Lesson video

Hi there.

My name is Ms. Lambell.

You've made a really good choice deciding to join me today to do some maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is "Solving problems with percentages and proportionality." By the end of this lesson, you'll be able to use your knowledge of percentages and proportionality to solve problems. Keyword to remember, proportion is a part-to-whole, sometimes part-to-part, comparison.

If two things are proportional, then the ratio of part to whole is maintained, and the multiplicative relationship between the parts is also maintained.

Today's lesson we're gonna split into two separate cycles.

The first one, we'll concentrate on percentages, and then on the second one, we'll look at proportionality, where we will use percentages and other forms of proportionality.

Let's get going with that first one then, using percentages to solve problems. This shape represents part of a different shape.

What might the whole shape look like? I'd like you to pause the video and draw for me what you think the shape may have looked like before I cut it up.

Wonder what you drew.

I'm intrigued to see how creative you were.

Here are Aisha and Jun's shapes.

So as Aisha's is the one on the left, and Jun's is the one on the right.

And we were using that that triangle there was equal to 25% of the whole shape.

Have they both drawn a correct shape? No, Aisha's shape is correct, but Jun's shape actually represents 125% as he has used five triangles.

So 5 multiplied by 25% would give us 125%, and the whole original shape would have been 100%.

So Aisha's is correct, but unfortunately, Jun had a good go but it wasn't quite right.

This shape represents part of a different shape.

What might the whole shape look like? Aisha says, "This will not work because if I put two together, that will be 120% and we need it to sum to 100%." So Aisha's recognised the original shape would have been 100%.

And if she has one of them, it's not enough.

If she has two, it's too much.

Jun says, "Aisha, we could use fractions of the shape." "Of course, we could split the rectangle into six equal parts.

Each part will be 10%, and then we would need one whole rectangle and 4/6 of another rectangle." And Jun says, "Or we could split the rectangle into three equal parts.

Each part would then be 20%.

So you would need one whole rectangle and 2/3 of another." We will use Jun's idea, but only because it means splitting the rectangle into fewer parts.

Both were correct methods.

I'm going to split it into three parts.

That means 60 split into three parts.

Each part is worth 20%.

If I want the whole shape, I only want 100%, so I need 40% to add to my 60%.

This is one way the original shape could have looked.

That 40% could have been on the top of the 60 to make it one long rectangle.

As long as we have a part that's 60% and 40%, that could be equivalent to the original shape.

This shape, again, represents part of a different shape.

What might the shape look like this time? Will the whole shape be bigger or smaller than this pentagon? The original shape must be smaller as it will be representing 100%.

How many parts should we split the shape into? If we split it into five parts, each part will represent 25%, and we would need to put four together to make the original shape.

125 divided by 5.

Yep, that's 25%.

We're going to split the shape into five equal parts.

Let's do that.

Each part represents 25%.

Remember, the whole shape was 125.

Divide that by 5, we get 25%.

How many parts do we need to put together to make the original shape? Well, 25 multiplied by 4 gives us 100%.

We need to put four together.

I could put them together like this.

I could put them together like this.

It doesn't matter as long as we have four of those triangles.

Because they each represent 25%, we need four of them to make the original shape, which is 100%.

Which of the following would be a correct method for this shape? So this represents 180%.

What would the original shape have been? I'd like you to decide which of those calculations is suitable for finding out the answer to this question.

Pause the video, and come back when you've got an answer.

How did you get on? What did you decide? B was a correct answer.

If I take 180, divide it by 18, I end up with 10% and I need 10 lots of 10% to make my 100%.

C was also correct.

180% divided by 9 gives me 20%.

I need five lots of 20% to make my 100%.

And D was also a correct method.

180 divided by 180 would give me 1%, and then I'd need to put 100 of those back together to make the shape.

Aisha watched a TV programme in which they buy items, mend or fix them, and then sell the items. She's going to calculate the percentage profit or loss.

