Lesson video

Hi there, my name is Ms. Lambel.

You've made a really good decision deciding 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 calculating compound interest rates, a nd by the end of this lesson, you'll be able to calculate compound interest rates when given the start and the end values.

We'll be using some keywords in today's lesson and these are rate, compound interest and rate of depreciation.

The rate of interest is the percentage by which an amount increases.

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

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

Today's lesson is split into two learning cycles.

In the first one, we're going to find the percentage, and in the second one, we're going to look at using a formula.

Let's get going with that first one.

So finding the percentage.

6,400 was invested in an account offering p% per annum compound interest.

At the end of two years, the account balance was £6,922.

24.

What was the interest rate? Andeep says, "Hi Sofia.

I'm wondering if you can help me with this question please." Sofia says, "I will certainly try! Well, we know how to calculate the starting or finishing balance if we know the interest rate." And Andeep says, "Yes, so we must be able to work out the interest rate." Sofia's reply, "Yes, if we know the starting and finishing balances.

Let's take a look." Andeep says, "And the number of time periods." Sofia's reply, "Oh yes," if we only have one piece of information missing, we can find it using the other three." We know that the final account balance is equal to the amount invested multiplied by the percentage increase as a multiplier to the power of n where n is the number of time periods.

We know here that the amount of money at the end of the two years is £6,922.

24.

That's the final account balance.

We know the amount invested is 6,400.

What we don't know is the percentage increase as a multiplier, so I'm going to call that x at the moment.

We do know the number of time periods.

It was for two years, so we do know that n is two.

We now have an equation with just one unknown, that unknown of x.

Andeep says we now need to solve this equation to find x.

Sofia says, "First, we will divide both sides of the equation by 6,400." Okay, so we're dividing.

Notice here, I've shown my divisions as fractions.

We are then going to simplify what we can.

So we know that 6,400 over 6,400 is one, which just leaves us with x squared.

I've just decided to put it on the left-hand side.

Don't know why, but I prefer my unknowns on the left.

You can leave it on the right if you want to.

And then £6,922.

24 divided by 6,400 is 1.

0816.

Andeep says, "And now we take the square root of both sides to find the value of x." Yeah, that's right, isn't it? To find x, if we know x squared, we do the inverse of x squared, which is the square root of x.

And Sofia says, "Oh, yes, of course." So the square root of x squared and the square root of 1.

0816 gives us x is 1.

04.

Does this mean the interest rate is 1.

04%? What do you think? And Sofia says, "No.

Remember that was the percentage increase as a multiplier." "Oh yes," says Andeep.

"1.

04 is the decimal equivalent of 104%, so the interest rate was 4%?" Sofia says, "Yes, I agree.

We can check this answer.

I love it when we can do that!" I also love it when I can check my answers.

So I know that I've got things right before I have it marked or check my answer with somebody else or in the back of the book.

Here's our check.

Starting balance we knew from the question, 6,400.

The finishing balance we also knew from the question.

£6,922.

24.

Interest rate.

Now, we didn't know the interest rate.

This is what we've calculated it to be, 4% per annum.

We knew the number of time periods from the question because the question says we were investing for two years.

I am going to use the starting balance, the interest rate, and the number of time periods as my check.

So the amount invested, 6,400, multiplied by 1.

04.

Remember, that's an increase of 4% to the power of two or squared because the number of time periods was two.

If we calculate that, we put that into our calculator, we get £6,922.

24, which was the finishing balance in the question.

We can see that these two match, so therefore, it must be right.

It was a 4% increase or interest rate I should say.

A car's value depreciates by p% per year.

Its current value is £12,288 and three years ago, it was worth £24,000.

By what percentage does the car depreciate each year? Notice here I'm using the same formula, but I just need to change percentage increase to decrease.

12,288 is my final value.

My original value is 24,000.

I don't know what my percentage multiplier is, so I'm gonna call that x, but I do know that it depreciated over three years, so I need to cube the x.

