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 Changing Compound Interest Rates, and that's a new unit, percentages.

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

Some keywords that we'll be using in today's lesson.

These should be fairly familiar to you by now, but it's worth just a quick recap.

And those are compound interest and this is where interest is calculated on the original amount and the interest accumulated over the previous period.

We'll also be writing things in exponential form.

Remember, exponential form is a way of showing a repeated multiplication.

So for example, two multiplied by two, multiplied by two multiplied by two can more simply be written as two to the power of four.

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

Today's lesson is split into two learning cycles, in the first of which we'll look at changing compound interest rates.

And then in the second one, we'll look at solving some problems with compound interest rates.

Let's get going with that first one, changing compound interest rates.

Sam's mum has 20,000 pounds to save, very lucky Sam's mum.

Which account should she choose if she plans to leave the money in the account for three years? She's got two options.

She's got option A, where the compound interest rate is 3.

8% for the first year, but then it drops to 1.

8% for every year after that.

Account B offers a lower interest rate of 2.

5%, but this doesn't change.

We need to work out which account should Sam's mum invest her money in.

Let's take a look at account A.

We're going to take the amount of money invested.

Remember we multiply that by the interest rate as an increase, which is for the first year, 1.

038 and for the second two years, 'cause she's investing her money for three years, the interest rates changes to 1.

8, which is 1.

018 and that needs to be squared because that's for two years.

And if we calculate that, we can see that Sam's mum at the end of three years would have 21,514 pounds and nine pence in her account.

Let's take a look at what happens if she invests it in account B.

Account B is 20,000 pounds investment multiplied by 1.

025.

That's our 2% interest rate, our 2%, sorry, 2.

5% interest rate, which is our 2.

5% increase.

This time we'll need to cube it because for all three years she's going to get 2.

5% interest.

And if we put that into our calculator, we get 21,537 pounds and 81 pence.

Which account should she invest her money in? Account A or account B? Account B is the better option.

We can see that she gets just over 20 pounds more with account B, but better in her pocket than the banks, I'm sure you'll agree.

In the first year, a car loses 35% of its value and then 14% each year after that.

How much is this car worth after three years? So the initial price for the car is 29,290 pounds.

Year one, a decrease of 35%.

So that's 100%, take away the 35 means it's 65% of the original value and my multiplier for that is 0.

65.

And years two and three, there is a 14% decrease each year and we know that that is a multiplier of 0.

86.

100 subtract 14, 86% and the decimal equivalent of that is 0.

86.

So we know the multiplier for years two and three is going to be 0.

86.

The value after three years therefore is going to be the initial value of the car, 29,290, multiplied by that decrease of 35% for year one.

So that's 0.

65 and that decrease of 14% for year two and three, so that's where the 0.

86 squared comes from.

And then we can put that into our calculator and it gives us a value of 14,080 pounds and 87 pence.

After three years, we can see that actually the car is worth less than half of its original value.

Cars do depreciate very quickly.

Like I said, that's what the car is worth now, 14,080 pounds and 87 pence.

The number of downloads of an app is growing each month.

In January, there were 30,000 downloads.

In February, the number of downloads increases by 25% and then the rate of increase is 10% a month over the next two months.

How many downloads of the app are there in April? Month one is an increase of 25%, so that is a multiplier of 1.

25.

Months two and three, we have an increase of 10% and our multiplier for that is 1.

1.

We need the initial number of downloads, 30,000, multiply that by our month one multiplier and our month two multiplier, remember we need to square that because we're multiplying by 1.

1 and then 1.

1 again.

And that gives us 45,375.

The number of downloads in April is 45,375.

Which of the following is the correct calculation for this problem? An investment of 25,000 pounds increases by 3% a month in the first two months and then decreases in the third month.

How much is the investment now worth? What I'd like you to do is decide whether you think the answer is A, B, or C to answer that question.

And for the two incorrect ones, I'd like you, please, to write me a question that that would answer.