Somebody on the programme bought a laptop for 50 pounds.

It cost them 30 pounds to fix it.

And then they sold it for 120 pounds.

Using our ratio table, we've got 100% is equal to 80 pounds.

That 80 pounds is the amount of money that it cost to get the laptop so it was suitable to sell, so what it was bought for, but also the cost to fix it.

Remember here, if you prefer, you can use a double number line or a bar model.

Like I said, that's the total amount spent.

Our multiplicative relationship between 80 and 120 is multiply by 1.

5.

These are directly proportional, and so therefore we're gonna multiply by 1.

5, giving us 150.

The change in percentage, remember, we're looking at the change in percentage, is 50%.

Therefore, they made a 50% profit.

We know it's profit because the selling price was greater than the purchase price.

I'd like you now to have a go at this question.

You're going to spot the mistake.

The cost of a house decreases from 260,000 pounds to 252,200.

What is the percentage decrease in price? So here, I've given you a ratio table.

I've put on my multipliers.

I've worked out my percentage decrease.

Please pause the video and decide for me what mistake I've made, and correct it, of course.

The percentage decrease is 3%.

You need to consider the difference in percentages.

100% is the original price of the house, which was 260,000.

252,200 is 97% of the original cost, and so therefore the change is 3%.

So my table was right.

I've just not interpreted that decrease at the end.

Now you're going to have a go at some questions like the ones that we did at the beginning of this lesson, so those examples.

I'd like you to pause the video, and I'd like you to draw out each of those shapes, and then you are going to draw what the original shape would have looked like.

Pause the video.

It'll take you a little while this one, I think, but come back when you're ready and I'll be sat here ready and waiting for you.

And question number two.

So we know that Aisha's watched this programme.

Okay, we looked at the laptop.

These are some other items that she saw on the programme.

So here, I've given you what the item was, how much it cost them to buy the item, any other costs, so that might have been to fix it or to mend it or to make it look beautiful by painting it.

We've got the amount of money that they sold it for.

I'd like you to decide firstly whether it is a profit or a loss, and then I'd like you to calculate that percentage change.

So quite a lot to do there.

Use your calculators, make sure you show all steps of your working, and I look forward to seeing you when you come back.

Here are some examples.

So in the next slides, I'm going to show you an example of what the shape might have looked like.

The important thing is that your shape has the same number of shapes, or all the same number of shapes and fraction of an additional shape.

So 1A, you should have four of those L shapes together.

B, 70% and 30%.

So you would have split your 70% into seven equal parts and then taken three of them.

20% here, you would just need to make sure that you've got five of those triangles together in any order.

It doesn't matter if it doesn't look the same as mine.

D, you'd need 10 of those.

I thought that was quite a cool shape to make, so that's why I went with that one.

E, you would need 60% and another 40%, and 40% is 2/3 of the 60%, so that's 2/3.

That arrowhead is 2/3 of the original triangle.

F, 5%.

Each one was 5%.

So you'd need 20 to make 100%, so there are 20 rectangles there.

And again, you may have arranged them in a different way.

G, here you would need that to be 5/6 of the original shape.

So split the original shape into six parts, meaning each part is 20%, and then put five of them back together to give us 100%.

H, you would need 2/3 of the original shape.

If we split this original shape into three parts, each part is worth 50%, and then we'd need two of those to make our 100%.

So your original shape would look something like that.

And then on to question two.

The vase was a loss, and it was a 5% change.

The watch was a profit with a 10% change.

The toy was a profit, 15% change.

The painting was a loss of 8%.

Chair was a loss of 15%.

And the necklace was a profit of 24%.

So in order, we're going to rank them in order from which was the worst to the best.

The order were chair, painting, vase, watch, toy, and necklace.

Now we're gonna look at the final part of today's lesson, which is using proportionality to solve problems, and we're gonna look at some different problems here.

Quite a wordy question here, okay, but let's break it down.

