Lesson video

Hi there, my name's Miss 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 "Writing Equations from Ratios." And no surprises, that's in the unit ratio.

By the end of this lesson, you will be able to write equations involving ratios.

Some keywords that we'll be using in today's lesson.

Obviously, one of those is ratio, we'll also refer to the numerical coefficient, and equation.

A quick recap of those then.

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

A numerical coefficient is a constant multiplier of the variables in a term.

An equation is used to show two expressions that are equal to each other.

Today's learning is split into two learning cycles.

In the first one, we will look at exploring equations and ratios, and we will concentrate on doing this with bar models.

And in the second learning cycle, we will look at being able to confidently convert between equations and ratios.

Let's get going with that first one then.

So we're going to look at exploring equations and ratios with bar models.

The ratio of x to y is one to four.

Let's take a look at what this looks like as a bar model.

We got x is one part, y, four.

So the ratio here we can see for x to y is one to four.

What would be the equation linking x and y? What do you think? Let's have a think about that together.

How many xs are equal to y? If we look at the y bar, we can clearly see that it is four times the length of the x bar.

There are four of those xs would make the y bar.

There are four xs to equal y.

Therefore, the equation linking these two of y in terms of x is y equals four x.

What about if we switch the ratio around? The ratio now is still x to y, but the ratio now is one to four.

Again, let's take at what this looks like with a bar model, x is four, and y is one.

What would be the equation linking y and x this time be? This time, we would need to consider what fraction of x is equal to y.

So what fraction of x is equal to y? It's a quarter.

We can see that y is one outta the four parts of x.

It's a quarter the size of x, therefore, the equation of y in terms of x is y equals one quarter x.

I think you're ready to have a go at one of these yourself now, 'cause you're really good at bar models.

I'd like to use the bar model to write the equation of y in terms of x.

Pause the video, make your decision, and then come back when you're ready.

What did you decide, a, b, or c? The correct answer was c.

We can see that the y is six times the length of x and that's what six x means.

How about this one? Again, pause the video.

And this time, you should have come up with c again.

So c, we can see that y is one fifth of x, is one outta the five parts of x.

Let's take a look then at how the ratios and the equations are linked.

We've got some ratios of x to y.

The equation in terms of y.

Jacob says, "I think I can see the link and would be able to complete the table apart from the last one." Can you fill in the rest of the table? Go on, give it a go.

I'm wondering if you've got these equations.

The correct equations are y equals three x, y equals one seventh x, and for the last one, I'm not too sure at the moment.

Maybe you've had a go at it.

But what we're going to do now, we're going to take a look at how we do this last one.

The ratio of x to y is two to five.

So let's take a look at what this looks like as a bar model.

I've got x is two, and y is five.

We are looking at the equation linking x and y.

How many xs are equal to y? A little bit harder this time.

I'm gonna give you a moment to think about it, and come up with your own answer before we go through it.

How many xs are equal to y? The purple box I've put around x.

I'm now going to see how many of those I can fit into y.

And I can see I've got one whole x, another whole x, and then I've got a fraction of x, and I can see that half of that box is shaded.

This means, that there are 2.

5 or five over two xs are equal to y.

Therefore, the equation of y in terms of x is y equals 2.

5x, or y equals five over two x.

Remember 2.

5 and five over two are equivalent.

One is a decimal, and one is a fraction.

(birds chirping) Let's take a look at another one then, x, three, and y is five.

The ratio is three to five, and that bar model represents that.

Again, we are looking for the equation linking x sorry, linking y and x.

How many xs are equal to y this time? There's our x.

So let's put the xs onto the y bar.

We can see we've got one whole one, and then what fraction of that final one is shaded grey, and that's two thirds.

Therefore, how many xs are equal to y one and two thirds, or you may prefer to write that as an improper fraction as five thirds.

Therefore, the equation of y in terms of x is y equals one and two thirds x, or y equals five thirds x.

