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Hello, Mr. Robson here, welcome to maths.
Super choice to join me today.
Today we're looking at compound interest calculations.
Sounds interesting.
Let's get started.
Our learning outcome, so we'll be able to calculate compound interest.
Some interesting keywords today.
Interest is money 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.
Look out for those keywords throughout our learning today.
Two parts to this lesson.
Will begin by calculating compound interest.
Alex is given 50 pounds and gets 10% simple interest every month.
The bar model shows how his money grows.
What bar model? This bar model.
For January, February, March, there's the bar representing our original amount, 50 pounds for January, and then the interest, 10% of 50 being 5.
So Alex gets an extra 5 pounds of interest that month.
In February we're starting with the original amount, finding 10% of the original amount, Alex gets another 5 pounds.
And the same for March, Alex gains another 5 pounds.
So after three months, Alex has 50 pounds plus 5 pounds, plus 5 pounds, plus 5 pounds, 65 pounds.
Jun is given 50 pounds and gets 10% compound interest every month.
This is compound interest, not simply interest, compound interest.
This bar model shows how his money grows.
January, February, March, start with January's original amount, 50 pounds.
And it's the exact same, we get 10% of 50 pounds, we get 5 pounds extra or rather Jun gets 5 pounds extra.
What happens next is the crucial difference between simple interest and compound interest.
We start February with 55 pounds and the interest is calculated on that 55 pounds.
10% of 55 is 5.
50 pounds, so Jun's gained an extra 5.
50 pounds.
What's gonna happen next? Well done.
March's starting amount is 60 pounds and 50 pence and then we gain 10% of that amount.
10% of that, 6 pounds and 5 pence.
That's what Jun's earned in interest that month.
Same after three months, with a compound interest of 10%, Jun has 50 plus five, plus 5 pound 50, plus 6 pounds and five pence, 66 pounds and 55 pence.
It's really important you understand the difference between simple interest and compound interest.
You might choose to pause and copy this down.
In the top half of the screen, those bar models represent simple interest.
Simple interest, always calculate on the original amount.
In the bottom half of the screen you can see the interest compounding upon the previous month's interest.
Compound interest is the interest calculated on the original amount and the interest accumulated over the previous period.
Quick check you've got that.
Two pupils are told they will each receive 15% interest each month on 180 pounds over the course of two months.
Sofia will receive simple interest and Sam will receive compound interest.
Fill in the blanks in these bar models and the calculations at the bottom of the screen.
Pause and give this a go now.
Welcome back.
Hopefully in month one you started Sofia off with the original amount of 180 pounds and added 15% interest to that.
That's 27 pounds.
And then because Sofia is getting simple interest, it's the same thing for month two.
By calculating the interest on the original amount, 15% of 180 is still 27 pounds.
So in total Sofia has 180 plus 27, plus 27.
She's gaining the same simple interest each month.
She has 234 pounds in total after two months.
Sam's case is different.
Sam's getting compound interest.
Sam starts with the same original 180 pounds and gets the same interest in the first month, but crucially, the previous month total, 207 pounds is what we're now calculating the next interest upon.
15% of 207, 31 pounds and five pence.
So in total Sam has 180 plus 27, plus 31.
05.
That is 238 pounds and five pence in total.
Andeep is evaluating this investment.
Starts with 200 pounds and gets 10% compound interest every month for three months.
So Andeep says, "Interest is easy, especially this one.
I just do 200 plus 20, plus 20, plus 20, which is 260 pounds." Now if I know that you've got this, you can tell me, what is Andeep's error? And more importantly you'll be able to write a sentence to explain Andeep's error to him.
Pause and do this now.
Welcome back.
You might have said, "You've calculated simple interest Andeep, whereas this investment says compound interest." Really important, you look out for that wording.
"That means in year two you earn 10% of 220 pounds, which is 22 pounds, and in year three you earn 10% of 242 pounds, which is 24.
20 pounds which will give you 266.
20 pounds in total." Let's have a look at another example.
800 pounds is invested at a rate of 6% compound interest p.
a.
for five years.
What's the p.
a.
mean? The p.
a.
stands for per annum per year, i.
e.
annually.
It's really important that you understand that abbreviation when you see it, per annum, per year.
So year one we have 6% of 800 pounds, that's 48 pounds, that's our interest.