So a little bit more challenging having to write me a question, but I know you are up for a challenge, so good luck with it.

Pause the video and when you come back, we'll see how you've got on.

Right, which was the correct one.

Let's start with that.

The correct one was B 25,000 multiplied by 1.

03 squared.

So that's a 3% increase over two months.

And then if we take a look at the first incorrect one, A, an investment of 25,000 increases by 3% in a month.

That was the original question, but this is the calculation.

So 0.

03 squared would mean actually a decrease of 97% because it's actually now only worth 3%, so it's decreased by 97%.

And now let's take a look at C, we had 0.

97 squared.

0.

97 is less than one, so therefore it must have been a decrease and the difference between 100 and 97 is 3%.

So here the word would need to change to decrease.

And then if we look at the second part of the calculation, 1.

05 is greater than one, so therefore it's an increase.

So this word here needed to change to increases.

The numbers were correct, it was just the words that were incorrect.

So it's really important you make sure if you are decreasing, your value is going to be less than one.

If you are increasing, your value is going to be greater than one, that's your multiplier.

Now for task A.

Sam's mom has 14,000 pounds to save.

Which account should she choose if she plans to leave the money in the account for four years, account A for year one, 4.

2% and then 2.

4% for each year, or account B, which is 2.

8% every year.

Question two, in the first year, a car loses 34% of its value and then it loses 13% each year after that.

How much is the car worth after three years? Pause the video, get those calculators out 'cause you'll definitely need your calculators for this.

Remember, in of these questions we're talking about money, so our answer will need to be given to an appropriate degree of accuracy.

So do think about that when you are given your answers and then when you are done, pop back and we'll have a look at the next set of questions.

Good luck.

Well done.

And question number three and four.

Question number three.

The number of downloads of an app is growing each month.

In January, there are 120,000 downloads.

In February, the number of downloads increases by 40% and then by 10% over the next three months.

How many downloads of the app are there in May? And question number four, an investment of 350,000 pound loses one fifth of its value in month one, but grows by 7.

7% a month over the next three months.

Has the investment managed to regain the money lost in one month? I remember for question number four, it is not okay just to say yes or no.

You need to convince me of your answer by working out what the value of the investment is now.

I've also made that a little bit more challenging because I have given you the loss as a fraction and not a percentage, but I know you know how to change a fraction to percentage.

So pause the video, good luck with these two, and then when you come back, we'll check those answers.

Fabulous, now let's check.

Question number one, account A, she would have 15,663 pounds 75 pence.

And in account B, she would have 15,635 pounds and nine pence, meaning Sam's mum should choose account A.

Question number two, the car is now worth 10,904 pounds, seventy six.

and the calculations there if you need to double check that.

Question 3, 223,608 downloads and again, if you need to pause the video and check the calculation you can.

And then finally question number four, one fifth was 20%.

Did you get that? Of course you did and then we've got our calculation.

So at the end of the four months, it is worth 349,788 pounds 19.

It was originally worth 350,000 pounds.

So no, unfortunately it is not quite managed to make it back to what it was originally.

Now let's move on to that second learning cycle, solving problems with changing interest rates.

A house is purchased for X pounds in 2020.

In 2021 it was sold for a profit of 15%.

In 2023 it was sold for 296,240 pounds at a loss of 8%.

Find the value of X.

Here we need to find the original value of the house in 2020.

We will form an equation.

The price in 2023, which was the final price, was the 2021 price with a loss of 8%, which is why I've multiplied by 0.

92 because a decrease of 8% leaves me with 92% of my value.

We know the price in 2023, so let's replace that.

We don't know the price in 2021, but we do now have an equation which has one unknown and so therefore we can solve it to find the price in 2021.

The price in 2021, it's 296,240 divided by 0.

92.

The price in 2021 then was 322,000.

Stop and think is that sensible? In 2023 it had lost some of its value, so therefore in 2021, yes, it was worth more and 322,000 is worth than the 296,240.

