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Hello, and welcome back to our lesson on growth and decay here, this is our second part of compound appreciation and depreciation.

I'm rearing to go, I hope you're rearing to go as well.

You've got your calculator, right? You've got your calculator, you got your notepad.

You got your pen, you're ready to go, as I am as well.

Make sure you're in that quiet room, so you can concentrate to the best of your ability.

Going to do some really, really big maths and really important maths today.

So, let's make sure we're ready to go.

So, without further ado, let's take it away with Mr. Thomas' lesson.

So, you've got a try this here that says, the population of Pythagoria in the year 2000 was, what's that number there? 600,000, right, it's population increased by blank percent in 10 years, and then by blank percent in the next 10 years.

Hm, the numbers underneath the ink stains add up to 60.

Find some possible populations of Pythagoria.

What do you notice? All right, pause the video there and play around with the maths there for eight minutes, off you go.

Let's come back to it then, so what did you get? Well, hopefully, you did notice something.

And what it was is that if you increased it by, say, 50% in 10 years, and then by 10% in the next 10 years, well, that would be 600,000 multiplied by 1.

5 to the power of 10, multiplied by 1.

1 to the power of 10.

So, if we do that, what do we get? Well we get, waiting for my calculator to load up the answer, we get 89,740,956.

17.

So if we do that, then we get this big, huge number here, which is almost 90 million.

Now, if I were to do that the other way round, what you'd find is these numbers are actually the same.

So, you can swap these round, and what you'll notice is they'll give you the exact same answer.

That's because of the cumulative law of multiplication we can see that, so you'll see you may get different answers per se, but when you do them reverse round, you get 1.

1 to the power of 10, then 1.

5 to the power of 10.

It's just simply switched them around.

Like you do 5 times 6, or 6 times 5 to get to 30.

It works exactly the same way.

So, it's really important to notice that.

So, for your connect, what I want us to consider is that exact idea is 1,200 increases by 4%.

So, think about it, 1,200 multiplied by 1.

04 and then by 7% for three years, so 1.

07 to the power of 3.

So, if we type that into our calculator, you'll probably go even quicker than me, can totally accept that, what's that going to be? You've beaten me, I imagine, yeah, very good.

So, that would be 1528.

85, so 85 pence, then there is pence, so that's what that gets us.

What we can then say is that we're going to have a 2000, 2,000 pounds decreases by 5% for four years.

So that is going to be 2,000 multiplied by 0.

95 to the power of 4, followed by 10%, so 0.

90 for two years.

And when we do that, what do we get? Well, let's plug it into a calculator, and have a look.

There's no way you could ever do this without a calculator, it'd be so difficult.

So, how amazing calculators really are, my fascination with calculators.

So, that will give us 1,319 pounds and 50 pence to the nearest pence.

So, really, really good that we've got that.

And then, finally, decreases by 6% for two years.

So, 9149 times 0.

94 to the power of 2 times by 1.

07 times by 1.

043 to the power of 5.

Now, if we do that, it's quite long winded, admittedly, but we're getting somewhere with this.

And it proves that if we have a compound investment of some sort, so investment goes up or down, there it shows us those cumulative returns, right? Cause it can fall as well as rise, as a result of various economic swings or whatever.

So, that's just a bit of applied math there to fill the time whilst I'm keying this into my calculator.

So, we get 10,676.

64 p there, very good.

So, that gives us an idea of how we can do this.

Now, what I'd like you to do for your independent task is to think about all this in one.

You've done those very similar examples, but this number four here, I want you to think about that a little bit more deeply.

You're going through cycles, you're increasing and decreasing, you're doing that five times over.

So, what's the percentage multiplier for the original amount? If you'd listened to a few previous ones, you may know how to do it straightaway.

So, pause the video now and have a go at that for 12 minutes, please, off you go.

Very good, so these are the answers you should have got.

Very similar to what we've done, just here.

I'm not necessarily going to go through these two in particular, but this one was a little bit more complex.

It decreases by 3% for six years, so 0.

97 to the power of 6, increase for a year, 1.

02 for 2%, and then an increase of 9.

1, 1.

091 to the power of 5 because it's five years.

Now, this one you're going to have 1.

12 times 0.

88.

Here we go, there we go, and that's going to be repeated five times over.

So, that's the one time, two times, that's the first cycle.

Then, you've got the second cycle here, et cetera.

So, then we eventually get up to three and then four, and then five.

So that then simplifies cause you've got 1.

1.

sorry, 1.

12, 1.

12, et cetera to this, right? And, eventually, we realise that the multiplier for the original amount is this one here.

So, if you got that without me having to help, very good, well done.

Now, for your explore today it's a little bit similar to that one that we had right at the end there, but I want you to think about that question there.

So, pause the video now and have a go for the next eight minutes please, off you go.

Very good, let's go through it, then.

So, a special deal offers those young people 10% increase on a 300 pound investment for the first year.

So, 300 times by 1.

10, and then 30% for the second year, 10%, 30, 10 and then 30.

So, what we notice is it's increasing one, two, three years in exactly the same amount.

So, it's the power of 3.

Now, we realised earlier in the lesson, didn't we, that it didn't matter what we multiply by that.

Cumulative law allow us to do, say, 5 times 6, and 6 times 5 in the same order.

So, we can have these 1.

10 bundled together as in to the power of 3 there because we see it three times.

Equally, we see 30% listed three times there.

So, 1.

30 to the power of 3.

Now, if I put that into my calculator, then, what do I get? Well, I get 300 times 1.

10 to the power 3 times, 1.

1.

sorry, 1.

30 to the power of 3, and I get 877 pounds and 26 p to nearest p.

Fantastic, if you were able to get that without me helping you, that's brilliant.

Really, really good work, well done.

So, unbelievably, that takes us the end of this little tiny little part series that we've got, but we've got more stuff to come.

We're going to be exploring those curves in the next lesson that we have.

So, do remember to.

don't forget, my apologies, don't forget to smash that extra quiz, and do a really good job of it, so you can prove to everyone just how much you've learned today.

For now, take care, and I shall see you soon, goodbye.