Lesson video

Hi there.

My name's 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 "Combining Ratios", and this is within the unit "Ratio", and no surprises, by the end of this lesson, you will be able to combine ratios.

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

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

If two things are proportional, then the ratio of the part to 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 whole.

LCM is our abbreviation for lowest common multiple, and the lowest common multiple is the lowest number that is a multiple of two or more numbers.

Today's lesson is split into two separate learning cycles, and in the second one, we will look at combining any ratio as long as it has a common component.

Let's get going with that first one then, combining simple ratios.

Here we go.

In a cafe one weekend, the ratio of cakes to ice cream sold is five to two, and the ratio of milkshakes to cakes is three to five.

We need to work out what is the ratio of milkshakes to ice creams. I'm going to draw a bar model, or a few bar models, to help us solve this problem.

Cakes to ice creams was five to two.

Five cakes, two ice creams. Milkshakes, the ratio was three to five.

It's got three milkshakes and five cakes.

Now, if I move my cakes so that they match in each ratio, so I've basically just slid that bar model along so that I can clearly see I've got the same number of cakes in both my ratios.

What is the ratio of milkshakes to ice creams? We can clearly see now, from our diagram, that there are three milkshakes and two ice creams. The ratio of milkshakes to ice creams is three to two.

In the Oak Academy PE department, the ratio of rugby balls to basketballs is two to three, and the ratio of footballs to basketballs is five to three.

What is the ratio of rugby balls to footballs? Again, let's draw some bar models.

Rugby balls to basketballs, the ratio is two to three.

Footballs to basketballs, the ratio is five to three.

Again, let's slide along that top bar model so that the basketballs match, and now I can clearly see from my bar model that the ratio of rugby balls to footballs is two to five.

It's a bit more abstract now, but remember if you needed to, you could think of these as rugby balls and footballs and basketballs.

The ratio A to B is five to three, and the ratio B to C is three to one.

What is the ratio of A to C? It's useful to write out the ratio for A to B to C so that we can see comparisons clearly.

A to B to C.

Now, from the first piece of information in the question, we know that A to B is five to three, and from the second ratio we know that B to C is three to one, so I've placed those into the ratio.

Now we can see that the ratio of A to B to C is five to three to one.

If we look at the question, the question asked us, "What is the ratio of A to C?" The ratio of A to C is five, and C was one.

So the ratio of A to C is five to one.

Let's look at this one.

The ratio of A to B is four to seven.

The ratio of C to B is six to seven.

Again, we're working out what is the ratio of A to C.

A to B to C.

We know that A to B is four to seven, and we know that C to B is six to seven.

So really important we get that order right, it says C to B.

We therefore know, the ratio of A to B to C is four to seven to six.

The question wanted to know what the ratio of A to C is.

The ratio of A to C is four to six.

And you could simplify that if you wanted to, and what would it simplify to? Yeah, it'd simplify to two to three.

Well done if you said that.

The ratio of A to B is two to five, and the ratio of C to B is three to five.

What is the ratio of A to C? Jacob says, "What do I put for B, as there is a five and a three?" In those previous two examples, B was the same in both of the ratios.

Jacob's made a mistake though.

What mistake has Jacob made? The question gave the ratio of C to B, but Jacob has written it as B to C.

It's really important we check the order of the ratio.

I'd like you now to correct Jacob's mistake, and find for me please the ratio of A to C.

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

And here we go.

Let's check.

A to B to C.

Now we know that A to B is two to five, and we know that C to B, so C to B is three to five.

The ratio, therefore, of A to B to C is two to five to three.

The ratio of A to C is two to three.

In a cafe one weekend, the ratio of cakes to ice cream sold is five to one, and the ratio of ice creams to milkshakes is two to three.

What is the ratio of milkshakes to cakes? So we're back to our cafe now.

Cakes to ice creams. Let's draw our bar model.

That's our ratio of five to one.

And ice creams to milkshake.

There's our bar model, it's two to three.

Again, just as I did previously, I'm going to move my bottom bar model so that the ice creams match.

What is different now to the previous problems we looked at? The number of ice creams is not the same in both ratios.

Hmm.

How can we make the number of ice creams the same in each ratio? We add another ice cream.

There's my extra ice cream.

But why can I not just add another ice cream? If we add another ice cream, the proportion's no longer the same.

The ratio will have changed.

