Lesson video

Hi there, my name is Ms. Lambel.

You've made a really good decision to decide you would 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 Advanced Problem Solving with Percentages.

And no surprises, that's within the Percentages Unit.

By the end of this lesson, you'll be able to use your knowledge of percentages to solve a variety of problems. Keywords that we'll be using in today's lesson, these should be really familiar by now, but it's always worth a recap, are interest.

And this is money that is added to savings or loans.

Simple interest, and that is always calculated on the original amount.

As opposed to compound interest, which is interest that is calculated on the original amount, plus any interest that has been accumulated over the previous period.

And exponential form.

When a number is multiplied by itself repeatedly, it can be written more simply in exponential form.

So for example, 2 x 2 x 2 could be written as two cubed.

Today's lesson is in two different learning cycles.

In the first one, we are going to look at mixed percentage problems. And in the second one, we are going to look at combining this with some work that you've done previously on area and volume.

Let's get started with that first one, mixed percentage problems. A special offer carton of juice contains 40% more than a normal.

Is the special offer carton better value for money? And here we've got a very, very simplistic representation of the two cartons, we've got the small one and we've got the large one.

Jun says, "Surely there's not enough information here to answer this problem.

We do not know the size of either carton." Laura says, "We don't need to know the size.

We can increase the price of the normal carton by 40% and compare this to 80 pence." The original carton represents 100%.

Here's my double number line.

So my original carton represents 100%, and we know the cost of the original normal carton is 56 pence.

We want to work out the price if the normal carton was 40% bigger, so that would be 140%.

I'm looking for my multiplier.

My multiplier that takes me from 100 to 140 is 1.

1, sorry, 1.

4.

Remember, if you're not sure, go backwards, 140 divided by 100 is 1.

4, that's where that's come from.

56 multiplied by 1.

4 is 78.

4.

If we scale up the price of the small carton, to buy one that was 40% bigger, would cost us 78.

4 pence.

The new special offer carton costs us 80 pence.

We can see that the normal carton is therefore better value for money.

Now let's take a look at this problem.

Is it better to choose option one or option two, which is best value for money? Option one, we'll use option one as our 100%.

100% is equal to 75 pence.

Option two is 30% smaller, so therefore it's 70% of the size of option one.

We will work out, using the option one price, what the equivalent price would be for something that is just 70% of the size.

We look for our multiplier, and our multiplier here is 0.

7, 75 pence multiplied by 0.

7 gives us an equivalent price of 52.

5 pence.

This means that the equivalent cost using option one would be 52.

5 pence.

And option two, we can see that it costs us 52 pence.

This means that option two is cheaper.

Option two is better value for money.

I'd like you to use multipliers to decide which of the following is the best value for money.

You've got option one, that's going to be your 100%, and then you've got whether it's smaller or larger, and which percentage by, and you've got the cost of that.

Pause the video and then come back when you're ready, and you can convince me whether you should choose option one or option two.

Pause the video now and good luck.

Super work, let's take a look.

1.

24, £1.

24 multiplied by 1.

2, because it was 20% larger.

So if I make my price 20% larger, we can see that it is less than £1.

50, so option one is best.

This time my multiplier is going to be 30% smaller, so that's 0.

7, that gives me 67.

2.

So option one is 67.

2, option two, 66 pence.

Onto the third one, we're gonna multiply by 1.

4, why? Because it's 40% larger, so an increase of 40% is represented by a multiplier of 1.

4.

Apply that to the cost, I get 8.

386, which we can see is less than £8.

40.

And then the final one, this time I'm gonna multiply it by 0.

9, 'cause 100 subtract 10% is 90%, as a multiplier, that's 0.

9.

And we can see that 3.

555 is smaller than £3.

56.

Now, you may have said on that final one that they were equivalent.

Although 3.

555 will round to £3.

56 when given to two decimal places, option one is actually cheaper, isn't it? Because 3.

555 is less than £3.

56.

