Lesson video

Hiya.

My name's Ms. Lambell.

I'm really pleased that you've decided to join me today to do some maths.

I'm really looking forward to working alongside you.

Welcome to today's lesson.

The title of today's lesson is "Dividing a Quantity in a Given Ratio," and that is within the unit Understanding Multiplicative Relationships with Fractions and Ratios.

By the end of today's lesson, you will be able to divide one quantity into a given ratio.

Two keywords that we will be using in today's lesson are proportion and ratio.

Just a quick recap of what those words mean.

Variables are in proportion if they have a constant multiplicative relationship.

And a ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

Today's lesson I've divided into two separate learning cycles.

In the first one, we will concentrate on using a bar model to divide in a given ratio.

And in the second, we will look at alternative and maybe more efficient ways of sharing in a ratio.

Let's get going on that first learning cycle.

Here we go.

Andeep and Sam decide to earn some pocket money.

They are paid 15 pounds.

Andeep worked for two hours, and Sam worked for three hours.

Andeep says, "We will each get 7.

50 pounds." Sam says, "That isn't fair.

I did more hours." Do you agree with Sam? I definitely agree with Sam.

It would work out well for Andeep, wouldn't it, but wouldn't be very fair on Sam.

How can Andeep and Sam share the money fairly? They could work out the pay for one hour.

Andeep says, "In total, we worked five hours." And Sam says, "That means we need to divide the pay by five." 15 pounds divided by 5 equals 3 pounds per hour.

So for each hour of work, they should get 3 pounds.

So Andeep will get 6 pounds 'cause he did 2 hours of work, and Sam did 3 hours of work, so Sam gets 9 pounds.

That seems much fairer now, doesn't it? We could have, however, represented this problem with a ratio.

Andeep and Sam worked in the ratio of 2 hours : 3 hours.

As a bar model, it would look like this.

We can see the white part is Andeep and the purple part is Sam.

We know that the total money earned for all of the work that they did was 15 pounds.

So we divide that by 5 to work out what goes into each part, and that's 3 pounds.

We can then see clearly Andeep's part is 2 multiplied by 3 pounds, which is 6 pounds, and Sam's part is 3 multiplied by 3 pounds, which is 9 pounds.

We get the same answer.

Now I want us to have a look at whether it would have mattered if we had set the bar model out in a slightly different way.

Here I've got Andeep.

We've got two parts for Andeep and three parts for Sam.

Notice my parts are the same size.

It's really important because each part's representing one hour, and an hour is a set length of time.

We got 15 pounds in total, so I take my 15 pounds and divide it by 5 because we can see that the total needs to be split into those 5 equal parts, giving us 3 pounds.

Then we can see Andeep's share, and we can see Sam's share.

So both bar models are useful for this type of question.

However, this one, the one that's on the screen now, is most useful in other situations.

So I'm actually going to, from now on, use the bar model that looks like the one on this slide.

Share 300 pounds in the ratio of 2 : 1 : 3.

Let's have a look at that then as a bar model.

Here is my ratio.

I've got 2 : 1 : 3.

Again, notice that all of my parts are exactly the same size.

That's really important.

We know the total amount that we want to share.

So we know that all of those parts is going to have to equal 300.

We're going to do 300 divided by 6, which is 50.

But why did we divide by 6? Because there are 6 equal parts in total and we are sharing the total amount.

That means we're going to put 50 in each of the parts.

So we can see that the first one is 100, the second one is 50, and the third one is 150.

And I always like to do a quick double check.

I would add all of those up and make sure that I have shared out 300, and 100 add 50 add 150 is 300.

So I've checked, and so I can be pretty confident that my answer is right.

I've shared out the right amount of money, or whatever.

Let's do this one together, and then I think you'll be ready to have a go at one on your own.

We're going to share 63 in the ratio of 5 : 2.

Here's my bar model to represent my ratio of 5 : 2, 5 parts and 2 parts.

I'm sharing 63.

That's the total.

I'm dividing by 7 because there are 7 parts in total in my bar model.

63 divided by 7 is 9.

We can put 9 in each of the boxes.

So now we can work out the total of the top bar.

That's 5 multiplied by 9, which is 45, and the bottom bar is 18.

And then a quick check, 45 and 18, the sum of those is 63.

Now I'd like you to have a go at this one.

