Lesson video

Hi there, my name is Ms. Lembel.

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 Compounding Trust Calculations, and that's within the unit, Percentages.

By the end of this lesson, you'll be able to carry out compound interest calculations.

Keywords that we'll be using in today's lesson, which you should be familiar with, but it's always worth a recap, isn't it? Interest, that's the money that is added to savings or loans.

Simple interest is always calculated on the original amount.

Compound interest is 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 is split into two learning cycles.

In the first one, we will be looking at calculating compound interest, and in the second one, we will look at how to find the original amount.

Let's get going on that first learning cycle.

Alex decides to save 200 pounds birthday money in this savings account for two years.

Lucky Alex.

It says the savings account interest rate is 2% PA.

Alex wants to know what does PA mean? Do you know what PA means? It means per annum and it comes from the Latin for yearly.

What this is really saying is Alex is going to get a 2% interest rate per year.

He's gonna get an extra 2% on the money in his account each year.

Alex says, so I'll get 2% each year, a total of 4% then.

Do you agree with Alex? This type of savings account will use compound interest.

This means that in the second year, he will get interest on his original balance and the interest made the previous year.

Therefore, Alex is incorrect.

Alex here is thinking about simple interest where the amount of interest is always paid on the original amount.

Let's take a look at how much money Alex will have in his account at the end of the two years.

Year one.

The best way to set your work out for this type of problem is to draw yourself a ratio table.

200 pounds is the amount of money that Alex is investing in the savings account.

What percentage is equal to the 200 pounds? That's right, yes, it's 100%, isn't it? It's the original amount, therefore 100%.

Let's pop that into our table then.

We'll find 1%.

How do we find 1%? Yeah, that's it.

We divide by 100, so we're gonna divide the number of pounds by 100, giving us that 1% is equivalent to two pounds.

The interest rate for this account is 2%, so let's now find 2%.

We're gonna multiply 1% by two to give us 2%.

Two multiplied by two is four.

The money in the account at the end of year one is the money in the account at the beginning of the year, which is 200 pounds, plus the interest gained over that year, which in this case is the four pounds, the 2% interest.

The money in the account at the end of year one is the sum of those, which is 204 pounds.

Let's now take a look at year two.

Again, a ratio table.

How much is equivalent to 100% now? 100% is the amount of money at the beginning of year two, which is also the same as the amount of money at the end of year one, and that's 204 pounds.

Now, our process is the same as on the previous ratio table because the interest rate hasn't changed.

So we'll find 1% by divided by 100.

That gives us two pounds and four pence, and then we'll be finding 2%, remember that's our interest rate, so we're gonna multiply by two, giving us four pounds and eight pence.

The money in the account at the end of year two is the money in the account at the beginning of the year, plus the interest.

The money in the account at the beginning of the year was the 204 pounds, and the interest was the 2%, which was four pounds and eight pence, giving a total of 208 pounds and eight pence.

So Alex, at the end of the year, sorry, at the end of the two years, has 208 pounds and eight pence in his account.

That means he's made interest of eight pounds and eight pence.

Let's now take a look at Sofia.

Sofia decides to save her 300 pounds birthday money in this savings account for two years.

So we can see here, Sofia's got a little bit more money for her birthday.

She's also managed to find an account which has an interest rate that's higher than the one that Alex was using, and it's 4% here.

Remember that PA just means per annum, per year.

Year one.

Again, what percentage is equal to 300 pounds? That's the initial investment.

So that is the original amount and 100%.

We'll find 1% by divided by 100, and that gives us three pounds.

So 1% is equal to three pounds.

Now the interest rate is 4%, so 4%, we multiply by four.

Three multiplied by four is 12.

The amount of money in the account at the end of year one is the money in the beginning of the account plus the interest.

How much money was in the account at the beginning of year one? Right, it was the 300 pounds.

How much interest was made on the 300 pounds? It was a 4% interest rate, so 4% was equal to 12 pounds.

The sum of those is 312 pounds.

At the end of year one, Sofia has 312 pounds in her account.

We'll take that then and we'll move on to year two.

How much is equivalent to 100% now? Yeah, that's right.

It's what was in the account at the end of year one, which is 312 pounds.

We'll find 1% and divided by 100, giving us three pounds and 12 pence.

The interest rate was 4%, so we're going to multiply by four, giving us 12 pounds, 48.

The money of the account at the end of year two is the money in the account at the beginning of the year, plus the interest.

Money in the account at the beginning of the year was 312 pounds and the interest was 12 pounds, 48.

