Lesson video

Hi there, my name is Ms. Lambell.

You've made such a fantastic choice deciding to join me today to do some maths.

Come on, let's get going.

Welcome to today's lesson.

The title of today's lesson is changing ratios, and that's within the unit ratio.

By the end of this lesson, you'll be able to find quantities given a change in a ratio.

Some keywords that will be using in today's lesson are proportion, ratio, LCM, and lowest common multiple.

Remember, proportion is a part to whole, and sometimes part-to-part comparison.

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

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

LCM is our abbreviation for lowest core multiple, and the lowest core multiple is the lowest number.

That is a multiple of two or more numbers.

Today's lesson is in two learning cycles.

In the first one, we will explore changing ratios; and in the second one, we will concentrate on problems where the ratio changes.

Let's get going with that first one and that's exploring changing ratios.

A bag contains 35 red and green sweets in the ratio of 5:2.

3 red sweets are eaten.

what is the ratio of red to green sweets now in its simplest form? Izzy says, I think the ratio of red to green sweets is now 2:2 or 1:1.

Simplify your ratio.

So well done, Izzy.

Sam says, I think the ratio of red to green sweets is 11:5.

Who do you agree with and why? Izzy has just subtracted 3 from the original ratio and not considered the actual number of sweets originally.

Sam is actually correct.

Let's take a look at how Sam got their answer.

So the ratio, the new ratio of red to green is 11:5.

We knew the ratio of red to green was 5:2.

Given us a total of 7 parts.

We also know from the question that the bag contained a total of 35 sweets.

What is the multiplicative relationship that takes us from 7 to 35? Of course, it's multiplied by 5, well done.

So I need to multiply everything in the top ratio by 5.

5 multiplied by 5 is 25 and 2 multiplied by 5 is 10.

Originally, there were 25 red sweets and 10 green sweets in the bag.

The question told us that 3 red sweets were eaten, meaning there were now 22 red sweets and 10 green sweets.

We know the ratio now is 22:10.

And Sam has spotted that both of those numbers are multiples of 2, so therefore we can divide both of them by 2, given us the ratio of 11:5.

Now, to start with, I didn't give Sam credit for simplifying his ratio, but I can see now that I should have.

Well done, Sam.

You found the correct answer and you also simplified your answer.

Like I said, we can now see that Sam's answer is correct.

Sofia's uncle sells t-shirts and glasses on his market stall.

On Saturday morning he has a mixed box of 90 items with t-shirts and sunglasses in the ratio of 3:2.

He sells 14 t-shirts and 6 pairs of sunglasses on Saturday.

What is the ratio of t-shirts to sunglasses at the end of Saturday? We know the original ratio is 3:2, which gives us a total of 5 parts.

We also know the number of items that are in the box, which is 90.

That's a total number of items that were in the box.

We are looking for that multiplicative relationship.

Remember, if it's not obvious, you could do 90 divided by 5, and that will give us the multiplier of 18.

3 multiplied by 18 is 54 and 2 multiplied by 18 is 36.

Originally, there were 54 t-shirts and 36 pairs of sunglasses in that box.

Now because 14 t-shirts were sold, there were 40 t-shirts; and because 6 pairs of sunglasses were sold, there are now 30 pairs of sunglasses.

We can then write that as a ratio in our ratio table and simplify.

This time, the highest growing factor of 40 and 30 is 10.

I'm going to divide both sides of my ratio by 10, giving me 4:3.

The ratio of t-shirts to sunglasses at the end of Saturday was 4:3.

There are 180 vehicles in a car park.

The ratio of cars to vans is 7:2.

4 cars arrive and 4 vans leave.

What is the ratio of cars to vans in the car park now? And you need to give your answer in its simplest form.

Please, could you now pause the video, work out your answer, and then when you've got that come back and we'll check.

What did you decide? What did you work out to be? Hopefully you said b.

There were originally 140 cars and 40 vans.

So now because 4 cars arrived, there are 144 cars; 4 fans leave, there are 36 vans.

That gives us the ratio of 144:36, which simplifies to 4:1.

Absolutely superb.

Well done if you've got that right.

Wow, that was a quick learning cycle, wasn't it? Now you're gonna have a go at these questions.

A running club has 104 members with adults and children in the ratio of 8:5.

6 more adults join.

