alex's score was 120,000 at the beginning of the year.

Let's do one more together and then you can have a go at one independently and you'll be then ready for task B.

A swimming pool has a leak and loses 15% of its volume a day.

It currently contains 36,125 litres.

Two days ago it was full.

What is the capacity of the swimming pool? Well, we're going to take our new amount, and we're dividing because we're going backwards.

Notice here, I've decided to miss out the first couple of steps, but remember you can include those if you need to.

So we were losing 15%.

That as a multiplier is 0.

85, and we know it was full two days ago.

So we're going to do 36,125 divided by 0.

85 squared, which gives us 50,000.

50,000 litres was the capacity of the swimming pool.

Now I'd like you to have a go at this one.

A company invests some money at a compound interest rate of 5% per annum.

After three years, their investment is worth 277,830 pounds.

How much did they invest? I like you to pause the video and then come back when you've got your answer.

Great work.

Let's check.

So you should have done 277,830 divided by 1.

05 to the power of 3, because they were investing for three years, giving us 240,000.

The company invested 240,000 pounds.

Now you can have a go at these questions here.

So I'd like you please to pause the video and then answer these four questions.

And then when you've got your answers, you can come back.

Well done.

Let's take a look at those answers then.

A was 24,300.

You can see the calculation there.

So if you've got the wrong answer, just check the calculation and make sure you can see why I've done what I've done.

B is 2,500, C is 1,139,850, and D is 9,792.

How did you get on with those? Superb work.

Well done.

Now we can summarise our learning from today's lesson.

Compound interest is the interest calculated on the original amount and the interest accumulated over the previous period, and we've been using this equation, final value = amount invested X by the percentage multiplier to the power of n, where n is the number of time periods.

This formula can be used to find the final or the original amounts.

So we went forwards and we went backwards.

Check carefully whether the question wants to know the final value or the interest or the change in values.

Go back and read that question through really, really carefully.

I'd like to thank you for joining me today to work through these compound interest problems. I hope you've enjoyed it and I hope you feel like you've made loads and loads of progress.

I look forward to seeing you again soon.

Bye.