We find that sum of those and that gives us 324 pounds, 48.

That's what Sofia has in her account at the end of two years, and if we want to just work out how much interest she's made, we would subtract her original investment from the amount of money in the account now, giving us the 24 pounds, 48.

I'd like you to have a go at this check for understanding now.

What mistake has been made here? So I've made a mistake when I've been answering this question.

500 pounds is invested in an account, offering an interest rate at 3% per PA, remember, per annum, per year, for two years.

And I've given you there a year one ratio table and a year two ratio table.

What I'm going to do now is ask you to pause the video, decide what you think the mistakes are and remember, correct those mistakes and then when you are ready, come back and we'll check those answers.

Good luck.

How did you get on? Did you spot the mistake and then correct it? Brilliant.

520 pounds.

That's the mistake.

The ratio table, that first ratio table is absolutely perfect, but they've made a mistake here.

This is the interest of 4%, not 3%.

So what this person has done or I've done to try and trick you is I have worked, I've added together all of the values in the ratio table, but that included a 3% and a 1%, but the interest rate here was just 3%.

So this should have been 515, beginning of the year one balance and the interest made during that year.

That will mean that my table two is now totally wrong, so I need to replace each of those values.

So 100% was 515, 1%, five pounds, 15, and 3%, 15 pounds, 45.

And let's not make the same mistake here.

We're having what's in the account the beginning of the year and what's in the account, sorry, and what the interest is, and if we add those together, we get 530 pounds and 45 pence.

So we're adding together just the 100% and the 3%.

A company invests 20,000 pounds in this account for three years.

How much interest is made over three years? Year one.

I'm gonna go through these ones a little bit quicker because I know that you are super good at this already, but I just want to have a look at what happens when we add in another year or we include another year, I should say.

So 100%, we're finding 1%, and then we're going to find the interest rate.

The interest rate here is 2%, and so we can see that their interest made would be 400 pounds, so it's going to be the initial investment at the interest, so we've now got 20,400 pounds.

So year two, that's going to be my starting balance.

I'm going to find 1% and I'm gonna find 2%, and then I'm gonna add together the initial balance at the beginning of year two, 20,400, with the interest made, which is 408 pounds, and that gives me 20,808 pounds.

And I need to repeat that because this question said the company invested for three years.

So we're going to repeat that process for year three, remembering that our starting balance for year three is our end balance from year two.

Again, let's calculate those percentages, and then we're gonna add together the balance at the beginning of year three and the interest made over that year, which is 21,224 pounds and 16 pence.

Let's just go back to the question a moment.

The question says how much interest is made over the three years, so we must make sure we are only given the answer of the interest, so that's the extra amount made.

So we started with 20,000 and we've now got 21,224.

16.

The actual interest was 1,224.

16.

Now you can have a go at this task.

Without a calculator, now, that's really important, so make sure you show all of your steps so when you come back, if you've made a mistake, you are know where you've gone wrong.

I'm gonna ask you now to pause the video and then when you've got your answers to these four questions, you're gonna come back.

Pause the video now.

Good luck.

How did you get on? Superb, let's check those answers.

First question was 80,000 pounds is invested at an interest rate of 4% PA for two years.

How much is in the account at the end of two years? And the correct answer was 86,528 pounds.

B, 300,000 pounds is invested at an interest rate of 2% PA for three years.

How much interest is made over the three years? Hopefully you spotted there at what this question wanted to know, just the interest, and that was 18,362 pounds, 40.

C, 600 pounds is invested at an interest rate of 3% PA for two years.

How much is in the account at the end of two years? And that's 636 pounds and 54 pence, and D, 1/2 million pounds is invested at an interest rate of 0.

5% PA for two years.

How much interest is made over the two years? Remember here, you would need to find out your value and then subtract the 1/2 million and you should get an answer of 5,012 pounds, 50 pence interest.

Now let's move on to that second learning cycle.

We now know how to find the new amount.

What we're gonna do in the second learning cycle is we're going to look at, if we know the newer amount and we know the interest rate, can we work out how much money was there to start with? Let's get going.

A new small business' sales increased by 50% a month over two consecutive months.

Wow, that's impressive increase, isn't it? At the end of the second month, their sales were 675 pounds.

What was the value of their sales at the beginning of the first month? So here, we can see we're going backwards in time.

When we go backwards, remember, we need to think about inverses.

Although this question is not about interest on a savings or a bank account, this is about increases in small business' sales, the percentages are compounded, as each month, you are calculating the percentage on the previous month and not the first month.

This method doesn't just apply to interest rates.