Divide both sides by 24,000 and we end up with x cubed equals 0.

512.

The 0.

512 comes from dividing 12,288 by 24,000.

Now I'm gonna cube root both sides 'cause I know what x cubed is.

If I take the cube root of both sides, I end up with 0.

8.

Andeep says, "This must mean a 20% change per year." Do you agree, Sofia? But do you agree? Yes, 0.

8 is the decimal equivalent to 80%, which is a decrease of 20%.

Did you also agree? Well done.

Again, let's check.

These were the values given in the question.

This is what we've worked out that percentage change to be.

Percentage decrease, and we were given the number of time periods in the question.

Let's imagine we didn't know this finish value and use the start value.

We know that a percentage decrease is 0.

8 to the power of three all cubed because it was for three years.

And if we calculate this, we get 12,288, which was the finishing value in the question.

So therefore, we know that our answer is right and we can move on competently.

I'd like you to have a go now at matching the percentage change to the correct multiplier.

Pause the video and then when you've matched all five, come back.

Okay, 3% increase, 1.

03.

70% decrease is 0.

3, 30% decrease, 0.

7, 7% decrease, 0.

93 and 0.

3% increase, 1.

003.

A company produces 250,000 yoghourts a month over a period of four months, their production increases by the same percentage each month.

They are now producing 518,400 yoghourts.

What is the percentage change per month? Here's our equation.

We know the final number of yoghourts being produced is 518,400.

We know that they were originally producing 250,000.

We don't know what that percentage multiplier is, but we do know it's going to be to the power of four because it was for a four-month period.

Let's solve this equation.

We divide both sides by 250,000, giving us x to the power of four equals 2.

0736.

"What do we do now to find x?" Andeep says.

How do we find x? I'm wondering if you know.

Well, Sofia says the inverse of squaring is square rooting and the inverse of cubing is cube rooting.

And Andeep says, "So we must be rooting." Sofia's response.

"We need to find the fourth root.

We can use our calculators to do that." We need to find the fourth root of 2.

0736.

Let's see how we can do this on our calculator.

This is the calculator that I am using and we need to find the fourth root.

And we do this by pressing Shift and then we press the root button and notice, that Shift + root gives me two boxes.

It gives me a box to the left of the root sign and it gives me a box underneath the root sign.

In the box to the left of the root sign.

We put in the number we're finding the root of, or no, sorry, which root we're finding.

And then underneath the root sign, we put what we're finding the root of.

We end up with that calculation and then we would change that to a decimal, which is x is 1.

2.

The production of yoghourts increased by 20% per month.

And again, we can check.

These are all of our values.

Let's imagine we don't know the final.

And check and we can see that the two match.

Therefore we must be right.

An investment of £6,000 is made in an account offering a compound interest rate of p% per annum.

At the end of five years, the investment is worth £8,029.

35.

What is the interest rate? I'm going to do 8,029.

35 is equal to 6,000 multiplied by x to the power of five.

I'm going to divide both sides by 6,000.

And then I'm going to take the fifth root and I end up with 1.

06.

But remember, we need to find out the percentage increase.

So my interest rate is 6%.

Your turn now.

An investment of £5,000 is made in an account offering a compound interest rate of p% per annum.

At the end of four years, the investment is worth £5,849.

29.

What was the interest rate? Good luck with this.

It is quite challenging, but I'm sure you can do it.

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

How did you get on? Let's check your answer.

We've got £5,849.

29 equals 5,000 multiplied by x to the power of four, giving us that x to the power of four is 1.

169858.

We take the fourth root, giving us 1.

04.

The interest rate therefore is 4%.

Now you're ready to have a go at task A.

I'd like you to pause the video, have a go at these questions, and then when you're ready, come back and we'll check the answers.

Good luck.

Question number two, find the missing percentage in each problem.

So for this first one, I've given you a little bit of a hint of where to get started.

So good luck with this one.