It's really important that you do not round these values, we are not giving an answer of money.

We are just comparing them, and we need to make sure that we have enough digits after the decimal point to compare our two values.

Company A invests X pound in account one, company B invests X pound in account two.

Account one, the compound interest rate is 2.

5% per year.

Account two, the compound interest rate is 2% per year for the first two years, and 3.

5% per year for each extra year.

After three years, company A's investment is worth £689,210.

Whose investment is worth more at the end of three years? Laura says, "I can see how we work out company A, but how are we gonna work out company B? We have no information about them except they're investing for three years." Jun says, "Both of them invest X pounds.

This must mean they invested the same amount of money." Do you agree with Jun? Well, Jun was right, both companies have made the same investment, because if we've got an unknown in our question and the unknown is the same letter, it has to represent the same value.

Account one, we know the finish value, we don't know the start value, but we do know the interest rate is 2.

5%.

So 2.

5% increase is a multiplier of 1.

025.

And we're cubing that, because we're investing for three years.

We can rearrange now and solve this equation, and we end up the X is £640,000.

Now we know, because in the question it told us company A invest X pounds and company B invests X pounds, that that's the value we need to use now for account two.

Account two, the final value is going to be the amount invested, that's our £640,000, multiplied by 1.

02, because that's an increase of 2%, squared, because that's for two years, multiplied by 1.

035, that's our increase of 3.

5%, and we were doing this for three years.

So we've already done the first two, so this is just going to be to the power one.

But remember, great mathematicians don't write a power of one.

Let's calculate this.

We get £689,160.

96.

Company A's investment is worth more after three years.

And here, remember you are comparing the final amount, which for account one was £689,210.

Now a check, X pounds is invested in an account for three years giving compound interest rate of 2.

4% per PA.

Remember that's per annum, per year.

At the end of three years, the account balance is £51,539.

61.

Which of the following is the correct calculation to find X? Pause the video and then when you are ready, come back and we'll check.

Great work, what did you decide? The correct answer was B, but also D.

If we look at D, all we've done in D is shown the division as a fraction, due to actually.

Now we are ready for task A.

A box of cereal costs £3.

62.

A larger box contains 30% more cereal and costs £4.

70.

Which box is best value for money? Make sure you show me all of your workings to convince me which box is best value for money.

You may decide here to use a double number line or a ratio table to help you visualise what's happening in this problem and help you to set your working out.

Good luck, you can pause the video now and then when you are ready, come back and I will reveal question number two.

Superb, question number two.

Which account should Laura's mum choose to invest her £500 in for five years? Remember, PA means per annum, per year.

Account one is simple interest at 2.

9% per annum.

Account two, compound interest, 3.

5% per annum for the first two years, and then 2.

2% per annum for each extra year.

Pause the video, get your answers, come back.

Question number three, company A invest X pounds in account one, and B, X pounds in account two.

After four years, company A's investment is worth £541,216.

08.

Whose investment is worth more at the end of four years? Pause the video, and then when you've got your answer, come back.

Super work, let's check those answers.

So here we are multiplying, so we had 100% was the small box, we are multiplying that by 1.

3 to give us the extra 30%.

So we take the cost of the small box, multiply it by 1.

3, giving us 4.

706.

But the larger box was £4.

70, so the larger box was better value for money.

4.

7 is less than 4.

706.

Question number two, account one, the interest would've been £72.

50.

Add that onto the amount invested, gives a balance of £572.

50 at the end of five years.

Using account two, the final balance would be £571.

75, meaning it's best for Laura's mum to choose option one.

And the calculations are there for each of those if you need to pause a video and take a look at them if you've made any errors.

And question number three, first thing we needed to do was work out how much money was invested by company A, which was 500,000.

Then we used this value to find the final value for company B, which was £541,149.

50.

Comparing that to the final amount for company A, which was £541,216.

08, company A's investment is worth more after four years.

Well done with those.

And now we'll move on to the second learning cycle, which is problems involving area and volume.