Share 55 in the ratio of 4 : 7.

Pause the video, and then come back when you're ready to check in with that answer.

Here we go.

Here's your answer.

So we've got 4 : 7.

So the first bar is split into 4 parts, the second bar is split into 7 parts, making sure all parts are the same.

I then take my 55, divide by 11 'cause there are 11 parts in total in my bar model, which is 5.

And then the top bar is 4 multiplied by 5.

That's 20.

And then the bottom bar is 7 multiplied by 5.

That's 35.

And then just a quick check.

Is the sum of those two values 55? It is.

Now you can have a go at these questions.

Pause the video, and I'll be here when you come back to check those answers.

Good luck.

Well done on those.

Now let's move on to question number two.

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

Great work.

Let's check those answers.

So the answers for question number one.

A was 8 : 16, B, 36 : 12, C, 4 : 20, D, 42 : 6, and E, 9 : 15.

And then question number two.

A, Andeep got 15 sweets, Sam 40, and Sofia 25.

B, the lengths of the sides were 17.

5 centimetres, 15 centimetres, and 10 centimetres.

And C, we needed 255 grammes of flour.

We needed 170 grammes of butter and 85 grammes of sugar.

Now let's move on to that next learning cycle, using efficient methods to share in a ratio.

Andeep and Sofia share 120 sweets in the ratio of 11 : 9.

Would you want to draw a bar model to represent this? Probably not.

That would be a lot of bars to draw out, wouldn't it? Let's think about what the ratio means.

What does it mean? It means they're not sharing the sweets equally, and that every time Andeep gets 11 sweets, Sofia only gets 9.

Here's a table we're going to use to help us solve this problem.

So Andeep gets 11, Sofia gets 9.

That's a total of 20 sweets.

If we repeat that, Andeep now has 22 and Sofia now has 18.

That gives us a total of 40.

We've not yet got to 120 sweets, so we're going to continue.

We're going to add 11 onto Andeep's and 9 onto Sofia's.

That gives us 60.

And we're going to repeat this process until we have given out 120 sweets in total.

We can now see that Andeep will get 66 sweets and Sofia is going to get 54 sweets.

What is the multiplicative relationship, though, between 20 and 120? That's right.

It's multiplied by 6.

20 multiplied by 6 is 120.

So actually, we could have done that in one step.

We're thinking about efficiency, so if we can do it in one step rather than five or six, we're going to do that if we possibly can.

We'll take a look at another example.

The ratio of orange juice to lemonade in a drink is 3 : 7.

Sofia wants to make a large jug of the drink.

The capacity of the jug is 4 litres.

We want to know how much orange juice and lemonade that she should use.

So she wants the drink to taste exactly right, so she must keep the ratio correct.

Otherwise, it's going to be not orangey enough or it's going to be too fizzy.

Here is a ratio table.

So in the left column, we've got our orange juice, and in the right column, we've got our lemonade.

So we've got in the ratio of 3 : 7, and that gives us a total of 10.

So if we use 3 litres of orange juice, we would use 7 litres of lemonade, we'd have 10 litres in total.

But Sofia's jug only holds 4 litres, so we know that we need the total to be 4 litres, not 10 litres.

So we're looking for the multiplicative relationship between 10 and 4.

You know what that is.

You're super good at finding the multiplier now, and that is multiplying by 4 over 10.

You can always just double check if you're not sure.

10 multiplied by 4 over 10 is 4.

Basically, that means we need 4/10 of the total, which means we need 4/10 of the orange juice and 4/10 of the lemonade.

So we're going to multiply the orange by 4/10, giving us 1.

2 litres, and we're gonna multiply the lemonade by 4/10, giving us 2.

8 litres.

And then like I said before, it's always worth that quick double check.

Is 1.

2 add 2.

84? Yes.

Sofia now knows that she needs to use 1.

2 litres of orange juice and 2.

8 litres of lemonade.

The ratio of flour to butter to sugar in a cookie recipe is 3 : 2 : 1.

You want to make a giant cookie weighing 450 grammes.

Wow.

Gosh, that is a giant cookie.

How much of each ingredient do you need? Let's draw the ratio table.

So we've got our flour, butter, and sugar in the ratio of 3 : 2 : 1.

Let's work out the total.

That would be 6.