Now let's form an equation using the percentage profit from 2020 to 2021.

The price in 2021 was the 2020 price, but that included a 15% profit.

So we need to multiply that by 1.

15.

We know the price in 2021.

We've just worked that out in the first part of the question, it was 322,000.

Now we can rearrange that equation to find the price in 2020, which is 280,000.

The house was worth 280,000 pounds in 2020.

Don't forget these questions we can check.

We know our start value, we know our finished value, we know it increases by 15% and then decreases by 8%.

Now I'm going to assume that I don't know the finishing value, that's the value I'm going to use to check, but what I do know is the start value is 280,000.

That's an increase of 15%, so that's why I'm multiplying by 1.

15 and a decrease of 8%, which is why I'm multiplying by 0.

92.

Now let's calculate that and we get 296,240 pounds so we can clearly see that it must have been right.

280,000 was the original starting price of the house.

An investment is worth X pounds in 2021.

In 2022, it makes a loss of 11%.

In 2023, it makes a profit of 8% and is now worth 43,254 pounds, find X.

Pause the video, I've only given you answers here, so you are going to need to write your calculations out clearly on your piece of paper or in your book.

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

It'd be really good if you could come back having checked your answer because then you'll know whether you've got this right or wrong, good luck.

What did you come up with? The correct answer was 45,000 and the calculation would be to do 43,254 divided by 1.

08 because that's the multiplier for a profit.

Remember that's an increase of 8% and then a loss or a decrease of 11% is the 0.

89.

200,000 pounds is invested for three years in an account paying compound interest.

In year one, the interest rate is 1.

8%.

And in years two and three, the rate is X percent.

At the end of the three years, the investment is worth 209,753 pounds and 81 pence.

Work out the value of X.

Right, there's quite a lot of information there to unpack, isn't there? So we know the initial amount invested and we know the finish amount.

We know the interest rate of year one.

What we don't know is the interest rate of years two and three.

That's the only thing we don't know, and we know that that is equivalent to X.

So therefore we can have an equation with one unknown and therefore solve it.

We're going to form an equation.

Here we go.

Here's our equation, wo we've got our finishing amount.

That's how much the investment was worth at the end, and that's equal to the principal amount or the initial amount, that's our 200,000.

We know the interest rate for year one, which is where the 1.

018 has come from.

But what we don't know is what the multiplier was for the second two years.

So we don't know what P is, but we do know we need to square that because year one it was 1.

8 and we were looking at the end of three years, we have that amount of money in the account.

So we can solve this equation by dividing both sides by 200,000, multiplied by 1.

018.

We can then take the square root of both sides, giving us that P is equal to 1.

015.

1.

015 shows an increase of 1.

5%, so therefore X is equal to 1.

5%.

And a check.

Here's our values, this is everything that we know.

Let's start with the finishing value is equal to the 200,000 pounds, which is the initial investment.

The percentage increase for year one, which we were given in the question.

And then what we think the percentage increase is for years two and three.

We can calculate that.

We just put that into our calculator and we can now see that actually they match.

So therefore our answer must be right.

Here, I'm gonna ask you to spot the mistake.

80,000 pounds is invested for four years in an account paying compound interest.

In year one, the interest rate is 3.

5% and then X percent in any further years.

At the end of the four years, the investment is worth 85,816 pounds and 71 pence.

Work out the value.

Now I'm not going to read out all of those calculations there.

I'm gonna ask you to pause the video and then work out what mistakes I've made.

Now often when I'm doing spot mistakes, what I do is I don't even look at what is written down on the page or on the screen.

I actually just cover that up and work it out myself and then I can compare the two.

Good luck with this.

It's quite challenging now, but I'm every faith that you are gonna get this right.

So good luck and pauses the video and I'll be waiting when you get back.

Well done, what mistake did you decide I'd made? Well, let's take a look.

Power four here.

The exponent should be three, as the first year is represented by the 1.

035.