This is going to combine what you've done previously with area and volume, but we're also going to include some percentages.

So I might be really testing your memories here.

Here are three squares, A, B, and C.

The area of square B is 10% greater than the area of square A.

The area of square C is 10% greater than square B.

By what percentage is the area of square C greater than square A? Hmm, lot of information there.

Laura says, "A will be the original, so that's 100%." Because we only know what B is based on A, so that's gonna be our 100%.

And Jun says, "That means that B is 110% and C is 121%.

Do you agree with Jun? Yes, he's correct.

100 multiplied by 1.

1 squared is 121%.

So he's noticed that each time they're getting 10% greater, and he's decided to do that as one calculation.

So 100 multiplied by 1.

1 squared, which gives 121%.

Here we've got our three squares with their percentages.

By what percentage is the area of square C greater than square A? And Laura says, "So C is 21% greater than A." Jun says, "Yes it is." We can see clearly that A was 100%, C is 121%, the difference between those is 21%, and C was greater.

So yes, Laura is correct.

There are three circles, A, B, and C.

The area of circle C is 25% greater than circle B.

The area of circle A is 10% less than circle C.

By what percentage is the area of circle A greater than circle B? I haven't got any diagrams here, but actually the diagrams didn't really help us in the last question anyway.

Which circle would it be best to assign to 100% in this question? And it would be B.

We look at the first sentence, or the second sentence, it says the area of C is 25% greater than B, so B is going to be our original.

Circle C is circle B multiplied by 1.

25, 'cause remember it's a 25% greater, that's represented by the multiplier 1.

25.

Circle C was 100, 100 multiplied by 1.

25 is 125%.

Now circle A is circle C, but it's 10% less than circle C, which is where that 0.

9 has come from.

10% less gives me 90%, and that as a multiplier is 0.

9.

Circle C, we've just worked out, it was 125.

So now we can work out using our calculators that A is 112.

5%.

This means that circle A's area is 12.

5% greater than circle B's.

Circle B, remember was 100, circle A is 112.

5, so we are looking at the difference between the two.

Now a check, the price of a TV is decreased by 20% and then increased by 30%.

What is the overall change in percentage from the beginning to the end? Pause the video, make your decision, and then come back when you are ready.

What did you decide, A, B, C, or D? The correct answer is B, but let's take a look at why.

The TV started off at 100% of its value.

It decreases by 20%.

We know the multiplier for a decrease of 20% is 0.

8, but then it increases by 30%, and we know that is represented by the multiplier 1.

3.

100 multiplied by 0.

8, multiplied by 1.

3 is 104%.

Remember, we started with the TV being 100% of its value, so we are looking at the difference, we can see it's increased by 4%.

A company packs small cube boxes with a side length of eight centimetres into this larger box.

How many smaller boxes can fit into the larger one? First thing we need to do then is we need to work out the volume of a small box.

Now we're told that they're cubes and we're told that they have a side length of eight centimetres.

So the volume is going to be eight cubed, which is 512 centimetres cubed.

The volume of the large box is 40 multiplied by 120 squared.

Why have I not used 1.

2 there? Why have I used 120? And the reason for that is I want to find the volume in centimetres cubed as the volume of the cube is in this unit, the volume of the large box is 576,000 centimetres cubed.

Here's the information from the previous slide so that we've got it to hand for the next part of the question.

How many of the smaller boxes can fit into the larger one? So we're going to take the volume of the large, divide it by the volume of the small, meaning that we can fit 1,125 small boxes into the large box.

The company now decides to increase the dimensions of the smaller boxes by 25%.

They're probably making some special offer boxes.

By what percentage has the number of boxes decreased by? So we're trying to work out what percentage the number of boxes has decreased by because the boxes are now bigger.

Well, the length of the size of the new cubes, it was 25%, they increased it by 25%, so it's eight multiplied by 1.

25, that's 10 centimetres.