A bag of sweets contains red, yellow, pink, and green sweets.

There are 48 sweets in the bag.

25% of them are red.

3/8 of them are yellow.

The rest are pink and green in the ratio of 7:2.

How many of each colour are there? So we've looked at percentages very, very recently.

You might not have done fractions of amounts and ratios for a while, so if you need to go back and jog your memory, you can.

But we will go through this example together one step at a time.

Let's start with red.

I have changed 25% into its decimal equivalent so that I can multiply 0.

25 by 48.

I could also here divide 48 by 4 because I know that 25% is equivalent to 1/4, giving me that there were 12 red sweets in the bag.

3/8 of them are yellow.

3/8, and remember, we can switch "of" for multiplication in our maths problems, 3/8 multiplied by 48.

Now here, you've got a choice.

You can use your calculator.

You can either do 3 multiplied by 48 and then divide by 8 or, personally, I would do 48 divided by 8 first because that's easier, which is 6.

And then you're going to do 3 multiplied by 6, which is 18.

Now, it says the rest are pink and green, so we need to work out how many there are, the rest.

So pink and green is going to be the total number of sweets in the bag, which is 48.

Subtract the sum of the red and yellow, which was 12 and 18, and that gives us 18.

We now know that there are 18 pink and green in total.

And we are going to work out how many pink and green there are because we know the ratio of pink to green.

This is my sort of bag of sweets.

More pink than greens.

Not keen on greens.

Here is our ratio table with our pink and our green and our total.

We need the total here because we know that the total number of pink and greens is 18.

So let's put that into our table.

Now we're looking for our multiplicative relationship.

9 multiplied by 2 makes 18.

So to go from the top line to the bottom line, I'm going to multiply everything by 2.

This means that there are 12 red, 18 yellow, 14 pink, and 4 green sweets in the bag.

Different type of problem now.

Andeep, Sofia, and Jacob record how long it takes them to complete a puzzle.

Andeep's time is 4/5.

Sorry, Jacob's time is 4/5 of Andeep's time.

Andeep's time is 5 times Sofia's time.

The total of Andeep and Sofia's times is 54 seconds.

What is the difference between Sofia and Jacob's times? Okay, lots of information there, isn't there? But the only time we know is the total of Andeep and Sofia's times is 54 seconds.

We also know that Andeep's time is 5 times Sofia's, so we could write that as a ratio.

Andeep's time is 5 times Sofia's.

So if Sofia's time was 1, Andeep's would be 5.

That's a total of 6.

Now, we know the total of Andeep and Sofia's times is 54.

So let's put that into our ratio table.

Now we're looking for that multiplicative relationship, which is multiply by, yeah, that's right, 9.

Now I'm gonna multiply Andeep's by 9 and Sofia's by 9.

So 45 and 9.

We now know Andeep's time and we know Sofia's time.

We can then look at finding Jacob's time.

We are told Jacob's time is 4/5 of Andeep's time.

So Jacob is 4/5 of Andeep.

And we now know that Andeep took 45 seconds.

So we're gonna do 4/5.

Remember, that "of" can be switched for that multiplication.

And here, I would do 45 divided by 5, which is 9.

4 multiplied by 9 is 36.

Jacob took 36 seconds.

The difference between Sofia's time, 'cause that's what ultimately the question was asking us to do, what is the difference between Sofia and Jacob's times? Sofia's time was 9, and Jacob's time was 36.

So the difference between those is 27 seconds.

36 subtract 9 is 27 seconds.

Let's do another one of those together 'cause that was quite tricky, quite challenging.

So we'll do another one together, and then I'm really confident you'll be able to do the one on the right-hand side by yourself.

We've got A, B, and C are three different numbers.

C is 3/4 of B.

C is 5 times A.

Find the value.

Sorry, the total of A and C is 144.

Find the value of B.

So the only thing we know at the moment is the total of A and C is 144.