What is the ratio of adults to children now in its simplest form? And the question number two, a bag contains 42 red and green sweets in the ratio of 4:3.

6 green sweets are eaten.

What is the ratio of red to green sweets now in its simplest form? Good luck with these.

Pause the video and then come back.

Remember to show all steps you've your work in and then we'll check those answers for you.

In fact, there's probably some more questions Well done, and no surprises, there was another question.

Question number three, in a car park, there are 231 vehicles.

The ratio of cars to vans is 9:2.

7 cars leave and 7 vans arrive.

What is the ratio of cars to vans now? Pause the video, give this one a good go.

And then when you come back, we'll check that for you.

Superb, question number one, the original ratio was 64:40.

Then the new ratio once the 6 adults joined was 70:40, which simplifies to 7:4.

Question number two, the original ratio was 24:18.

Somebody ate 6 green sweets, and so therefore the ratio was then 24:12, simplifies to 2:1.

Question number three, the original ratio was 189:42.

That meant that the new ratio once the 7 cars left and the 7 vans arrived was 182:49, and that simplified to 26:7.

Well done, if you've got all of those right, which I'm sure you did.

Now we can move on to our second learning cycle, problems where ratio changes.

The ratio of adults to children at an athletics club is 6:5.

6 new adults join.

The new ratio is now 4:3.

How many children are members of the club? So we've got an original ratio and we've got a new ratio.

Let's a look and see what Izzy and Sam have got to say about this.

All right, so Izzy draws a table of equivalent ratios of 6:5.

She's drawn an equivalent ratio table with all of the equivalents all the way up to 72:60.

Not sure why she decided to stop there.

Maybe in a minute we'll find out.

Sam says, I'll draw out an equivalent ratio table for 4:3.

Thanks for that, Sam.

If Sam does this, what will we need to look for? Any ideas? Remember, it's okay not to really have an idea at the moment because this is the first time we've looked at anything like this, but I'm pretty certain you might have an idea.

The number of children doesn't change.

Therefore, we need to find the ratio in each table where the number of children is the same.

Here's the original ratio table.

So the equivalent ratios for 6:5, and here is the new ratio.

And this one, Sam has had to go into two separate columns.

We now look for a ratio in each table where the number of children is the same.

So let's have a look.

I've got 5 in the first one, no, 10, no.

And so I have to go to 15 and I end up with 15 children in both of those ratios.

Now let's check that the number of adults has increased by 6 because that's what we were told the new ratio was because there were 6 new adults that had joined the club.

Now, if we look at the adults, the increase has gone from 18:20.

And that's only an increase of 2, not 6.

Now we look for the next time the number of children is the same, and that's 30.

Again, we need to check that number of adults is increased by 6.

So originally there were 36 adults, there are now 40.

That's unfortunately an increase of 4 rather than 6.

So we're going to go to the next pair of ratios where the number of children is the same.

So we're gonna go to the next pair, and that's 45 and 45.

Let's check that the number of adults is increased by 6, and it does.

54:60 is an increase of 6, and we can see that that means there are 45 children.

The number of children didn't change.

No children left or joined the club.

It was just the number of adults that changed.

So we can see that in both of these ratios, we've got 45 children.

So that's how many children must have been at the athletics club.

Sam says, I've noticed that we could have saved loads of time.

Can you see what Sam is talking about? Let's see what he's got to say.

They say the children are increasing by 15 each time in both tables.

Let's take a look.

Yeah, that's right, isn't it? It went 15, 30, 45, 15, 30, 45.

So Sam's right.

It is increasing by 15 each time.

Why do you think the children are increasing by 15 each time? Sam says that's because the LCM, remember that's lowest common multiple of 5 and 3 is 15.

All right, yes, that makes sense, doesn't it? Why it's going up by 15 each time? We could have saved ourselves some time then.

So we could.

We know that each time it is going to be increasing by 15.

So we're looking for our first equivalent ratio, 5:15.

Our multiplicative relationship is multiplied by 3.

6 multiplied by 3 is 18.

And we know it's increasing by 15s 'cause that's what the LCM is.

So the next one is gonna be 30.

So 5 multiplied by 6 is 30, 6 multiplied by 6 is 36.

So the next one is going to be 45.

5 multiplied by what? Is 45, that's 9.

6 multiplied by 9 is 54.