So after the end of year one, we've got our original amount, 800 pounds plus 48 pounds.
We have 848 pounds in the bank after year one.
So in year two, we gain 6% interest on the 848 pounds, that's 50 pounds and 88 pence.
And then we added to the original amount giving us 898 pounds and 88 pence.
And then in year three, we start with 898 pounds and 88 pence and we find 6% of that, which is 53 pounds and 93 pence.
And then we add them together, we end year three with 952 pounds and 81 pence.
Then we start year four finding 6% of that amount, which is 57 pounds and 17 pence.
Then we add all that together, we get 1,009 pounds and 98 pence.
Then in year five, we find 6% of that amount, which is 60 pounds and 60 pence.
Add it all together, and we end with 1070 pounds and 58 pence.
Phew, I'm exhausted and I've done all that maths and I hope it's right, but I might have made an error.
Calculating compound interest need not be so slow and cumbersome, and need I say painful.
There are more efficient ways to do this.
If you have previously calculated compound interest by doing all that mathematics.
The good news is you don't need to, we can make you more efficient.
Let's have a look at how we do that.
Same problem, 800 pound invested at a compound interest rate of 6% for five years.
We start with a hundred percent of our money and we gain 6%.
That gives us 106% Every year we end up with 106% of what we started the year with.
That means we can find that 106% by using a multiplier.
A multiplier for 106% is 1.
06.
It's a bit like me saying what's 106% as a decimal and you would say 1.
06.
We'll call this a multiplier.
So the quicker way to do it, we start with 800 pounds and we use our multiplier.
Multiply by 1.
06, and we find that end of year one amount, 848 pounds.
And then for year two we start with 848 pounds, we use the same multiplier, 0.
06, we get our end of year two amount.
That becomes the next input for year three, multiply it by 1.
06.
That becomes the input for year four, multiply by 1.
06 again.
And you can see what's gonna happen for year five I'm sure.
isn't that much more efficient and simple to write it like so.
Quick check, you've got that.
2000 pounds is invested at a rate of 3.
5% compound interest per annum for four years.
What multiplier will we use in our calculations? You just saw me turn 6% interest rate into 106% into a multiplier of 1.
06.
What are you gonna do with an interest rate of 3.
5%? Pause, choose which one of these options you think it is.
Welcome back.
Hopefully you said it's D, multiplied by 1.
035 for a multiplier of 1.
035.
We start with a hundred percent of our amount, we add 3.
5%, we have 103.
5% of what we started the year with by the end of the year, 1.
035 is therefore the multiplier.
Another check now.
I'd like to check that you can use that multiplier.
It's the same problem.
2000 pound investor is rate of 3.
5% compound interest per annum for four years.
But I'd like you to complete these calculations.
What's going on in year one, year two, year three, year four, this problem how you use that multiplier, fill in these blanks.
See you in a moment to check your answers.
Welcome back.
Hopefully at the end of year one you are 2070 pounds.
That becomes the input for year two.
We're adding 3.
5% to 2070 pounds now.
We end up with 2,142 pounds of 45 pence which becomes our input for year three.
Apply the exact same multiplier.
We get 2,217 pounds and 44 pence, which becomes the input for year four.
I do hope you would multiply by 1.
035 again to give us 2,295 pounds and five pence at the end of four years.
After four years, the investment is worth 2,295 pounds and five pence.
Using a multiplier makes calculating compound interest much more efficient but we can get more efficient again.
We've seen this problem before.
800 pound invested at 6% rate of compound interest per annum for five years.
We've seen it modelled like so.
Crucially, have you noticed, we're performing a repeated multiplication? Every single time, I'm multiplying by 1.
06, so I don't need to write all five years out.
I could simply type this into my calculator.
800 times 1.
06, times 1.
06, times 1.
06, times 1.
06, times 1.
06, and I get straight to my final answer.
The investment is worth 1,070 pounds and 58 pence at the end of five years.
Wonderful, that was more efficient.
But I'm sure you are shouting at the screen now, "You can get even more efficient again, Mr. Robson." Well done.
We can.
In maths we love to simplify and in this case, we can simplify this repeated multiplication.
1.
06 times 1.
06, et cetera.
It's 1.
06 repeatedly multiplied by itself five times, well that's what powers indices are for, 1.
06 to the power of five.
So instead of writing all those multiply by 1.