We wanted to know it was invested to four years, but actually year one we'd already accounted for with the 1.

035.

So it was just an additional three years.

The cost of a car depreciates 35% in the first year and the next percent each year for the next four years.

After five years, the car is worth 10,500 pounds.

Wow, what a change in price.

Find the value of X, give your answer correct to two decimal places.

We're going to form that equation.

We know the final price is equal to the original price, multiplied by that decrease of 35%, which gives me 0.

7, sorry, 0.

65, multiplied by P, which I'm using to stand for the multiplier percentage for the next four years.

So that's why it's the power of four.

We can then rearrange this equation.

We can then take the fourth route and we get a value of P.

P is equal to 88.

11%.

Remember, we are looking for the change in percentage.

So it started off being worth 100% of its value.

It's now only worth 88.

1% of its original value.

So the loss is 11.

89%.

So it loses 11.

89% per year over those three years.

And again, we'll check.

We've got our start value, our finish value, and what we know to be the first year, what we think for the next year is, and the number of time periods.

So imagine we don't know the finishing value, substitute everything else in, okay, and here, because we rounded the interest rate to two decimal places, we don't get exactly 10,500, we get 10,499 pounds and two pence, but that's close enough and it was because we rounded our interest rate.

Now this one, the cost of a car depreciates at 32% in the first year and then X percent each year for the next three years.

After four years, this car is worth 12,800.

Find the value of X.

Give your answer to two decimal places.

I'd like you to identify for me A, B, C, or D, which is the correct first step to solve that problem.

So basically, what should your the first equation look like? Pause the video, and when you've decided which is the correct answer, you can come back.

What did you decide? The correct answer was B.

If we look at A, we can see here that we are multiplying by 1.

32%.

That would be an increase of 32% not decrease.

Here we can see that the start and final values are the wrong way round.

The final value of the car was 12,800.

So that should be on the left hand side of the equation.

So the values need to be switched.

And then D, we can see here it's got to the power of four, but it was X percent for the next three years.

So that should have been to the power of three.

Task B now then, you're ready for this.

Question one, a house is purchased for X pounds in 2020.

In 2021, it was sold for a loss of 7% and in 2023 it was sold for 322,896 pounds at a profit of 12% on the previous year.

Find the value of X.

Pause the video and then come back when you're ready.

Superb and question number two.

3,500 pounds is invested in this account for three years.

The compound interest rate for the first year is 3% and then X percent for each further year.

At the end of three years, the account balance is 3,809 pounds and 71 pence.

Find the value of X.

Pause the video and then pop back when you're ready.

Question number three, 50,000 pounds is invested in this account for four years.

Compound interest rate for the first year is 2%, then X percent each further year.

At the end of four years, the account balance is 53,172 pounds and 13 pence.

Find the value of X.

Again, you can pause the video and come when you're ready.

And question number four, the cost of a card depreciates, remember that means goes down by 32% in the first year and then X percent each year for the next three years.

After four years, this car is worth 14,500 pounds.

Find the value of X and give your answer correct to two decimal places.

Remember, you can check your answer before you come back and check it with me.

Good luck and pause the video now.

Superb and we can check our answers.

So with all of these, if you need to pause the video to check the methods, you can, but I'm just going to read out the answers.

So X is 310,000.

Question two was 2.

8%.

Question three, 1.

4%.

And question four was 11.

15%.

How did you get on with those? Of course you got them all right.

Now let's summarise what we've been looking at this lesson.

Often interest rates change over time.

The formula can still be used in these situations to find missing values and we had our example here.

Account A had a compound rate for the first year of 4.

2% and then 2.

4% each year beyond that.

We were also, if we were given finishing values and start values and one of the percentage changes, we were able to form an equation to find the other percentage changes.

Today's lesson has been quite challenging, so I'm really glad that you've managed to stick with me right till the end and I hope you found everything we've done today really useful.

I look forward to seeing you again really soon.

Take care of yourself, bye.