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

Using this bar model, I'd like you to write the equation of y in terms of x.

Pause the video, give it a go, and then when you're done, pop back, and we'll check that answer for you.

Let's take a look.

We've got x.

How many times does x fit on y? And we can see it's one, two holes, and then one third left over.

So this means that y is equal to one and two thirds x, or like I said, if you prefer, you could write that seven thirds x.

We'll take a look at this one now.

The ratio of x to y is five to three.

There's my bar model to represent this problem.

Again, we are looking for the equation linking y to x.

What fraction of x is y? So this time, y is a fraction of x, but what fraction? Let's take a look, there's x, and there's y.

What fraction of that purple box is covered by y, which are the grey boxes? And that is three fifths.

Therefore, the equation of y in terms of x is y equals three fifths of x, or you may choose to write that as a decimal as it's a terminating decimal.

We can do that, y equals 0.

6 x.

Your turn now.

Check for understanding.

Have a go at this one.

Please could you write the equation of y in terms of x.

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

Okay, how did you get on? Super.

There's x, let's put it down on the y.

What fraction is coloured grey in the purple box? And we can clearly see that's two fifths x, or again, it's a terminating decimal, so you could write it as 0.

4 x.

How did you get on? Brilliant, well done.

Your turn now then.

Task A.

So very, very quick learning cycle, this one.

So Task A, you're gonna use the bar models to write the equation of y in terms of x.

Pause a video, and come back for the next set of questions when you're ready.

It's questions three and four.

Five and six.

Here we go with the answers.

Question number one, y equals three x, two, y equals one six x, three, y equals 1.

5 x, or y equals three over two x, four, y equals 1.

75 x, or y equals seven over four x, number five, y equals five seventh x, and six is y equals four ninth x.

How did you get on with those? I think you would agree.

When you draw those bar models it becomes much, much easier.

So keep drawing the bar models if you need to.

Now let's take a look at converting between equations and ratios.

We've got this table back now, so exactly the same table as we had earlier, but we've also included that answer to the bottom one, which is, y equals five over two x.

I've written it as a fraction, rather than a decimal, but I could write it as one, sorry, 2.

5 x if I wanted to.

Jacob says, "It looks like the coefficient of x comes from the ratio and is y over x." Do you agree with Jacob? What did you think? Well, if Jacob's right then the ratio two to seven gives an equation of y equals seven over two x.

Let's check that out with our bar models.

The ratio of x to y is two to seven.

We want to know y in terms of x, how many xs are equal to y? Let's have a look.

There's x, let's put our xs along the y bar, and we can see we've got one, two, three holes and a half.

So was Jacob right? We've got three and a half, which actually is equivalent to seven halfs, isn't it? If we count up there just the individual grey boxes, we get seven halfs.

Therefore, the equation of y in terms of x is y equals 3.

5 x, or y equals seven over two x.

Jacob was right.

I'd like you now please to have a go at matching each ratio.

Remember, it's written in the form x to y with the correct equation.

So y in terms of x.

Pause the video, go back through all of your notes or even go back and rewatch parts of the video if you need to.

And then when you are done, I'll be waiting for you to go through the answers.

Good luck.

Well done.

Let's check those answers then.

Six to one is y equals one six to x, one to six is y equals six x, three to four is y equals four thirds x.

And we wouldn't wanna change that into a decimal, 'cause it's a recurring decimal.

Four to three is 0.

75 x, or you may have three quarters x, that's fine, 'cause it's a terminating decimal.

And then the final one, four to six is 1.

5 x.

You may have six over four x or three over two x, but you really should try and simplify your answer.

So if you had six over four, just make a note that you need to simplify that to three over two in the future.

Given the a to b equals four to five.

So given that the ratio of a to b is equal to the ratio four to five, we need to write an equation linking a and b.

Aisha says, "It's b equals five over four a." And Jacob says, "It is a equals four over five b." Can both Aisha and Jacob be correct? Yes, they can.