06s, we can simplify it to multiply by 1.
06 to the power of five.
So it's the simplest way we can calculate the value of that investment at the end of five years by typing into our calculator, 800 multiplied by 1.
06 to the power of five.
Your calculator display should look like that and your answer should match mine.
So there's our problem solved in one simple, efficient calculation.
I wanna make you aware of something.
You might see this calculation expressed in a formula for compound interest.
What's happening in this calculation? Well, we start with 800 pounds.
That's our principle amount.
The amount of money we started with or the amount of money we invested.
What's the 1.
06? It's our multiplier.
That's one plus the rate.
The exponent of five is our time period as in we repeatedly multiplied five times, or we saw this investment increased by that amount five times.
1070 pounds and 58 pence was our final amount.
Written as a formula that's gonna look like this.
P multiplied by (1 plus r) to the power of t equals A.
That's your formula for compound interest.
One important note about r, r the rate as a decimal.
So we had a rate of 6% here, but please don't use six.
If you did 800 multiplied by one plus six, that's seven.
800 multiplied by seven to the power of five.
Well that would be some really powerful interest.
That's not in reality what's happening.
We need to take r the rate, 6%, and turn it into a decimal 0.
06.
So inside that bracket there, add one plus 0.
06.
That's where the 1.
06 multiplier came from.
Quick check, you've got all this.
250 pounds invested at a rate of 12% compound interest per annum for six years.
Fill in the blanks.
What are we gonna put in that sentence and what are we gonna put in that calculation? Pause and write these things down now.
Welcome back.
Hopefully your sentence read: This problem can be solved using repeated multiplication.
The multiplier in this case is 1.
12 and this will be repeated six times.
This is most efficiently calculated by doing 250 pounds multiplied by 1.
12 to the power of six and that would equal 493 pounds and 46 pence.
Your calculator a display would look just like that.
Another quick check you've got that.
I'd like you to match the scenarios to their calculations.
Four scenarios on the left.
Four very quick efficient calculations on the right hand side, which one belongs to which.
Pause and decide now.
Welcome back.
Hopefully you matched the first scenario to the third calculation, 600 times 1.
02 to the power of eight.
The second scenario to 600 times 1.
08 to the power of two.
The third scenario to 200 multiplied by 1.
06 to the power of eight.
And the fourth scenario to 600 times 1.
002 to the power of eight.
Well done.
Comparing simple interest to compound interest reveals something about the nature of compound interest.
A hundred pound invested as rate of 40% simple interest and a hundred pound invested at a rate of 40% compound interest.
Gonna populate those tables to represent what happens over the course of five years.
The simple interest table looks like so.
The compound interest table looks like so.
Let's take a closer look at those values, those amounts.
Can you see what's happening in the simple interest table? 100, 140, 180, 220, well spotted, common additive difference of positive 40 each year.
That makes it an arithmetic sequence.
You might also see that called a linear sequence, a common additive difference.
So what's happening in the compound interest table? 100 to 140, to 196, to 274 pounds and 40 pence.
How are we making those steps? Well done.
We're multiplying each time.
You know we use a multiplier, you know we use the same multiplier each time.
It's a common multiplicative difference.
What kind of sequence do we call this? Geometric, of course, common multiplicative difference.
So an arithmetic sequence is formed by simple interest 'cause of the common additive difference between successive values.
Compound interest though, forms a geometric sequence because the common multiplicative difference between successive values.
We can graph these to visualise a comparison.
Here's the graph of simple interest and compound interest.
Can you see which one is which? The top one, the curve exponential graph, that's a hundred pound invested at a rate of 40% compound interest.
Our straight line is simple interest.
For compound interest, we've grafted A equals 100 multiplied by 1.
4 to the power of t and we've got a geometric sequence forming that curve.
For the simple interest, we graphed a equals a hundred plus 40t.
That's a linear equation, hence a straight line.
It's an arithmetic sequence.
Quick check you've got that.
True or false, compound interest over time forms an arithmetic sequence.
Is that true? Is it false? Once you've decided, please select one of the two statements at the bottom to justify your answer.
Pause and do this now.
Welcome back.
I'm sure you said false and justified that with compound interest has a common multiplicative difference between each value giving us a geometric sequence.
Well done.
Practise time now.
Question one, I'd like you to complete these compound interest calculations.