Pause the video again, and then when you come back, I'll reveal the next questions.

And parts B and C.

Again, pause the video, make sure you show all your steps of working.

Set your work out nice and neatly and tidily, making sure that you double check your answers.

Use that check so that when you come back, you don't need to look at my answers because you know yours are right 'cause you did the check.

Good luck and pause the video.

Great work.

Now let's check those answers.

Number one, A, 3,400, B, 1.

8%, C, three and D, £186.

92.

2a.

The investment lost 20% of its value each month and the workings are there if you need to pause the video and have a look if you didn't get 20%.

B, the answer is 3%.

And C, the answer was 2.

8%.

Again, there's part of the working there to show you where that comes from if you need to pause the video and take a look.

Now we can move on to the second learning cycle for today's lesson, which is using the formula.

Let's go.

The final account balance is equal to the amount invested multiplied by the percentage increase as a multiplier to the power of n where n is the number of time periods.

We're really used to using that now, aren't we? We've used that a lot, not just necessarily in this lesson.

You may have used that previously in other lessons.

Andeep says, "We have been using this formula, but it's very wordy." Sofia's response.

"You're right, Andeep.

Normally formulae are written using algebra and in the simplest form." So yes, they're right, aren't they? Often when we see formulae, and I can think of one in particular, area of a circle is A equals pi R squared, no words in there, just symbols and numbers.

So Andeep says, "Maybe we can create a more concise formula." Let's take a look at what they're going to do.

"We can choose a letter to represent the original amount." So Sofia says, "Shall we choose A for amount?" Well, that sounds sensible, doesn't it? Andeep says, "Okay, we need to multiply this by the percentage increase as a multiplier." And Sofia says, "We found that by adding the interest rate onto 100% and then dividing by 100." So we had 100% plus the interest rate and we've used R to represent the interest rate divided by 100 because we needed that remember as a multiplier, as its decimal equivalent.

And to change a percentage to a decimal, we divide by 100.

We then raise that to the power of n, which was the number of time periods, and that gave us F, which I've used to represent the final amount.

Andeep says, "Now I've thought about it, I think I can remember seeing a formula." Sofia says, "You're right.

I now remember my teacher mentioning a formula." Let's compare our formula to the formula that Andeep and Sofia are referring to.

Here's our formula.

F equals A multiplied by 100% plus R% over 100, all to the power of N.

This is the formula that they're referring to.

Total accrued equals P, and then one plus r over 100 in brackets to the power n.

Andeep says, "Our formula is almost the same.

It's the part in the brackets that is most different." And Sofia says, "Actually, they are the same thing.

The bottom one has just simplified the 100% over 100 to one." Do you agree with Sofia? And Sofia is correct.

So here's the formula, and this is where total accrued is the final amount.

P is the principal amount.

That's basically the starting value.

R is the interest rate over a given period and n is the number of times the interest is compounded.

This formula can also be used to calculate any repeated percentage change.

Obviously it's entirely up to you which version of the formula you use.

An investment increases its value by 8.

2% a month.

Its initial value is £134,000.

How much is the investment worth after four months? We're going to use this formula but remember, you can go back if you prefer and use the previous formula, but I just want to show you for a few how you use this particular formula as this is the one that you may see in your lessons or in textbooks, et cetera.

P is the initial amount invested, which in this case is £134,000.

R is the interest rate and n is the number of time periods.

Here then we've got P, we've substituted in P, R, and N.

And then we can calculate this giving us £183,659.

69.

The value of a caravan depreciates by 15% per year.

It cost 18,500 when it was new.

What is the caravan worth after three years? Here's our formula for total accrued.

Will this formula need to be adapted to answer this question? Yes, the addition will need to change to a subtraction.

Why will the addition need to change to a subtraction? Because we're depreciating.

The value of the caravan is going down.

It's losing 15%.

P equals 18,500.

That's how much the caravan cost when it was new.

