Lesson video

Hiya, my name's Ms. Lambel.

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 Problem Solving with Fractions and Ratios.

This is within the unit Understanding Multiplicative Relationships, Fractions and Ratios.

You've been working really hard recently on using fractions and ratios and we've got lots and lots of skills.

Today we're gonna put all of those things that we've learned together.

By the end of this lesson, you will be able to use your knowledge of fractions and ratios to solve problems. Two keywords that we are going to be using in today's lesson are proportion and ratio.

If you need a reminder of what these are, you could pause the video now and read through those definitions.

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

In the first one, we'll look at strategies to solve ratio problems, and in the second one we'll solve some problems with fractions and ratios.

Let's get started on that first one.

So we're looking at strategies to solve ratio problems. The ratio of red to blue counters in a bag is three to five.

There are 30 blue counters.

Where should the 30 be labelled in a bar model of this problem? Would it be here, here or here? Which is the correct one? And well done if you said the final one.

Five is the part of the ratio that represents the blue counters.

So the part has a value of 30.

If we look at the ratio, the words were red to blue and the ratio was three to five.

This means the red has three parts and the blue is five parts.

We will now take a look at what problems the other bar models were representing.

What question do you think this bar model is asking? Let's see what Sam's got to say.

Sam says, "I think the bar model is saying that there are 30 red counters and how many counters are there altogether?" Do you agree with Sam? Yes.

The 30 is over the red bar and the question mark is around the whole model.

So the 30 is with the red bar and we wanted to know how many counters there are altogether.

So that's why the question mark is around the whole of the model.

30 divided by 3 is 10.

And remember, each part of our bar model is equal.

And now I can find the whole, I've got 8 multiplied by 10, which is 80.

There were 80 counters altogether.

What about this one? I'll give you a moment to have a think before we see what Sam and Jacob have got to say.

Sam says, "I don't think I've seen a bar model with a bracket there before." Have you seen a bar model with a bracket there before? What's Jacob got to say? Jacob says, "Yes, I think you are right, Sam, but I'm sure that we can work it out." I'm loving Jacob's confidence, absolutely loving it.

Jacob is so confident with bar models and what he knows about them already that he feels that even though he's not seen question like this, they'd be able to work it out.

Sam says "The 30 is above the extra blue parts." And Jacob says, "Therefore it must be that there are 30 more blue than red counters in the bag and we want to know how many counters there are in total." Do you think Jacob's right? Yes, he's right.

The 30 is over the two extra blue parts and we can see the question mark is around the whole model.

The 30 is above two extra parts for blue, so we're gonna do 30 divided by 2 to give us 15.

That goes into all of the boxes we need to find the whole, and we can see that there are eight boxes each in 15, so we need to do 8 multiplied by 15 and then we get 120.

There are 120 counters in the bag to start with, Jacob was right to be confident.

He used his bar modelling skills to solve the problem.

Sam and Jacob share some sweets in the ratio of three to seven.

Jacob says, "I'm going to get 12 more than you." Sam wants to know how many he's going to get then? Here's the bar model.

We've got Sam and we've got Jacob.

And we want to know how many Sam is going to get? What other piece of information do we know? We know that Jacob is going to get 12 more, so we need to put the 12 over the extra parts that Jacob has in the bar model and then we need to divide that by four to give us three.

We're gonna put three in each part.

Remember, all of the parts are equal and so then three goes in all of them and now we can see clearly how many Sam is going to get.

Sam is going to get three multiplied by three, which is nine, three lots of three.

Sam gets nine sweets.

We can solve different types of ratio problems with bar models and here are few of the most common.

When we're given the total.

Have you been told the total amount that needs to be shared or has been split? Given one part, have you been given only one of the parts of ratio is? For example, just one colour of counters or just one person's share, or if you're given the difference? Have you been given how much more or less one part is than the other? Look for the words more or less or difference.

In this check for understanding, what I'd like you to do is to decide which type of question each of these are.

Have you been given the total? Have you been given one part or have you been given the difference? Pause the video and when you've got your answers come back.

Well done, let's take a look at those.

Let's have a look.

The first one is difference.

We can see it says there are 15 more red cars.

The second one was one part.

It says we use 25 millilitres of squash.

The third one is the total, two charities divide 350 pounds.

350 pounds is the total.

And then number four is one part.

We are told there are 12 children in the club.

The ratio of red to blue cars in a car park is five to two.

There are 15 more red cars.

How many blue cars are there? Although in this question we are finding the other part to set up the question, we only know the difference between the two parts.

Here is what the bar model is going to look like.

So we've got our ratio of red to blue is five to two.

We are trying to find the blue part.

We want to know how many blue cars there are.

We know that the difference is there are 15 more red cars.

So 15 is the extra parts for red.