And we also know the ratio of A to C because we're told that C is 5 times A.

Here's our ratio table.

So we know that C is 5 times A.

So if A is 1, C is 5.

And we know the total of those is 6.

We know the total, actual total, is 144.

Again, it's that multiplicative relationship, which is multiplied by 24.

1 multiplied by 24 is 24.

5 multiplied by 24 is 120.

I now know the value of A and the value of C.

It says C is 3/4 of B.

So C was 120, and that is 3/4 of B.

Ah, but we don't know what B is.

But we can come up with this calculation, and then we can solve this to find B.

So we are going to do 120 divided by 3/4 'cause we need to do the inverse of multiply by 3/4.

So we're gonna do 120 divided by 3/4, and that is 160.

So B has a value of 160.

Now it's your turn.

A, B, and C are three different numbers.

C is 2/3 of B.

C is 4 times A.

The total of A and C is 140.

Find the value of B.

Pause the video.

Good luck with this one, and come back when you're ready.

How did you get on? Great work.

Here we go, there's our ratio table.

Now we're told that C is 4 times A, so the ratio of A to C is 1:4, and that gives us a total of 5.

We know the actual total of A and C was 140.

Let's look for that multiplicative relationship, and that's multiply by 28.

Multiply 1 by 28, I get 28, and 4 multiplied by 28 is 112.

Now we need to find the value of B.

It says C is 2/3 of B.

C is 2/3 of B.

So here we can see C was 112, and that's 2/3 of B.

But we don't what B is, so we've just left it as B for now.

So we're going to find B.

We're gonna do 112 divided by 2/3, which gives us 168.

How did you get on? Brilliant.

Now you're ready to have a go at these questions independently.

Question number one.

A bag of sweets contains red, yellow, pink, and green sweets.

There are 56 sweets in the bag.

25% of them are red.

2/7 of them are yellow.

The rest are pink and green in the ratio of 5:8.

How many of each colour are there? I suggest you use your calculators, but make sure you show all steps of your working.

Come back when you can tell me how many of each colour sweet there are in the bag.

Good luck.

And question number two.

Aisha, Jun, Izzy, Sam, and Laura raise money by doing a sponsored run.

They raise 600 pounds.

Aisha raised 15%, Jun raised 4/15, and Izzy raised 3/8 of the total.

Sam and Laura collected the rest in the ratio of 2:3.

How much money did each of them raise? Again, here, you may decide to do this with bar models, or you may decide a ratio table.

Do whatever you feel most comfortable with, and I'll be waiting when you get back.

Question number three.

Andeep, Sofia, and Jacob record how long it takes them to complete a puzzle.

Sofia's time is 2/3 of Andeep's time.

Jacob's time is 3 times Sofia's time.

The total of Jacob and Sofia's time is 88 seconds.

What is the difference between Andeep and Jacob's time? Pause the video, and then when you get your answer, pop back and see if you're right, which I know you will be.

Let's check those answers then.

So number one, red was 14, yellow, 16, pink, 10, and green, 16.

Two, Aisha raised 90 pounds, Jun raised 160 pounds, Izzy raised 225 pounds, Sam, 50 pounds, and Laura, 75 pounds.

Now, did you add all of those values up and check they made 600 so you could be sure your answer was right? You did? Brilliant, I taught you well.

And question number three.

Andeep took 33 seconds, Sofia, 22 seconds, Jacob, 66 seconds.

The difference between Andeep and Jacob's times was 33 seconds.

Now we can summarise the learning.

You will often come across questions which involve fractions, percentages, and ratio.

Drawing a diagram such as a bar model or a double number line may help you.

And using a ratio table to organise your working will keep your errors to a minimum, so just as we did during this lesson.

This is especially important if the question has lots of information.

Writing down or drawing what you know using a representation can help.

Fantastic work today.

Well done.

I really look forward to working with you again, and hopefully that will be really soon.

Goodbye.