The next one is 60.

5 multiplied by what? Is 60, that's 12.

6 by 12 is 72.

We do the same thing now then for the new ratio.

The multiplicative relationship between 3 and 15 is multiplied by 5.

4 multiplied by 5 is 20.

Now let's look what happens.

We need to go to 39.

The multiplicative relationship between 3 and 30 is multiplied by 10.

4 multiplied by 10 is 40.

And then on to 45, the multiplicative relationship multiplied by 15.

4 multiplied by 15 is 60.

And we can stop there as the number of adults has increased by 6 and the number of children has stayed the same in those two rows.

So 54 became 60 and the number of children stayed the same.

I think you'll agree with Sam that that's a much more efficient way of answering that question.

We still get the same answer that there were 45 children at the club.

Actually, there's an even more efficient method than that one, and we're going to take a look at that now.

Remember, any three of these methods is absolutely fine for you to use.

You just choose the one that you feel most confident with.

Same question then.

We know the ratio, the original ratio is 6:5 and we know the new ratio is 4:3.

What do we know doesn't change between the two ratios? And that's the number of children.

It stays the same.

Therefore, we are going to make equivalent ratios where the value for the children is the same in both.

That's why we use the LCM of 5 and 3, and what was it? That's right, it was 15.

So we put 15 for the number of children in both.

Now let's create our equivalent ratios using those multiplicative relationships.

So multiply by 3, that gives us 18.

And with this one, it's multiplied by 5, which gives us 20.

We can now clearly see that the ratio shows the number of children staying the same.

So in the original ratio there were 15 children and in the new ratio there are also 15 children, because no children joined or left the club.

What's happened between the adults? Well, we can see that it shows 2 more adults.

20 is 2 more than 18, but we wanted an increase of 6 adults.

It told us 6 new adults join the club.

What is my multiplicative relationship between 2 and 6? That's multiplied by 3.

I'm therefore going to multiply by 3.

18 multiplied by 3 is 54 and 15 multiplied by 3 is 45.

20 multiplied by 3 is 60 and 15 multiplied by 3 is 45.

Now we can see the 6 new adults.

So there were 45 children at the club.

Now like I said, there are three methods there getting slightly more efficient as we went through, but there is absolutely no problem with sticking with that very first method if you need to.

Let's take a look at another example.

And from this point on, I'm going to use that most efficient method.

But remember, you can choose to use one of the other two if you prefer.

The ratio of money in Izzy and Sam's savings accounts is 3:5.

Izzy spends 5 pounds and the ratio is now 1:2.

How much money did they have between them originally? Here's my ratio table for the original ratio, and here's my ratio table for the new ratio.

What doesn't change between the two ratios? And you should have said the amount of money that Sam has.

Sam doesn't add any money to his account and he doesn't spend any money.

He has the same amount of money in the first and the second ratio.

Therefore, we will make equivalent ratios where the value for Sam's money is the same in both ratios.

So we're looking at Sam, that means we want to find the LCM, lowest core multiple of 5 and 2.

What is the LCM of 5 and 2? Yeah, of course it's 10, isn't it? It's 10.

We now need to create those equivalent ratios with S being 10, Sam's money being 10 in each of those ratios.

So it's that multiplicative relationship again.

5 multiplied by 2 is 10, 3 multiplied by 2 is 6.

Relationship this time is multiplied by 5.

1 multiplied by 5 is 5.

We now can see that Sam's money has stayed the same.

Let's consider what's happened to Izzy's money.

This shows a decrease of a pound.

But actually we know that Izzy spent 5 pounds, so we want that change to be 5 pounds.

What's the multiplicative relationship between 1 and 5? That's a nice easy one, isn't it? It's multiplied by 5.

We just need to then add third row onto our table or fourth row if I count the top headings.

And I need to multiply by 5.

6 multiplied by 5 is 30, 10 multiply by 5 is 50, 5 multiplied by 5 is 25, and 10 multiplied by 5 is 50.

And then we can see that Izzy has spent 5 pounds.

She started with 30 pounds, she now has 25 pounds.

So we can clearly see that does represent the 5 pound that Izzy spent.

Originally, they had 80 pounds between them.

So originally, it's the first of the two ratio tables and we can see that between them they have 30 pounds and 50 pounds each, which in total is the 80 pounds.