So we're gonna do 15 divided by 3 to work out what goes in each part, which is 5, put 5 into all of the parts and now we can see clearly how many blues there are.

And that is 10.

There are 10 blue cars.

The ratio of squash to water in a drink is one to six.

You use 25 millilitres of squash.

How much water should you use? This is the other part in the ratio.

So how much water should we use? We know how much squash.

It's our bar model and we know it is 25 millilitres of squash, so we need to assign the 25 to the squash.

We've only got one part, so we know that that's equal to 25.

And remember, we're trying to find the water.

That's why that's where the question mark is.

So 25.

Here now I can see that I've got six lots of 25, 6 multiplied by 25, so I would use 150 millilitres of water.

Two charities divide 350 pounds in the ratio of two to five.

This time we know the total.

Here's my bar model and I know the total amount to be shared.

So I know the total, my whole entire bar model is 350.

Here, it doesn't specifically ask us to find one of the parts, so we need to find both of them.

We need to find Charity A and Charity B.

We need to take the 350 and divide it into seven parts because the whole bar model has seven parts, which is 50, put 50 into each of the parts.

We can now see that charity A gets 100 pounds and charity B gets 250 pounds.

And remember here it's useful to check that you have shared out 350 pounds.

And yes, the sum of 100 and 250 is 350.

Charity A will receive 100 pounds and Charity B received 250 pounds.

The ratio of children to adults in a play centre is two to three.

If there are 12 children, how many are in the play centre in total? Here, we need to find the total and we know one part, we know how many children there are.

Here's our bar model representing the ratio two to three.

We know that there are 12 children and we're asked to find the total, 12 divided 2 two is 6.

Put six in all of the parts.

We want the whole bar model, which is five parts.

So we there are 30 people in the play centre in total.

Now, I'd like you to have a go at matching each of these problems to the correct bar model.

Pause the video and when you've matched 'em up, come back and check your answers.

How did you get on with that? Let's take a look and see how you got on with those.

I'm sure you got 'em all right.

First one, A was three.

B was four.

C was two and D was one.

Let's just take a quick look at why.

If we look at the first bar model, we can see that 120 is the difference between the parts.

And we're told in the question there are 120 less blue.

Bar model B, we are told that there are 120 blue, that's why the 120 is on the blue section.

If we look at number two, it says there are 120 in total, so that's why the 120 is around the entire bar model.

And then the final one, it says there are 120 red, which is why the 120 is over the red part.

Bar models can be really useful and you don't need to draw them exactly to scale, but there are certain things you need to make sure of to solve a problem correctly.

Can you spot the error in the bar model of this problem? The ratio of lemon to chocolate cakes left on a stall is four to seven.

If there are 12 more chocolate cakes, how many cakes are sold in total? And here we can see a bar model and we can see the 12.

And so there are 66 cakes in total.

What's the mistake? What did you come up with? This is what it should have looked like.

We look at the one on the left, we can see that the four parts actually went over five parts of the bottom bar.

We must make sure that the parts line up in each of our bars.

Then we can see that the 12 now is actually equivalent to the three extra parts.

So although we don't need to draw them exactly to scale, here, we need to make sure that the four parts and the four parts are aligned.

We can also use the ratio to check that we have labelled the correct number of parts.

For example, we know that the difference between four and seven is three parts.

So make sure that we have drawn our brace over three parts.

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

You might want to draw a bar model to help you with these.

What I'd like you to do is to pause the video, give these questions a go, and then when you're ready you can come back and we'll check those answers for you.

Good luck with these.

Great work.

Let's check our answers.

1a, the answer is 525 millilitres of white paint.

B, is 60 millilitres of blue paint and 150 millilitres of white paint.

C, is 490 millilitres in total.

And D, you would need 84 millilitres of blue paint.

We're now gonna move on to our second learning cycle.

We're going to look at problem solving with fractions and ratios together.

Sofia counters 300 red, blue and white cars in a car park.

The ratio of red to blue cars is two to three.

3/10 of the cars are white.

How many of each colour are there in the car park? Andeep says, "This is my first step to solve this problem." So Andeep has drawn a bar model to represent the ratio.

Has Andeep chosen the correct way to start solving this problem? What do you think? No, he has forgotten about the white cars.

The 300 is the total of the red, blue, and white cars.

Let's take a look at how we should start this problem.

The first step is to find the number of white cars.

We know that 3/10 of the car are white, 3/10 of 300.

We know that of a multiplication in maths are equivalent.

So we can rewrite this as 3/10 multiplied by 300, which is 3 multiplied by 30.

I've just done the 300 divided by 10 in my head, which is 90.

There are 90 white cars.

The number of red and blue cars then is going to be 300.

Subtract the number of white cars.

So there are 210.

