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 Problem Solving with Algebraic Ratios, and this is within the unit Ratio.

By the end of this lesson, you'll be able to use your knowledge of ratio to solve problems. Here are some keywords that we'll be using in today's lesson, and they are ratio, coordinates, expression, and probability.

If you're feeling that you need to recap any of those, I suggest you pause the video now and then carefully read through the definitions to refresh your memory.

We are going to do today's learning in two separate cycles.

In the first one, we are going to concentrate on ratios in geometry, so in shape or solids or on straight lines.

And in the second one, we will look at and concentrate on ratios with probability.

So let's get going with the first one, and that is ratios in geometry.

Here we go.

Here we have a problem and it says, the points A, B, C, and D are in order on a straight line.

The ratio of A to B to B to D is three to five, and the ratio of A to C to C to D is 11 to five.

We need to find the ratio of AB to BC to CD.

Let's take a look at how we go about solving this problem.

So here are my lines, A, B, C, and D.

Doesn't matter how I place them on the line.

This is just a representation to help us solve this problem.

Now we know that the ratio AB to BD is three to five.

So we know that AB is three, BD is five.

We also know that the ratio of AC to CD is 11 to five, so we can put that onto our diagram.

Now we'll look the total length of the line AB.

Using the larger of those two totals is 16.

Let's go back to that first line.

The total of the first line at the moment is eight.

We need to create an equivalent ratio that has a total of 16.

What's my multiplier that takes me from eight to 16? That's right.

It's multiplied by two.

This means we need to multiply everything by two in that top ratio.

Three multiplied by two is six and five multiplied by two is 10.

Now let's just double check.

Yes, that line now does add up to 16.

We're looking for the ratio AB to BC to CD.

Now we can clearly see from this diagram that AB is six.

So we can put that into our ratio.

We can also clearly see that CD is five.

We can put that into our ratio.

We now need to find the length of the line BC.

So we need to find this distance here.

How are we going to find that distance? What do you think? Well, we know that the distance from C to D is five.

We also know that the distance from B to D is 10.

So therefore, to find BC, which is where I put the question mark, we take 10 and we subtract the five to give us five.

We now know that the ratio AB to BC to CD is six to five to five.

Let's take a look at another example, exactly the same but with different ratios.

So let's set our problem up with our diagram.

We have our line with A, B, C, and D on it.

Now I've tried to make the ratios in the right proportion, but you don't need to.

You just need to place A, B, C, and D on a straight line.

Let's fill in what we know then.

We know that AB is two and we know that BD is three.

We also know that AC is eight, and CD is seven.

The total of those is 15.

So let's go back to our first line and we can see the total of two and three is five.

We need to find that multiplicative relationship between five and 15.

What's my multiplier that takes me from five to 15? That's right.

It's multiplied by three.

I need to multiply everything by three in that top ratio to create an equivalent ratio that has a total of 15.

Two multiplied by three is six and three multiplied by three is nine.

Now here I always like to double check that I've made my line the correct length.

Six add nine is 15.

So now all of my lines are a total length of 15.

We're finding the ratio AB to BC to CD.

Again, AB is nice and easy.

It's clearly labelled on the diagram.

So AB is six.

Also, CD is clearly labelled on our diagram as seven.

We're now looking to find the distance between B and C.

We know that CD is seven.

I've put that onto the diagram.

So the question mark must be equal to the length of B to D, which is nine, subtract the length from C to D, which is seven.

So the length of the question mark, which is BC, is two.

We've now got our ratio AB to BC to CD is the ratio six to two to seven.

I'd like you now to pause the video and have a go at this question.

So I've drawn a diagram for you.

I'd like you to fill in the missing values.

Good luck with this.

Like I said, pause the video and then when you're ready, come back and we'll check that answer.

Great work on that.

Let's check through our answer then.

So from the first ratio, AB to BD is three to eight.

Using the second ratio AC to CD, we've got 15 and seven.

The total length of the lines is 22.

But the total length of the top line is only 11.

So our multiplier to get to 22 was multiplied by two.

So I'm going to multiply my top ratio by two.

This will create an equivalent ratio where the total is 22.

Three multiplied by two is six and eight multiplied by two is 16.

Just that quick double check.

Is the sum of 16 and six 22? It is.

Now to find the question mark, I'm going to take the length of BD and I'm going to subtract the length of CD, giving me 16 subtract seven, which is nine.

The ratio AB to BC to CD is six to nine to seven.

How did you get on with that? Brilliant.