The rate it depreciates by is 15% and n is the number of time periods, which in this case is three for three years.

I can substitute each of those in to my formula and I can work out that actually, the caravan is now worth £11,361.

31 of its original value.

And we'll take a look at this one.

£3,000 is placed into a savings account offering a compound interest rate.

After two years, the account balance is £3,244.

80.

What is the interest rate? So this time, I'm missing the interest rate.

We've got our formula and we know what T, P and n are.

Substitute in the things we know.

This time, the missing piece of information is r, the rate, the rate of interest.

I'm going to divide both sides of my equation by 3,000.

And then I'm going to take the square root of both sides of my equation because on the right-hand side, I've got one add r over 100 in brackets squared.

So I need to take the square root of that, which gives me one plus R over 100 is equal to 1.

04.

That 1.

04 comes from working out the square root of 3,244.

80 over 3,000.

Now I need to solve this, so I need to subtract one from both sides.

So I end up with r over 100 equals 1.

04 minus one.

R over 100 equals 0.

04.

So r is 4%.

The interest rate is 4%.

Now you're gonna have a look at this question.

Which of the following is the correct formula to answer this question? The value of a TV decreases by 10% a month over three months.

It now costs £349.

92.

How much was it originally? I'd like you to pause the video, decide which of those three calculations is the correct first step for solving this problem.

Pause the video and I'll be waiting when you get back.

Now, what did you decide? A, B, or C? The correct answer was C.

The value of the TV decreases.

So therefore, we need to have a subtraction in the bracket.

So therefore, A was definitely wrong.

Perhaps we can see there why that one was wrong.

£349.

92 is not the principal of value.

It's not the amount that we started with.

It's not the original amount.

It's T, which leaves us with C, which is correct.

Now I'd like you to have a go at these questions.

Andeep wants to know which account will be best to put his £250 birthday money into for 10 years.

Which account should he choose? And two, the value of an investment is initially 680,000.

At the end of three years, its value is £655,812.

59.

The value decreases by x% per year.

Find x.

Good luck with these.

Make sure you show me all your steps of working and then when you're ready, you can pop back.

Good luck.

And questions three and four.

Three, a motorbike depreciates at a rate of 8% per year.

Three years after it was purchased, it is worth £5,061.

47.

What was the purchase price? And four, £60 is invested in an account offering a compound interest rate of 6.

7%.

How many years will it take for the investment to more than double? Again, good luck with these and I'll be waiting when you get back.

Super work.

Let's check those answers.

Question number one, account A gives £92.

50 interest, and account B gives £92.

56 interest.

So account B will be better, but only by six pence, but lots of six pences would soon mount up, wouldn't they? The calculations are there if you need to pause the video and take a look but I'm sure you got it right anyway.

Question number two.

Again, if you need to pause the video to take a look at each of the steps of the working, obviously you can, okay? But if you're happy and you've got the answer of 1.

2%, you can move on with me.

Question three, the answer was £6,500.

And question number four, it would take 11 years to more than double.

So I needed to make sure that 60 multiplied by 1.

067 to the power of something was greater than 120.

And so this is a little bit of a trial and improvement.

I tried 10 years and that only gave me 114.

76.

So I tried 11 years and that gave me 122.

45.

So that was when it doubled.

It doubled after 11 years.

At 10 years, it hadn't quite got there.

Let's summarise our learning from today's lesson then.

So we've looked at this formula.

Total accrued, remember is the total final amount.

P is the principal value, that's the amount of the money originally invested.

And then we find our multiplier by doing one, add the interest rate over 100.

And that's in brackets because we need to raise that to the power of n where n is the number of time periods.

We can also form equations of the known information and rearrange to find the interest rate.

And that's the one that we've looked at during this lesson.

So if you need to pause the video and carefully look back through that example, you can.

You've done really well today.

There was some really challenging stuff there, but I was really impressed with you and you stuck with me right till the end.

So well done.

I look forward to seeing you soon.

Take care.