The volume of the new cube is 10 cubed, which is 1,000 centimetres cubed.

So the number of new boxes is the volume of the large box divided by the volume of the small boxes, and that gives us 576 boxes.

We now need to calculate the percentage change.

We know 1,125 was the original number of boxes, but now we can only fit 576 boxes in.

We want to work out what percentage of the original number of boxes, that's the 1,125, is equal to the new number of boxes, 576.

We know that in maths, of can be exchanged for multiplication, let's solve this equation, we're gonna rearrange it to find P.

We end up with P is 576 divided by 1,125.

If I calculate that on my calculator, I get 0.

512, convert this to a percentage, so P as a percentage, we multiply by 100, is 51.

2%.

576 is 51.

2% of the original number.

Therefore, the number of boxes has decreased by 48.

8%.

Let's have a go at this one.

An increase of 10% followed by a decrease of 10% returns you to the original value.

Now is that true or false? And I'd like you to justify your answer.

So is the justification A or B? Pause the video and then when you're ready, come back.

Super, what did you decide, true or false? Correct answer is false.

Lots of people think actually, because you've gone up by 10, down by 10, a lot of people think that A is the correct justification.

But actually, remember we're talking about a percentage change.

We're talking about a percentage of what it was, and then the percentage of that new amount, so it's not the same.

So we're gonna do 100% multiplied by 1.

1, that's that increase of 10%, followed by that decrease of 10%, which is multiplied by 0.

9.

And actually if we calculate that, it's only 99% of the original value.

So actually it's still 1% less.

Task B, question number one, you're going to match each repeated percentage change to the correct overall percentage change.

So just as you did in that previous check for understanding, pause the video, match them up and then when you're ready, I'll be waiting for you when you get back.

Great work, question number two.

There are three rectangles, A, B, and C.

The area of B is 20% smaller than rectangle A.

The area of rectangle C is 30% greater than rectangle B.

By what percentage is the area of rectangle C greater than rectangle A.

And three, there are three circles, A, B, and C.

The area of circle C is 25% greater than circle B.

The area of circle A is 20% less than circle C.

And what is special about those circles? Pause the video now and then when you're ready, come back and we will have a look at question four.

And finally then, question number four, a company packs small cube boxes with a side length of 12 centimetres into this large box.

A special offer box increases the dimensions of the smaller boxes by 25%.

By what percentage has the number of boxes decreased when used for the special offer boxes? Quite a lot to do there, pause the video, make sure you show all steps of your working.

If you need to go back and re-watch the example, which is similar to this, you could do.

But I'd like you to try and give it a go by yourself first.

Pause the video now, good luck, and we'll check those answers when you get back.

Superb work, well done.

Question number one, the first one was 15.

5% increase.

The second one was a 16% decrease.

The third one, a 5% increase.

The fourth one was an 8% decrease.

And the final one was an 8% increase.

Question number two, rectangle A was 100%, which meant B was 80%, C was 104%, meaning that C was 4% greater than A.

C, oh, sorry, question three, circle B was 100%, circle C was 125%, circle A was 100%, so what was special about circles A and B? They were actually the same.

And then question number four, the final answer was 48.

8.

All of the steps for working are there.

If you haven't got 48.

8, I suggest you pause the video, have a look at each of the steps and see if you can identify where you've gone wrong.

And then when you're done, come back, 'cause we are going to look at summarising our learning from today's lesson.

Fantastic work today, well done.

We've looked at quite a lot of different things.

The main things though are that multipliers can be used to make comparisons.

So it didn't matter that we didn't know the size of the different things we were finding for best buys, we would just look at using that multiplier.

Interest rates can be used when comparing savings or investments.

Multiple repeated percentage changes can be represented by one overall.

So for example, the TV was increased, sorry, decreased by 20% and increased by 30%, so the overall change in price was 4%.

Superb work today, well done.

I'm really glad that you decided to join me.

Take care of yourself.

I look forward to seeing you really soon, goodbye.