Aisha has written b in terms of a, and Jacob has written a in terms of b.

So they're both correct.

They've just one of them's decided to write b equals and the other has written a equals.

Let's take a look at this then.

We can look at this with a bar model.

So the ratio of a to b is four to five.

We need to make the bars equal, because notice now we've got that equal symbol.

So there's one a, and there's one b.

So I'm going to keep going until both of my bars are the same length.

From this then, we can clearly see that five of the a bars is equal to the same length as four of the b bars.

So we end up with five a equals four b.

If I divide both sides of my equation by five, I end up with a equals four fifths b.

And we can see, that was the answer that Jacob got.

Starting with the same equation, five a equals four b from the bar model.

If I decide to actually write it b in terms of a, I would divide both sides of my equation by four, giving me that five over four a is equal to b.

And that was Aisha's answer.

So we can now see from that bar model that those two answers are both correct, their equivalents of each other, they mean the same thing.

They're just written in a slightly different way.

The ratio of hearts to stars in a bag of sweets is two to nine.

We need to write an equation link in hearts and stars.

Aisha says, "The initial equation is nine h equals two s." And Jacob says, "The initial equation is two h equals nine s." Right, so they've got the two and the nine from the ratio, but they've got them different ways around.

Who do you agree with? Let's take a look.

The original ratio was two to nine.

Aisha is saying that nine h, s, so let's work out what nine h, s are.

That's 18, is equal to two ss.

That's 18.

That looks good, doesn't it? 18 equals 18.

Let's take a look at Jacob's initial equation.

Jacob's initial equations, we've got the same starting ratio, but the equation linking them, Jacob thinks is two h.

So two lots of h is four, and nine s, nine multiplied by nine is 81, is four equal to 81.

No, definitely not.

So we can see that Aisha was correct.

She has equated h and s correctly.

So it's really important, and it might be worth doing this little double-check to make sure when you're writing your own equations that you've got the numbers the right way ranked, or I should say, the coefficient's the right way ranked.

We now know that Aisha's was the correct equation, nine h equals two s.

Now we can rearrange this so that we've got h in terms of s or s in terms of h.

So I'm going to start by dividing both sides of my equation by nine, giving me nine equals two ninth s.

Or alternatively, I could divide both sides of my equation by two, giving me nine over two h equals s.

So we can end up with two different equations from the same ratio.

One is h in terms of s, and the other is s in terms of h.

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

The ratio of adults to children at a running club is five to three.

Which of the following are correct equation linking adults and children? There are two answers.

You just need to decide which of the two correct ones.

So work it through, draw the bar models if you need to.

I'll be here waiting when you get back, you can pause the video now.

Let's take a look.

What did you decide on? You should have chosen b and c.

Well done, if you chose b and c.

Now we're going to go back the other way.

We're going to start from an equation and we are going to write a ratio.

We're told that e is equal to two f, and we want to know the ratio of e to f.

There's e, and that is equal to two fs.

So this bar model represents that.

We can see clearly from this bar model that e is two times the size of f, so therefore, the ratio is two to one, e is twice the size of f, e is two double one.

Yes, it is.

How about this one? Three x equals 0.

5 y.

And again, we want to write this as a ratio of x to y.

Three x equals 0.

5y.

What I'm going to do is I'm going to divide both sides of my equation by 0.

5.

So I end up with an equation, y equals.

So we get six x equals y.

Remember, I could write that the other way round if I wanted to, y equals six x.

There's my six xs, and there's my y.

We know they're the same length, because of that equals symbol.

Y is six times the size of x.

Six xs make up one y.

So therefore, the ratio of x to y is one to six.

Double-check.

If x is one, y is six.

Yes, that's correct.

And this one, three x equals two y.

Again, we want the ratio of x to y.

Three x equals two y.

I'm now going to split my bars, so that each of the sections are the same.

What is the lowest common multiple of three and two? And that's six.

