Lesson video

Hi there, my name is Ms. Lambel.

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 Problem Solving with Percentages, and that's in 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 are going to be using in today's lesson you should be familiar with because you've been working on percentages fairly recently.

Interest is the money added to savings or loans.

Simple interest is always calculated on the original amount.

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

And when a number is multiplied by itself multiple times, it can be more simply written in exponent form.

So for example, two multiplied by two multiplied by two, we could write as two cubed.

This lesson in the first learning cycle, we are going to look at mixed percentage problems and then what we're going to do is we're going to pair that up in the second learning cycle with solving problems with percentages, area and volume.

Let's get going with that first one then, which is mixed percentage problems. A special offer carton of orange juice contains 40% more.

Is the special offer carton better value for money? So I've got a representation here of the two cartons.

So we've got the small one, 56 pence and the larger one is 40% bigger and it costs you 80 pence.

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

We do not know the size of either carton." And Laura's response is as percentages show proportion, we could choose the size for the small carton and use this to work it out.

And Jun says, "Yes, Laura, I suppose that's right." So Laura is suggesting it doesn't really matter whether it's 100 millilitres in the small one or a litre or 500 litres because the bigger one is 40% bigger and percentages are about proportion, then it isn't going to matter.

What do you think? Let's take a look.

We'll assume that the normal carton contains 250 millilitres of juice.

The special offer carton contains 40% more and we know our multiplier for an increase of 40% is 1.

4.

So we're going to take 250 millilitres, multiply it by 1.

4, meaning that the larger carton is 350 millilitres.

We're now going to use our double number line to compare the prices and I always like to start with my normal one on my double number line.

So my normal carton is 250 millilitres at a cost of 56 pence.

I'm going to ignore the large carton for the moment that I know the price of.

What I need to do is to work out an equivalent price for the same number of millilitres using the cost of the small carton.

So 350.

Now you know how to use your double number line.

We're looking for our multiplier that takes us from 250 to 350.

If you're not sure, remember go backwards, 350 divided by 250 is 1.

4.

So we know that moving across in our double number line we're multiplying by 1.

4 giving us 78.

4%.

We're now going to compare that.

So that would be, if I could buy 350 millilitres of the small one, it would cost me 78.

4 pence.

However, 350 millilitres bought in the big carton cost me 80 pence.

We can clearly see then that the normal carton is 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.

75.

75 pence multiplied by 0.

75 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 now to use multipliers to decide which of the following is the best value for money.

And remember, if you need to, you can draw out a double number line.

You are going to pause the video and decide whether option one or option two is the best value for money.

Option one is going to be your 100%, that's your normal.

It's not the one that is increased or decreased by a percentage.

Good luck with this.

Remember no guessing, I want you to work out the exact answer and then when you are done, we'll check your answers, good luck.

Great work, well done.

Let's take a look then.

Option one 20% larger.

So I'm going to take my price and I'm going to multiply it by 1.

2, which is my multiplier for a 20% increase, that gives me 1.

488.

Option two was one pounds 50.

So we can see option one is less, so therefore better value for money.

Second one, this time it was 30% smaller, so I'm going to multiply the price by 0.

7 giving me 67.

2.

If we look at option two, it was only 66 pence, so option two was cheaper.

Now five pounds 99 multiplied by 1.

4 because it was 40% larger.

So 1.

4 represents an increase of 40%, which is 8.

386 and this option two was eight pounds 40.

So we can see this time option one was cheaper.

And then finally 3.

95 multiplied by 0.

9 because that represents a decrease of 10%, is 3.

555 which is smaller than three pounds 56, which is option one.

Now, you may have decided that option one and option two were both equal, although 3.

555 will round to three pound 56 when given to two decimal places, if we look, option one is actually cheaper, isn't it? Because 3.

555 is less than 3.

56.

So it's really important when you look at this type of question that you don't round to two decimal places, okay? If you were asked to move on with this question and give an answer in terms of money, then you would need to, but here we're just using these values for comparison between the two.

So it was option one.

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 pounds.

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 going to work out company B? We have no information about them except they're investing for three years." Hmm.

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 that X is 640,000 pounds.

Now we know because in the question it told us company A invests 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 pounds 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 power one.

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

Let's calculate this.

We get 689,160 pounds and 96 pence.

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

Okay, and here, remember you are comparing the final amount, which for account one was 689,210 pounds.

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 pounds and 61 pence.

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.

Now task A.

Question number one, a 750 gramme box of cereal costs three pounds 62.

A larger box contains 30% more cereal and costs four pounds 70.

Which box is best value for money? I'd like you to draw your double number line or use a ratio table, whichever you prefer.

We've done it with double number lines in today's lesson and then you're going to work out the answer.

Remember, it is not okay just to say the big or the small box.

I want to see all those calculations.

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

And question number two, which account should Laura's mum choose to invest her 500 pounds in for five years? Account one or account two.

Again, I want to see all of your working out.

I want to see exact values so we know exactly how much money Laura's mum is going to make.

Good luck, pause the video and then come back when you're ready.

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

After four years Company A's investment is worth 541,216 pounds and eight pence.

Whose investment is worth more at the end of four years? Pause the video and then come back when you're ready.

Great work.

Now we can check our answers.

Question number one, the larger box would've had 975 grammes in it.

So if we scale up the small box to find the equivalent cost of 975 grammes using that price, we get 4.

706.

If we look at the bigger box in the question, it was four pounds 70.

So the larger box is best value for money.

So again, not by much, but it is better value for money.

And question number two, account one, the interest was 500 multiplied by the interest rate multiplied by five because it was simple interest.

So the balance at the end of five years is 572 pounds 50.