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

The ratio of red to blue beads in a bag is 7:8, 2 red beads are taken out.

The ratio is now 5:8.

I'd like you to decide, is that true or false? And as always, with these true and false questions, I don't just want a true or false.

I want justification for your answer.

Please could you pause the video and then when you come back, we'll check and see whether you've got it right, which of course I know you will have.

What did you decide? The correct answer was false.

It was false.

And the reason for that is because ratio share a multiplicative relationship, not an additive relationship.

We cannot simply just subtract 2 from the 7, because we don't know how many there were in the bag to start with.

Okay, we need to be looking at that multiplicative relationship.

Izzy, Sam and Andeep are playing a game.

The points they each have are in the ratio of 4:5:3.

Izzy loses some of her points to Sam and Andeep.

The ratio of points is now 3:5:2.

What is the lowest number of points Izzy loses? Let's draw a ratio table.

Here is the ratio table for their original points, 4:5:3, and that's a total of 12.

And here is our ratio table for the new proportions of points, 3:5:2, and that's a total of 10.

What does not change between the two ratios this time? And it's the total number of points.

The total number of points is the same.

This means we need to make the total number of points the same in both of our ratios.

So what is the LCM of 12 and 10? And that is 60.

Remember, we're trying to use the LCM if we can just because it makes life a little bit easier.

So we're gonna put 60 in both of them.

So it's that multiplicative relationship.

12 multiplied by 5 is 60.

So I'm gonna multiply everything by 5.

And then if we look at the second table, my multiplicative relationship is multiplied by 6, so I need to multiply everything by 6.

Therefore, we can now see that Izzy loses 2 points.

She started with 20, she now has 18.

She has lost 2 points.

Why is the question worded? What is the lowest number of points? And the reason for that is because we don't know the original number of points.

So the total could have been any multiple of 60.

We don't know the total number of points.

We're now ready for task B.

Number one, the ratio of cows to sheep at a farm is 3:4.

The farmer sells 2 sheep, and the ratio is now 4:5.

How many sheep are there now? Question two, the ratio of money in Izzy and Sam's savings account is 6:5.

Izzy pays her birthday money of 34 pounds into the account.

The ratio now is 7:3.

How much money did Sam have to start with? Pause the video, give these questions a go.

Make sure you show me all steps of your work in and you can use a calculator for these.

Good luck, pause the video now, and then we'll move on when you get back.

And questions three and four.

Question three, a bag contains heart and star shaped sweets.

The ratio of hearts to stars in the bag is 2:3.

Sam eats 2 heart-shaped sweets, the ratio is now 4:9.

How many stars are in the bag? And question four, the ratio of the number of 1-pound coins to 2-pound coins in Sam's money box is 5:6.

That grand gives them 2 more 1-pound coins.

The ratio is now 7:8.

How much money was in Sam's money box originally? Pause the video.

Good luck with these, and I'll see you when you get back.

And question number five, Izzy, Sam, and Andeep are playing a game.

The points they each have are in the ratio 2:7:6.

Izzy and Sam lose some of their points to Andeep.

The ratio of points is now 1:3:6.

a, Jun says, "Andeep's score does not change," explain why Jun is wrong; and b, what is the minimum number of points that Andeep gains? Pause the video.

Good luck with these ones.

And then when you are ready, we'll come back and check those answers.

We're almost there.

Great work, now let's check those answers.

Question number one, there were 30 sheep.

Question number two was 30 pounds.

Number three, there were 9 stars.

Number four, Sam originally had 88 pounds.

Number five, this is an example of a response.

So remember you will have something, you know, it might not be exactly the same but should be along the same lines.

The proportion of points changes between the two ratios.

So although the value of Andeep's parts in the ratio has the same value, the number of parts has changed.

And part b, the minimum number of points that Andeep gains is 6 points.

Summarising our learning from today's lesson.

Then when considering changes in ratios, the most important thing to remember is that the ratios have a multiplicative relationship.

If the original total is known, the original value should be found, adjusted, and the new ratio found.

It is however not necessary to know the original total if you are given the old and new ratio as long as one component remains unchanged.

And here, we use the LCM to find the equivalent ratios.

Great work today, well done.

I look forward to seeing you again very, very soon.

You will join me, won't you? Yeah, of course you will.

Take care, goodbye.