The ratio of cakes to ice cream needs to stay as five to one.

If we add, and notice I've put that in inverted commas, because we're not actually strictly adding it.

We're introducing it.

If we add another ice cream, how many cakes do we need to "add", in its loose sense? The ratio means for every one ice cream there are five cakes, therefore we need to add five cakes.

We can now check, is our ratio the same? So in the grey bar model, I've got one ice cream and five cakes, and then in the purple bar model, I've also got one ice cream and five cakes.

The ratio of milkshakes to cakes is? Three milkshakes for 10 cakes.

The ratio of milkshakes to cakes is three to 10.

Your turn.

Check for understanding.

Given the ratio of hearts to faces is three to five, and clouds to faces is two to one, find the ratio of hearts to clouds.

Jacob says, "I think I can replace each face in the top ratio with two clouds to help me find the ratio of hearts to clouds." Do you agree with Jacob? Pause the video, make your decision.

Don't forget to have some justification of that decision, and then when you're ready, you can come back.

And did you agree with Jacob? And you should have agreed with Jacob.

Since for every one face, if we look at that bottom bar model, for every one face there are two clouds.

So I could replace every face with two clouds, and then I would be able to see what the ratio is.

What is the ratio now of hearts to clouds? That's right, it's three to 10.

Well done.

Your turn now then.

Task A.

Pause the video, give these questions a go, come back when you're ready, and I will reveal the next set of questions.

And questions three and four.

Again, pause the video, work through these carefully, and then come back when you're ready.

And finally, question five.

Pause the video, and no surprises, I'll be waiting here when you get back.

Now let's check our answers.

So number one, the ratio of squares to triangles was three to one.

Two, the ratio of hearts to faces is two to five.

Question three, the ratio of A to B to C was two to three to seven.

Number four, the ratio of A to C is three to nine.

Now if you need to pause the video through those work-ins because you've made an error, obviously you can do that, but I know that you didn't make any errors.

And finally, question five, the answer is three to eight, squares to triangles is three to eight.

Now we can move on to our second learning cycle.

We're now going to be looking at combining any ratio with a common component.

The ratio of A to B is one to six, and the ratio of B to C is three to five.

We need to find the ratio of A to B to C.

We're going to write out the ratio of A to B to C.

Now we know that the ratio of A to B is one to six, so this is just what we did previously, and we know that B to C is three to five.

The common component in both of those ratios is B.

We need to make B the same in both ratios.

What is the LCM, the lowest common multiple, of six and three? And hopefully you've said six.

The lowest common multiple of six and three is six.

We therefore now need to find the equivalent ratio of three to five, where B is six.

So I've got my ratio of B to C, and we know that it originally is three to five.

But we want the common component of B to be six.

So I'm looking for that multiplicative relationship, and what is it? Multiply by two, good.

So I'm gonna multiply by two, giving me 10.

I now know that C is 10, and then A is one.

I didn't need to change that first ratio, because B already had a value of six.

We therefore know the ratio of A to B to C is one to six to 10.

Let's look at another one.

A to B to C.

We know that A to B is three to two, and we know that B to C is seven to three.

Again, the common component here is B.

We need to make B the same in both ratios, and to be most efficient at that, we need to work out the lowest common multiple of two and seven.

What is the lowest common multiple of two and seven? Yeah, 14.

So 14 is going to be our new B.

This time, neither of the original ratios have a B value of 14, so we are going to need to convert both of them.

We're gonna need to find an equivalent ratio where B is 14.

Let's take a look at the ratio of A to B.

It started as three to two, but we know that we want that B value to be 14.

We're looking for that multiplicative relationship, and it is? Multiply by seven.

So we multiply the right hand side of our ratio by seven.

What do we need to do to the left? We need to do the same.

Three multiplied by seven is 21.

Now let's take a look at our original ratio of B to C, and that was seven to three.

We want B to be 14, so that we can combine these ratios.

My multiplicative relationship here is? Yeah, multiply by two.

Or you might have said double.

And so I'm gonna double that side, giving me six.

So now I can put those values of A and C into my ratio.

A was 21 and C was six.

This means the ratio of A to B to C is 21 to 14 to six.

So the important thing here is that we are making the common component the same in both ratios by using equivalent ratios.

Slightly different question here, but actually, we're gonna do exactly the same thing.

It's just our answer that's going to be slightly different.