We now know that there are 210 red and blue cars in the car park.

We could go back and use Andeep's bar model now, but Andeep is saying, "As I've used fractions to solve the first part, I think I'll continue with fractions." What fraction of the cars are red? Look at the ratio.

What fraction of the cars are red? 2/5.

Two parts are red out of five parts in total, 2/5 of the cars are red.

Now we can find 2/5 of 210.

We know that that's equivalent to multiplication, which gives us 2 multiplied by 42, which is 84.

We now know there are 84 red cars.

The number of blue cars, therefore it's going to be 210.

Subtract the 84 red cars and that's 126.

We now know that there are 90 white cars, 84 red cars and 126 blue cars in the car park.

Aisha is three times as old as her younger brother, Benji.

Benji is 2/5 of their cat's age.

The total of Aisha and Benji's ages is 16.

What's the total of their ages? So that's all three of them together, Aisha, Benji, and the cat.

When I'm solving a problem like this, I quite like to write down the people that I'm going to be finding the ages of.

So that's what I've done here.

We've got Aisha, Benji, and the cat.

The ratio of Aisha's age to Benji's age is what? Aisha is three times as old.

So if we've got Aisha to Benji, Aisha is three times as old.

So if Benji was one, Aisha would be three.

So that's the ratio of their ages.

What fraction of their total age is Aisha? Aisha is three quarters of their total age.

Remember here you may choose and prefer to draw out a bar model, that's absolutely fine.

But Aisha is three parts out of the total four parts in the ratio.

Aisha is three quarters of their total age.

Aisha's age is three quarters of 16.

Then because we know we're told in the question that Aisha and Benji's ages is 16.

3 multiplied by 4 equals 12, Aisha is 12.

I'm gonna pop that over there.

Now we can work out Benji's age.

We know that Benji's age must be the 16 subtract Aisha's age because we knew the total of Aisha and Benji's ages was 16.

We know that as Aisha is 12, so therefore Benji is four.

Now let's take a look at how we're going to find the age of the cat.

We're told that Benji is 2/5 of their cat's age.

We don't know how old the cat is, but we know that Benji's age is 2/5 of it and I've changed my odd there for that multiplication.

We know Benji's age, so let's substitute that into our equation and then we can rearrange our equation to give us 4 divided by 2/5, which is 10.

Have we answered the question? Not quite because the question wanted to know the total of all of their ages.

So I need to add together all of their ages.

The total of all of their ages is 26.

I'd like you here to have a go at spotting the mistake.

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

1/3 of the bag are red.

The rest of the sweets are either green or yellow.

In the ratio of three to five, there are 72 sweets in the bag.

How many of each colour are there? Pause the video, decide what the mistake is and then you can come back when you're ready and we'll check and see if you've spotted the mistake.

Well done.

Let's check.

The red sweets are not subtracted from the total before finding the green.

Whoever's answered this question has remembered to find the number of red by finding a third of 72, but then they've used 72 in the bar model.

But actually they needed to subtract the 24 from 72 to find out how many were left that were green and yellow.

What could have been done to check the answer and spot their own mistake? Finding the total of red, green and yellow sweets gives 96 sweets.

So if we add together the red 24, the green 27, and the yellow 45, that gives us 96 sweets.

But we know that there are only 72 sweets in the bag, so it can't possibly have been right.

Always worth checking whether your answer is sensible or not.

Now you can have a go at some of these questions independently.

So pause the video, have a go at this one, come back and then I'll put the next question up on the screen.

Good luck with this.

You may use a calculator.

Do make sure though that you write down all of your steps of working.

You can pause the video now.

And question number two and number three.

Great work on those.

We're nearly there.

We're nearly at the end.

Well done for sticking with me.

Let's check these answers.

Number one, the answer was 24 red, 18 green and 30 yellow.

And just remember to do that check.

Number two was 200 white, 60 red and 90 blue.

Number three, Andeep was 4, Belle was 20 and their cat was 6.

Have we answered the question? Not quite.

Because the question wanted to know the total of all of their ages.

So I need to add together all of their ages.

The total of all of their ages is 30.

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

In a ratio problem, you may be given the total, one part or the difference between the part.

You must look at the question really, really carefully to decide which parts you know so that you can label your bar model correctly.

Using and labelling the bar model can help you solve problems involving ratio.

So that's really just highlighting what I just said there.

Because fractions and ratios are alternative ways of showing proportion.

They can be interchange when solving problems. And we did that, didn't we? In that second learning cycle, we changed our ratio into a fraction.

And remember, it is not always necessary to find all of the parts.

Look at the question carefully.

Make sure that you don't do any more work than you need to.

You've done really, really well today and I've enjoyed working through these problems with you.

I'd look forward to seeing you again really, really soon.

Goodbye.