Account two, the final balance is going to be 571 pounds 75.

We can therefore see that account one is the best option for Laura's mum.

Question number three, account one, we could work out how much money was invested by both companies, which is 500,000.

Then we can use the interest rates to work out the final value for account two, which is 541,149 pounds 50.

Compare that to account one whose final balance was 541,261 pounds and eight pence.

We can see company A's investment is worth more after the four years.

And now onto our final learning cycle where we're going to look at some problems which also involve area and perimeter.

Now you might not have looked at area and perimeter for a while, so I could be really testing your memories here.

Here we have three squares, A, B, and C.

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

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

By what percentage is the area of square C greater than the square A? Laura says A will be the original.

So that will be 100%.

Jun says, "That means that B is 110% and C is 121%." Do you agree with Jun? Jun is correct.

100 multiplied by 1.

1 squared equals 121% because it was 10% greater and then another 10% greater.

And we know that an increase of 10% is represented by the multiplier zero point, sorry, 1.

1.

And we were doing that twice over the two squares, which is why it's squared.

We now know that A is 100%, B is 110% and C is 121%.

Now we can answer the question, by what percentage is the area of square C greater than A? Laura says, "So C is 20% greater than A." And Jun agrees.

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

Now we don't have any diagrams here.

Does it matter that we don't have any diagrams? Absolutely not.

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

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

By what percentage is the area of circle A greater than the area of circle B? I'd like you to have a think.

Which circle would it be best to assign to 100%? Circle A, circle B or circle C.

It's going to be circle B because if we look at the first sentence, it says C is 25% greater than B.

So we're going to be comparing to B to start with.

To calculate C, we take circle B, which we know is 100% and we multiply that by 1.

25.

Circle C is 125%.

Circle A is 10% less than circle C, so it's going to be circle C multiplied by 0.

9.

That's our multiplier for 10% less.

Circle C we've just worked out was 125.

If we multiply that by 0.

9, we get 112.

5.

Therefore circle A's area is 12.

5% greater than circle B's.

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, think about what we just did in the previous question and then when you're ready, come back, and make sure you can justify your answer.

Good luck.

You can pause the video now.

Great work.

And your answer? The correct answer was B.

We start with 100%, the original price of the TV.

We multiply that by 0.

8, that represents our decrease of 20% and then we multiply that by 1.

3, which represents our increase of 30%, giving us 104% of the original value, which shows a 4% increase.

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.

1 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 divided 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% and 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.

With 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 going to rearrange it to find P.

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

If I calculate that on my calculator, I got 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 to justify your answer.

So it is A justification A or B? Pause the video and then when you are 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 going to 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, you're going to match each repeated percentage change to the correct overall percentage change.

You need to work out the answers and then match them up.

And then when you've done that, you can come back and I will reveal question number two.

Good luck.

Super, question number two.

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

The area of rectangle 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 question number 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.

What is special about circles A and B? You can pause the video now and I will be here waiting when you get back.

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

You are going to answer these questions.

A, how many small cube boxes can fit into the larger box? B, the length of the sides of the small box increases by 25%.

What is the length of the new boxes? C, how many of the new boxes can fit into the larger box? And D, really challenging, but give it a go.

By what percentage has the number of boxes decreased when using the larger cubes? Like I said, there's quite a lot there.

It's going to take you a little while to do, but just be patient.

Work your way through it step by step and I know you'll be fine.

I'll be waiting when you get back.

Great work.

Let's check the answers.

Question number one.

The first one should match the 15.

5% increase.

The second one is a 16% decrease.

The third one, a 5% increase.

The fourth one a 8% decrease and then no surprises, the final one is the only one that's left, which is an 8% increase.

Question number two, rectangle A would've been 100%, meaning that B would've been 80%, it was 20% smaller.

C would be 80% multiplied by 1.

3 'cause it was 1.

3, sorry, it was 30% greater than B, given 104.

So rectangle C is 4% greater than rectangle A.

Question number three, circle B was 100.

Circle C would've been 100 multiplied by 1.

25, representing that 25% increase, which is 125%.

Circle A is circle C, but 20% less.

So 125 multiplied by 0.

8.

That represents our 20% less, which is 100.

So actually circles A and B were the same.

Did you get that? They were the same.

And finally, question four.

A, volume of the small cube boxes is 1,728 centimetres cubed.

The volume of the large box is 1,296,000 centimetres cubed.

Therefore, the number of boxes that we can fit in is 750.

Part B, what was the length of the sides of the new boxes? So it was 12 originally, they increased by 25%, giving us 15 centimetres.

The new boxes were 15 centimetres.

C, the volume of the new box is 3,375 centimetres cubed.

The number of boxes that we would be able to fit in then are 384.

And then for part D, I've actually done it there just as one calculation.

You probably have drawn out a double number line or a ratio table or an equation showing more than one step, but the final answer should be 48.

8.

Now let's summarise what we've done during today's lesson.

There's been some really quite challenging stuff there.

So well done If you've managed to get through all of that.

Multipliers can be used to make comparisons.

So here, remember we didn't need to know the size of the different things.

We just compare the percentages.

So start with the original as 100% and then adjust that using the percentage increase or decrease.

Do this to the price and then compare.

Interest rates can be used when comparing savings or investments.

Multiple repeated percentage changes can be represented by one overall percentage change.

So start with 100 and then use those multipliers.

And then remember, you are looking for the difference between that and one or 100 depending on whether you've changed it to a decimal, kept it as a decimal, changed it to a percentage.

And there's an example there of one that we looked at during this lesson.

Like I said, there was some really quite challenging stuff there.

There always is in these problem solving lessons.

So well done.

I look forward to working with you again really soon.

Take care of yourself, goodbye.