A to B is seven to five.

B to C is seven to two.

The common component is B.

We need to make B the same in both ratios.

We're looking then, to be most efficient at this.

You don't have to use those common multiple, it just means you'll need to do some simplifying at the end.

What is the LCM of five and seven? And that's 35.

So we need to find the equivalent ratios, our original two ratios, seven to five and seven to two, where B is 35.

Let's start with the ratio A to B.

The multiplicative relationship this time is multiplied by seven.

So we're gonna multiply seven by seven to give me 49.

Now we're going to look at the ratio of B to C, which originally was seven to two.

Now it's still going to be seven to two, we're just going to create an equivalent, where B is 35.

The multiplicative relationship this time? Multiply by five.

Two multiplied by five is 10.

I can now put those values into my ratio.

The question this time though, asks for us to express A as a fraction of the total of A, B, and C.

Well we can see that A is 49, and the total of A, B, and C is 94.

So the answer is 49 over 94.

Please could you pause the video and have a go at this one? Good luck.

Super work on that.

Let's check our answers then.

So we should have started off with recognising that the lowest common multiple of five and two is 10.

So B, we want to be 10 in both of our ratio tables.

My multiplicative relationship between five and 10 is multiplied by two.

So I'm gonna do the same to the left side of the ratio, giving a six, and repeat that for the ratio of B to C.

My multiplicative relationship here is multiplied by five.

Seven multiplied by five is 35.

A as a fraction of the total, or before I do that, I'm gonna go back, I'm gonna put my values of A and C into my ratio, so we end up with six to 10 to 35.

A is six, and the total of all of the parts of the ratio is 51.

So my answer is six over 51.

A box of sweets contains 145 sweets, which are either hearts, stars, or bottles.

The ratio of hearts to stars is four to three.

The ratio of stars to bottles is two to five.

How many hearts are there in the box? Hearts to stars to bottles.

This is just what we've done on the previous questions.

So hearts to stars is four to three, and stars to bottles is two to five.

There's my common component.

My common component this time is stars.

My LCM, lowest common multiple of three and two is six.

My ratio table, I'm going to convert the ratio four to three, so that it has an S value of six.

My multiplicative relationship is multiplied by two.

Four multiplied by two is eight.

Now we'll deal with the second ratio, the ratio of two to five.

We know that we want the number of stars to be six.

So our multiplicative relationship is multiplied by three.

Five multiplied by three is 15.

Now we can go back and complete our ratio.

So the ratio, we've got hearts was eight, stars we already knew is six, and bottles is 15.

Now we know that that is the ratio of hearts to stars to bottles, and we know that the total of those parts is 29.

The box of sweets contains 145 sweets, so we know the total number of sweets is 145.

We're looking again, for that multiplicative relationship.

Here it's not so obvious, but remember you can do 145 divided by 29, and that will give you our multiplier of five.

I'm gonna multiply everything in my top ratio by five.

Eight multiplied by five is 40.

Now, we could find those two, but we don't have to, 'cause actually, the question just wanted us to answer how many hearts are there in the box? And we can clearly see now how many hearts are in the box.

There are 40 heart-shaped sweets in the box.

Moving on to a question now, we've got some algebra.

Y=3x.

Z=4y.

We need to find the ratio of X to Z.

What is the ratio of X to Y? So using the equation, what is the ratio of X to Y? Y is three times larger than X.

That's what it means, doesn't it? Y=3x means that Y is three times the size of X, therefore the ratio is one to three.

Let's fill that in.

What is the ratio of Y to Z? So using that second equation.

Z is four times larger than Y, therefore the ratio is one to four.

One to four.

Let's make sure that our common component, in this case that's Y, is the same in both ratios.

Lowest common multiple of three and one is three.

So we don't need to change the top ratio.

It's already got a Y component of three, but we do need to change the Y to Z ratio, one to four.

My multiplicative relationship, multiplied by three, multiplied by three gives me 12.

So I can now fill in, we know that X is one, and Z is 12.

The question asked us to find the ratio of X to Z, and I can now clearly see from that chart that the ratio of X to Z is one to 12.

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

Now you've only done one example, so you might feel you're not quite ready yet, and you might need to just go back and re-watch that first example.

Either way, pause the video now, and then come back when you're ready.

How did you get on? We'll see, shall we? What is the ratio of X to Y? Y will be four times larger than X.