How efficiently can you solve these three problems? Pause and do this now.
Ooh, don't forget to write down your method.
See you in a moment.
For question two, you're gonna write scenarios for each of these compound interest calculations.
The first is done for you.
I've turned 200 multiplied by 1.
05 to the power of three into the scenario, 200 pounds invested at a rate of 5% compound interest per annum for three years.
I'd like you to write a similar statement for B, C and D.
Pause and do this now.
Question three, we're gonna check Lucas's work here.
Lucas has performed this compound interest calculation.
He's written an awful lot of mathematics.
I'd like you to check his work for errors and then suggest a better method to him.
Pause and do this now.
Welcome back, feedback time.
Question one, we're completing these compound interest calculations.
Part A, you should have written 500 multiplied by 1.
12 to the power of three, giving you 702 pounds and 46 pence.
For B, 5,000 multiplied by 1.
02 to the power of eight is 5,858 pounds, 30 pence.
For C, 3,250 multiplied by 1.
035 to the power of five giving you 3,859 pounds and 98 pence.
Question two, we're writing scenarios for these compound interest calculations.
A was done for us.
For B, you might have written 350 pounds invested at rate of 3% compound interest per annum for six years.
For C, you might have written 2,500 pounds invested at a rate of 5.
5% compound interest per anum for 12 years.
For D, 50 pounds invested at a rate of 0.
95% compound interest per annum for nine years.
Question three, we were checking Lucas's work.
Hopefully you spotted the error at the end of year three.
Possibly a calculator misread.
If you compare what should have come out the calculator to what was written.
The problem we then have is this error compounds in every calculation beyond it.
Once we have this error at the end of year three, we continue to error afterwards.
A suggestion you might have said, it's more efficient and less prone to error to use the formula for compound interest and then showed Lucas that calculation.
Onto the second part of our lesson now.
We're gonna look at calculator proficiency.
An organisation invests 10,000 pounds in a green energy company and the shares of forecast to grow 8% each year.
The organisation plan to hold onto these shares for five years.
We can quickly and easily calculate how much they'll be worth at the end of this time period.
We could put that into our calculator.
It's our formula for compound interest.
So we know we're gonna end up with 14,693 pounds and 28 pence.
Unfortunately, the accountants need to know the value of this asset, the shares, at the end of every year for reporting purposes.
So we cannot just do one simple calculation, we need to do every year step by step.
If the accountants need to know how much they're worth every year, then we're gonna have to do something different.
We could list year one, year two, year, three, year four, year five, 10,000 times 1.
08.
That's our year one value.
We type in 10,800 pounds, multiply by the same multiplier to get our year two value.
11,664 becomes the next input.
Year three value and so on and so forth.
We can record year by year what's happening in our calculator there.
This works, it's fine.
However, perhaps accurate, this method was slow and cumbersome.
I had to press 61 buttons on the calculator to obtain all of these values.
It took me quite a while.
So what can we do differently? We're always seeking efficiency as mathematicians and our calculator is going to enable this on this occasion.
Every time we used our calculator, all we did was take the previous output and multiply it by 1.
08 to get the next input.
11,664 became the next input.
12,597 pounds and 12 pence became the next input and so on.
What can we do differently then? Start with the initial value 10,000 and press the EXE button on the calculator.
EXE stands for execute.
If you have an inferior calculator to the Casio ClassWiz, you might have an equals button, but my Casio ClassWiz as an EXE button.
So I type 10,000, press EXE and my calculator tells me the answer is 10,000.
Of course it is.
Why do I want to do that? Well, my calculator now has an answer of 10,000 in it.
I can now use the answer button.
I'm gonna type Ans times 1.
08.
My Ans button looks like that.
Ans is short for answer.
Find yours on your calculator and then make your calculator display look like that.
Ans times 1.
08.
Oh look, when I pressed execute, I got 10,800 pounds.
The joy of this now is I can just press the execute button again and record the value.
I get 11,664 and I just keep doing that.
Press the execute button, I get my year three value.
Execute button, get my year four value.
Execute button, I get my year five value.
Using this method, your calculator automatically uses each output as the next input into this repeated calculation.
That's the joy of using the Ans button.
Our efficiency improved hugely.
I only required 16 presses of buttons on my calculator to do all that maths.
How much more efficient is that? Quick check you've got this.