Well done.

Once you've set up that problem with a diagram, it becomes much, much easier to do, doesn't it? Let's move on then.

Slightly harder question now because we've got some coordinates.

So we've got A, B, C are on a straight line.

The coordinates of A are seven, four.

The coordinates of B are one, two.

Given that AB to BC is one to two, so the ratio of AB to BC is one to two, we need to find the coordinates of C.

Again, here we're going to sketch ourselves a diagram.

Now it's really important here that I consider where my points are going to go with relationship to the coordinates.

So if we look at point A, point A is seven, four.

At point B has the coordinates of one, three, which it means it's further to the left than A and it's also lower than A.

So that's how I know that my B needs to go into the left and down.

And that's going to be important in a moment when we're deciding whether we're going to be adding or subtracting our values to find the coordinates of C.

So here's B and we are trying to find C.

So I've just labelled that x, y for the xy coordinates at the moment.

We know that the ratio AB to BC is one to two.

Now what we're going to do is we're going to look at the change.

So I'm going to start with my horizontal change.

So I'm going to find the change in x.

The x-coordinate for B is one and for A is seven.

So that is a change of six.

We're going to repeat that now for the vertical change.

Remember that's the y-coordinate, the second coordinate.

So B is at two and the y-coordinate of A is four.

So the change in y is two.

We now know that the ratio of AB to BC is one to two.

So therefore, we're going to take that six and two and we're going to multiply by two, giving us 12 and four.

I now know that those are my changes from B to C.

Now this is why I said it's really important to get a sense of where point B is in relation to A because now we need to find point C.

So we're going to start with the x-coordinate.

The x-coordinate.

So let's look.

The x-coordinate of B was one and we can see that C is 12 to the left of one.

So if I'm go moving to the left, I'm moving towards my negative number, so I'm going to be subtracting 12.

So I'm going to do one subtract the 12, which gives me negative 11.

We're now going to repeat that for the y-coordinate.

The y-coordinate of B is two.

And we can see that the y-coordinate of C is four below two.

So again, we need to subtract that, giving us two subtract four, which is negative two.

The coordinates of point C are negative 11, negative two.

Now there's quite a lot there.

You may feel now that you need to pause the video and just go through back each of those steps slowly and make sure that you've made a note of each of them.

But if you think you're ready to move on, we're going to now with a different question.

Okay, the same principle but slightly different question.

We've got the same scenario.

Points A, B, and C, and they are on a straight line.

The coordinates of A this time are two, negative nine and B is negative two, negative one.

The ratio this time of AB to BC is four to three.

Again, we need to find the coordinates of point C.

Now notice my line looks different this time.

This time, it has a negative gradient, and we can see that A is in the bottom right-hand corner.

How did I know that? Well, I know that A had coordinates of two, nine.

Now if I look at B, you can see the x-coordinate is smaller than two, so it must have been to the left, and the y-coordinate is larger, so it must have been up.

That's how I know that B needs to be to the left and up.

It doesn't matter in its exact position, just as long as we've got an idea of the gradient of this line, and then we're trying to find C.

Now let's go and use our ratio.

So we've got the ratio AB to BC is four to three.

Now we're looking for the change in x.

The A coordinate x was two and in B was negative two.

So the change in x, two to negative two is a change of four.

Now let's consider the y change, the change in y.

A's y-coordinate was negative nine and B was negative one.

That's a change of eight.

The difference between those two is eight.

Now last time, because we had the unit ratio, I could just work out and multiply both of those by two.

This time, I don't have a unit ratio.

My ratio is four to three.

So we're going to draw myself a little ratio table.

AB to BC is four to three.

And I'm looking for my multiplicative relationship between four and three.

What is the multiplicative relationship between four and three? That's right.

It's multiplied by 0.

75 or you might have multiplied by three-quarters.

Remember, they mean the same thing.

So now I know that I need to multiply both of those values by 0.

75 or by three-quarters.

Four multiplied by 0.

75 is three and eight multiplied by 0.

75 is six.

I now know that my change in x is going to be three and my change in y is going to be six.

So now we can work out the x and the y-coordinate of point C.

Starting at B, my x-coordinate is negative two, and we can see that three is to the left of that.

So I'm going to take my negative two and I'm going to subtract that three, giving me negative five.

Now let's take a look at the y-coordinate.

The y-coordinate of B is negative one.

And we can see that the six is going upwards.

So therefore, we're going to add that six.

A negative one add six is five.

This means that the coordinates of the point C are negative five, five.