So the ratio is one to four.

What's the ratio of Y to Z? Z is nine times larger than Y, so the ratio is one to nine.

We need to make the common component for both ratios the same.

LCM of four and one is four.

We need to make sure that the Y component is four.

My multiplicative relationship here is multiplied by four.

Nine multiplied by four is 36.

So we end up with X is one, Y is four, Z is 36.

We needed to find the ratio of X to Z.

So that is one to 36.

Well done if you got that right.

The ratio of cars to vans in a car park is seven to two.

The ratio of black vans to white vans is two to five.

What fraction of the vehicles are white vans? Let's draw some bar models to help us with this.

Here's my bar model for the number of cars to vans.

And here's my bar model for the number of black vans to white vans.

We're looking for the fraction of the vehicles that are white vans.

What fraction of the vehicles are vans? There's two vans, and there are nine parts in total.

So two out of the nine parts are vans.

We want to know white vans.

What fraction of the vehicles are white? There's our white parts.

That's five out of seven.

There are seven parts in total.

That means two ninths of five sevenths.

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

Two multiplied by five is 10, and nine multiplied by seven is 63.

The fractions of vehicles that are white vans is 10 over 63.

In a box of chocolates, there are twice as many plain as white, and four times as many milk as plain.

What is the ratio of milk to plain to white? Let's start with, what is the ratio of plain to white? It says there are twice as many plain as white.

This means P will be two times larger than W, therefore the ratio P to W is two to one.

Let's put that in.

What is the ratio of milk to plain? It says four times as many milk as plain.

M will be four times larger than P, 'cause there's four times as many, so therefore the ratio is four to one.

Four to one.

This time, the common component in both ratios is plain, is P.

My LCM of two and one is two.

This means I don't need to change the top ratio, but I need to change the bottom one, so that its P value is two.

We've got milk to plain is four to one, but we want the P value to be two.

My multiplicative relationship here is multiplied by two.

So I end up with eight.

I can now put that information into my ratio.

M is eight, and W is still one.

We have now got that the ratio of milk to plain to white is eight to two to one.

Have a go at this one for me, please.

I'd like you to match each of the following to the correct ratio, and the ratios are given in the form A to B.

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

And how did you get on with those? Did you manage to match them all up? Great work.

There are three times as many A than B, so that means that A is three times the size of B.

So it gives us the ratio three to one.

A is half the size of B.

That gives us the ratio of one to two.

Just check, is A half the size of B? Yes.

The next one is written as an equation.

A=2B.

So that means A is twice the size of B.

So that gives us the ratio two to one.

Is A twice the size of B? Yeah, B is one, and A is two.

That's twice one.

And finally, B=3A.

So this means that B is three times the size of A, so it's the ratio one to three.

Is B, the number on the right hand side, three times A, which is the number on the left hand side? Yes.

So it's always worth doing that quick little double check to make sure you've not made a mistake, and got the numbers around the wrong way.

Ready now then, for the task A.

Have a go at these questions, so pause the video, and then when you're ready, come back.

I'm moving on to questions four and five.

And question number six.

Super work.

Let's check those answers.

One, the answer was 18 to three to seven.

Two was 63 to 16.

And three was 28 over 109.

Question number four was 600 bottles, and five was one to 21.

I'm wondering how you got on.

Well done.

Question number six.

The ratio is 15 to three to one.

And did you get that one right also? Yeah, of course you did.

Now we can summarise the learning that we've done during today's lesson, and we did quite a lot.

Ratios can be combined if they share a common component.

So there's one of the easier examples we looked at, where we used a bar model, and they shared a common component.

The common component was cakes, and it was the same number of cakes in each.

So we were able to easily work out the ratio of milkshakes to ice creams. If the common component does not have the same value in each ratio, equivalent ratios are used to create a new ratio where the common component has the same value in both ratios.

And we've got an example there that we've been through during today's lesson.

So the common component was B, we had five and two, so we needed to make that the same in both ratios using the lowest common multiple, which was 10.

Sometimes the ratios can be written as words, or as an equation.

So for example, remember that Y=4X means that the ratio of X to Y is one to four.

Absolutely superb work today.

There's quite a lot to get your head around during that lesson, and you've done fantastically well to stick through right to the end.

Take care of yourself, and I look forward to seeing you again really soon.

Goodbye.