An organisation invests 16,000 pounds in a tech firm and the shares of forecast to grow at 12% each year.
The accountants need to know the value every year for the first five years.
So what's the first thing you'll type into a calculator? Pause and decide now.
Well done, it's A, 16,000 EXE.
It's not B because typing a new calculation step by step's is possible, but it's inefficient.
If you did this five times, you'd get the answer, but you'd be pressing 60 plus buttons on your calculator.
I don't like that.
I wanna be more efficient.
C gives you the final value, but it only gives you the final value.
Crucially, we were told the accountants need the value after each year.
We don't get that with option C.
Option A sets 16,000 up as our next input.
That's important.
Same problem.
Our investment of 16,000 forecast will go 12% per annum.
The accountants need to know every value of every year.
We've started by typing 16,000 EXE into our calculator.
What's our next step? Three options to choose from at the bottom.
Which one do you think it is? Pause, take your pick.
Welcome back.
Well done.
Of course you said option C.
Why didn't we say option A? It utilises the input 16,000, but the accountants need the value for all five years, that only gives us the final value.
For B, well, we've retyped 16,000 so we can do it step by step, calculation by calculation.
It's nowhere near as efficient as C.
C gives us our year one value, 17,920 and leaves it in place as the next input.
Next check.
It's the same problem again, but which series of button presses delivers the solution in the most efficient way? If we look at option A, that's me pressing 1, 6, 0, 0, 0, EXE and multiply 1, decimal point, 1, 2, EXE, EXE, EXE, EXE.
You are reading left to right and then top to bottom for those calculator button presses.
So which one of those three options delivers our solution in the most efficient way? Pause and take your pick.
Welcome back.
Hopefully you said B.
Why was it not option A? Well, it successfully sets up our input/output that we want on our calculator, but it only gave us four years of results.
At the end, there were only four presses of the execute button.
We only got four years.
For C, it would give us the correct value, but there's 35 presses of the button.
Don't need to do it.
Option B did it in just 17 presses of calculator buttons.
So if that's the fastest way we can do it on our calculator, do it.
Pause.
Use this sequence of button presses to find these values.
See you in a moment.
Welcome back.
Hopefully your calculator looked like that initially and then Ans, multiply by 1.
12, and then we get the year one value.
EXE to get the year two value.
EXE again for the year three value.
EXE again for the year four value.
EXE again for that final year five value.
Practise time now.
Question one, an organisation invests 250,000 pounds in a tech firm and the shares of forecast to grow 8% each year.
The accountants need to know their value every year for the first four years.
For part A, I'd like you to populate the table of values.
For part B, a little challenge for you, a little competition.
I know what the lowest number of calculator button presses I can do this problem in, but what's the lowest number you can do it in? Have a good play with that.
See how low you can go? Pause, see you in a moment.
Question two, you put 350 pounds in a savings account, which will grow at 4.
5% each year.
After how many years will you have 500 pounds? Then again, I've got this challenge for you.
What's the lowest number of calculator button presses you can answer part A in? Pause, get this problem a go.
Feedback time.
Question one, part A asked, populate the table of values for this scenario.
Your table should have looked like so.
Incredibly, those shares after four years are worth 340,000 pounds.
Well, 340,122 pounds, 24 pence.
What's the lowest number of calculator button presses we could do that in? Well, those are the presses I used.
I did it with just 17 calculator button presses.
Question two, after how many years will we have 500 pounds? If you press that sequence of buttons, you reach this value, 497 pound and 74 pence.
That's eight repetitions, eight years, i.
e.
, the execute button was pressed eight times.
When I press it a ninth time, I see that value.
After nine repetitions, nine years, we get to 520 pounds and 13 pence.
So after how many years will we have 500 pound? We'll get there in the ninth year.
What's the lowest number of calculator button presses we did that in? 20.
We're at the end of the lesson now, sadly, what have we learned? We've learned that we compound interest and understand that it forms a geometric sequence because the multiplicative relationship between successive values.
The formula for compound interest.
P multiplied by (1 plus r) to the power t equals A is used to efficiently calculate the value of an investment if the principle value, the rate of interest and the term are known.
We also know how to efficiently use the 'ans' button on our calculator should we need to see the value after every repeated calculation.
Hope you enjoyed this lesson as much as I did and I look forward to seeing you again soon for more mathematics.
Goodbye for now.