I'm going to split my total xs into six parts, and the total ys into six parts.

Now I should be able to see that x is equal to two thirds of y.

So we've got x, and that's two thirds of the y box.

Now we can write the ratio, x is two thirds of y.

So if y is one, x is two thirds, but we need to make sure that we have an integer value for each of our components of our ratio.

So we are going to multiply both sides of the equation by three, giving us the ratio, two to three.

True or false.

If five x equals two y, the ratio of x to y is five to two.

So is that True or False? And as always, please make sure you come up with your justification.

Pause the video, and come back when you're ready to check your answer.

And your decision was? Hopefully, you said False.

Common mistake is people just look and they keep the numbers the same way around.

So if we look at our justification and it's this one, y is 2.

5 times x, therefore, the ratio of x to y is one to 2.

5, but we don't want a decimal as part of our ratio.

So if we multiply both sides of our ratio by two, we end up with two to five.

So drawing the bar models should help you avoid that mistake.

And a little bit harder again, but you're okay.

I know you're gonna be okay.

You can do this.

Seven x, subtract four y, equals four x, subtract two y.

First thing we're going to do is we're going to rearrange the equation, so that we have the xs on the left, and the y is on the right.

I'd like you to do that before I reveal it, please.

I'm gonna firstly subtract four x from both sides, and then we're going to add four y to both sides, and we end up with three x equals two y.

So the ratio of x to y is two to three.

And again there, if you need to, you can draw out the bar models so that you can clearly see that.

Your turn.

Three x, plus eight y, equals x, plus 13y.

What is the ratio of x to y? Pause the video and come back when you've got your answer.

How did you get on? So the first step, remember, was to rearrange the equation with x is on the left, and y is on the right.

And the reason for that is that the ratio has x is on the left and y is on the right.

So we subtract x from both sides, then we're gonna subtract eight y from both sides.

So remember, we're doing inverses here.

So we end up with two x equals five y, therefore, the ratio of x to y is five to two.

Now your final task for today's lesson is Task B, and the first part of this is question number one.

I'd like you to match each of the ratios written in the form x to y, to its correct equation.

Pause the video and then when you've matched those five, come back and I will reveal the next question.

Good luck.

Great work.

And question number two, I'd like you to fill in the gaps.

Notice it says, "Please give your answer in the simplest form." Super.

And question number three.

Pause the video again, give these a go.

C, we haven't done an example like c, but I've put it in there for a challenge.

But if you're a little bit stuck with c, I wouldn't worry too much.

Okay, good luck with these and I'll be here waiting when you get back.

Super work, well done.

How did you get on? Well done.

Question number one then, eight to one matches with y equals one eight to x, one to eight matches with y equals eight x, six to seven matches with y equals seven over six x, seven to six matches with y equals six over seven x, and eight to six matches with the 0.

75x.

Or you may have three quarters x.

Remember, you could write it either way round.

Say, in that, you could also, for eight to one, you could have that as y equals 0.

125 x.

Question two, I'm gonna ask you now to pause the video and mark your answers to this one 'cause we could get a bit confused about whether we're filling in a ratio or an equation.

So pause the video.

Once you've marked it, come back, and we'll move on to question three.

And question number three, a, was one to five, b, was four to eight, but I asked for it in its simplest forms, that's one to two, and c, was eight to 29.

Well done, particularly if you've got that final one right.

But well done anyway.

Now we can summarise the learning from today's lesson.

We looked at values written in a ratio, and we linked them using an equation.

For example, the ratio of x to y is two to five, therefore, equations linking those two are y equals five over two x, or x equals two over five y.

An equation may need rearranging in order to write it as a ratio.

So as an example of the one that we went through.

Remember, because the question wanted our answer as a ratio of x to y, we collect the xs on the left side of the equation and the ys on the right side of the equation.

Well done with everything you've achieved during today's lesson.

You've worked really, really well and I look forward to joining you again very soon to do some more maths.

Take care, goodbye.