If it was 3 grammes of flour, 2 grammes of butter, 1 gramme of sugar, that would be a total of 6 grammes.

But we don't want the cookie to weigh 6 grammes, we want it to weigh 450 grammes.

We need then the multiplicative relationship between those two.

And you know how to find that.

You do 450 divided by 6, which is 75.

So my multiplicative relationship between the total of the ratio and the total weight of my cookie is 75.

3 multiplied by 75 is 225, 2 multiplied by 75 is 150, and 1 multiplied by 75 is 75.

We now know how much flour, butter, and sugar we need to make a cookie weighing 450 grammes so that the cookie turns out right.

We're going to share 64 pounds in the ratio of 3 : 10 : 7.

Here we can see our ratio table.

It doesn't matter that we don't have a column heading for the 3, the 10, and the 7.

We know the total of those values in the ratio are 20, and we want to share 64.

Again, we're looking for the multiplicative relationship between 20 and 64.

So we do 64 divided by 20, and that gives us a multiplier of 16 over 5.

We now need to multiply all of our values in our ratio table by 16 over 5, and that gives us 9.

60 pounds, 32 pounds, and 22.

40 pounds.

And then I would always just do a quick double check to make sure I've definitely shared out 64 pounds because then I can be quite confident that my answer must be right.

You are definitely ready now to have a go at this check for understanding.

Andeep is mixing pink paint using red and white paint in the ratio of 3 : 5.

He needs to make 20 litres of pink paint.

Here is Andeep's working.

He's made a mistake.

I'd like you please to decide what mistake Andeep has made.

I'm gonna pause the video, and I'll be right here when you get back.

Did you spot Andeep's mistake? You did? Brilliant.

And his mistake is a really common mistake, but I'm gonna show you that by checking, we could have spotted that mistake.

So here, he has found the additive relationship between 8 and 20.

He's added 12 onto 8 to get 20, so therefore he's added 12 onto the red and 12 onto the white.

What could he have done so that he would have known that his answer was wrong? And that is add together the red paint and the white paint.

15 add 17 is 32, not 20.

So at that point, Andeep would have known he'd made a mistake and I'm sure would have then realised that he was needing to find the multiplicative relationship between the two totals.

We're now going to.

Oh no, actually, you're going to correct Andeep's mistake.

So pause the video, and I'll be here when you get back.

Okay, how did you get on? So you found the multiplicative relationship, which was 2.

5.

And I'm okay here, it's the terminating decimal, so I can use that.

You may have decided there to put 2 1/2, or you may have put 5 over 2.

Then we need to multiply the red and the white by 2.

5, giving us 7.

5 and 12.

5.

And then remember, I want us to do that check so that we don't make a mistake like Andeep did.

7.

5 add 12.

5 is 20.

He's gonna have 20 litres of paint.

Now you're ready to have a go at this task.

I'd like you to use the ratio table to solve these problems, so to share in these given ratios.

Pause the video, and I'll be here when you come back to check all those right answers.

Good luck.

Well done.

And now you can have a go at question number two.

You're going to use a ratio table to answer the following.

We're nearly done now, so pause the video, give those questions a go.

You've got all of the skills to be really successful at this, and I look forward to seeing you when you come back.

You can pause the video now.

Great work.

How did you get on with those? You found them easy? See, I told you.

I knew you would.

Here are our answers then.

1A was 12 : 24, B, 48 : 16, C, 6 : 30, D, 56 : 8, and E, 13.

5 : 12.

5.

And then question number two, A, Andeep got 45, Sam 120, and Sofia 75.

The lengths of each side in the triangle were 44.

1 centimetres, 37.

8 centimetres, 25.

2 centimetres.

And the final one, part C, you would need 360 grammes of flour, 240 grammes of butter, and 120 grammes of sugar.

Well done on those.

Now we can summarise the learning that we've done during today's lesson.

You've done fantastically well.

I've been really impressed.

We first looked at using a bar model to share in a given ratio, and there is an example of one that we looked at earlier.

We then looked at maybe a more efficient way to share in a given ratio, and that is to use that ratio table.

Remember, you are looking for the multiplicative relationship between the totals, and then you are applying that to all of the ratio parts.

Like I said, you've done fantastically well during today's lesson.

I've been super impressed.

I've really enjoyed working through these ratio problems with you, and I look forward to seeing you again very, very soon.

Goodbye.