It can also apply to any percentage change that is going to be compounded.

So that means worked out on the previous, not the first.

Here's our original amount.

So we're going to use a bar model to help us visualise what's going on with this problem.

There's the original amount.

That's the number of sales they had at the beginning.

They increase their sales by 50%.

We now have the original month plus one month's increase.

We want to know what it looks like at the end of the second month.

We need to find 50% of the new amount, and my new bar is 150%.

What is 50% of 150%? Yeah, that's right.

It's 75%.

We now need to add another 75% onto our bar model.

Our entire bar now represents the original amount plus the month one and the month two increase.

What percentage is the bar worth? It's worth 225%, our original amount, plus the 50% increase, plus the 50% increase on the first two bars, giving us 225%.

And we knew that that was equal to 675 pounds.

We'll pop this into our ratio table.

We know that 225% is equal to 675 pounds.

We want to know the value of the sales at the beginning, which is the original amount, so we want to find 100%.

In order to do that, we're gonna find 1% first by dividing by 225.

We do that to the number of pounds, we get three pounds.

We're finding the original sales, which is 100%, so we're gonna multiply by 100, giving us 300.

At the beginning of the first month, their sales were 300 pounds.

The number of bagels a bakery produces increases by 20% a month.

At the end of the second month, they would produce 720 bagels.

How many bagels did they produce at the beginning of the first month? Again, we'll look at this with a bar model.

That's the original number of bagels that the bakery produces.

They increase that by 20%.

My entire bar now is the original plus the month one increase.

We now need to find 20% of the new amount.

What is 20% of 120%? Yeah, so 10%, we know, divide by 10, that's 12, so 20% is gonna be double 12, so that's 24%.

So we're going to add on the 24%.

My bar is now representing what percentage? It's representing the original plus month one and month two increase, which is a total of 144%.

We know that at the end of the second month, they were producing 720 bagels.

So we know that the bar is representing 720 bagels, which means that 144% is equivalent to 720 bagels.

We find 1% and then we find 100% because that's what we wanted to find, the original number of bagels that they were producing.

And so we multiply both sides by 100, giving us 500.

At the beginning of the first month, they produced 500 bagels.

Now you are going to have a go at this one.

Sofia improves her score on a computer game by 10% a month over two months.

Her score is now 1,815.

What was her original score? You're gonna solve this problem.

You need to start by finding out what A, B and C are.

Good luck with this.

You can pause the video and then come back when you are ready.

And how did you get on? Great work.

And here we go, we've now checked our answers.

So A was 10%, B was 11%, C was 121%, the entire length of the bar, and that was equivalent to 1,815, so we find 1% by dividing by 121, and then we find the original by multiplying by 100.

So this score was originally 1,500.

Now you can have a go at this final task.

Without a calculator for all of these questions, please.

The cost of a laptop increases by 10% each month for two months.

It now costs 484 pounds.

How much was it originally? And I've given you here a bar model and a ratio table to help start you off.

Pause the video, come back when you're ready and I'll reveal the next question.

Well done, and part B, C and D.

B, the number of T-shirts sold by shop increases by 20% a month over two months.

This month, they sold 360 T-shirts.

How many T-shirts were sold two months ago? Part C, Alex improves his score on a computer game by 50% a month over two months.

His score is now 1,800.

What was his score originally? And part D, the number of litres of squash a machine produces increases by 10% a week over two weeks.

It now produces 30,250 litres of squash.

How many litres did it produce two weeks ago? Pause the video and then when you've got your answers to these, you can come back.

Great work.

Let's check those answers now then.

Without a calculator we were doing these, remember? We had 100% as our original and then it was increasing by 10%.

So we had 10% in that first bar.

And then, remember, we were finding 10% of the new bar, which was 110% and 10% of 110% is 11%.

We then put this into our ratio table.

We knew that 121% was equivalent to what the cost is now, which was 484 pounds.

Find 1% by dividing by 121 and then find 100% by multiplying by 100.

The laptop originally cost 400 pounds.

B, the correct answer was 250 T-shirts.

C was 800 points and D was 25,000 litres.

We can now summarise our learning from today's lesson.

What we've looked at is compound interest.

It's the interest that is calculated on the original amount and the interest accumulated over the previous period.

Other situations can represent compound percentage problems. So for example, a company sales, or the price of something increasing or decreasing.

Ratio tables and bar models can be used to calculate new and original values.

And we saw some examples of that during today's lesson.

Thank you for joining me again.

I really enjoyed working along with you today and I look forward to seeing you again really soon.

Take care, goodbye.