Now your turn.

Have a go at this question for me.

Pause the video.

Like I said before, if you need to go back and re-watch either of those two examples or even both of them, do that until you are confident that you'll be able to answer this question independently.

I'll be waiting for you right here when you get back and we'll go through that answer.

Good luck with this and you can pause the video now.

Great work.

Shall we check that answer? Yeah, of course.

Come on.

We had A to B was a change in two, zero to two is two, and then negative eight to negative seven is a change of one.

Here it was a unit ratio, so I could just multiply those values by three, and giving me a change in x of six and a change in y of three.

I'm now going to go six to the right of two, which gives me eight, and I'm going to go three above negative seven, which is negative four.

The coordinates of point C are eight, negative four.

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

So you're going to pause the video and then you're going to have a go at question one and question two.

So this is like the first two examples we went through.

So don't forget to sketch your line with A, B, C, and D on it.

It doesn't need to be to scale.

Good luck with these and then when you're ready, come back and I will reveal the next set of questions.

Good luck.

And moving on then to questions three and four.

Again, good luck with these.

Don't forget.

You will need to draw that diagram.

Well done.

And let's have a look at those answers.

Question one, the ratio was three to eight to four.

Question two was six to seven to eight.

Question three was negative nine, negative eight.

And question four was 15, negative four.

How did you get on? Well done.

Now let's move on to that second learning cycle.

So we're going to be looking at other problems involving ratio.

98 beads are contained in three bags.

Each bag contains a different number of beads.

We've got there bags A, B, and C.

Have some information about these bags now.

Bag A contains two fewer beads than bag B.

Bag C contains six times the number in bag A.

We need to find the ratio of beads in bag A to bag B to bag C and give that in its simplest form.

Here we've got A, B, and C.

Now I'm going to assign B a letter.

So B, I'm going to call b, because it's bag B.

And the reason for this is because A is linked to B and then C is linked to A.

So our base bag really is bag B.

It says that A contains two fewer than bag B.

Well, bag B has b and two fewer than that would be b subtract two.

It then says bag C has six times the number in bag A.

So it's going to have six lots of A, so six lots of b subtract two.

Remember that needs to go in brackets because it's six lots of the whole of bag A.

We can now form an equation in terms of b.

We know that bag A add bag B add bag C equals 98 because we're told that in the question.

It says 98 beads are contained in the three bags.

Firstly, we're going to expand the brackets in the expression for C.

So I'd like you to expand six bracket b subtract two, close bracket.

What did you get? You should have got six b subtract 12.

Six multiplied by b and six multiplied by negative two.

We can now replace A, B, and C with its expression in terms of b.

A is b subtract two, B is b, and C is six b subtract 12, and we know that equals 98.

We can now collect together the like terms. Collect together the like terms for me, please.

And you should have got eight b subtract 14 is 98.

So b add b add six b is eight b and negative two subtract 12 is negative 14.

We're now going to solve this equation.

Let's solve it.

What do we do first? We're going to add 14 to both sides of our equation, giving us eight b equals 112.

And then we're going to divide both sides of the equation by eight to give us what b is.

So we get 112 over eight.

And if we evaluate that, we get that b is 14.

We now know that b is 14.

So bag B has 14 beads.

Bag A has two fewer beads.

Two fewer than 14.

So 14 subtract two is 12 beads.

And we also are told that bag C has six times the number of bag A.

So we now know what bag A has got.

So we know that it's six times A, so it's six multiplied by 12, which is 72.

I can now check that the bags sum to 98.

Is 12 add 14 add 72, 98? Yes, it is.

So I know that I must have done this problem correctly or pretty certain I've done it correctly.

Now have we answered the question? No, not quite, have we? We need to give the ratio and the ratio needs to be in its simplest form.

We now know there are 12 beads in A, 14 in B, and 72 in C.

So we can now write that as a ratio, but I want it in its simplest form.

So I'm going to divide each of the components by two, giving me six, seven, and 36.

The ratio of beads in the bags is six to seven to 36.

Now it's your turn to have a go.

I've broken this problem down, so it's a very similar to problem to the one that we did, but you are going to at each step decide whether the answer is A, B, or C.

120 beads are contained in three bags and each bag contains a different number of beads.

Bag A contains five more than bag B.

Bag C contains three times the number of bag A.

Which of the following shows the correct expressions for this problem? Pause the video, make your decision, and then when you've got your answer, come back and we'll move on to the next part of this question.

And what did you decide? Hopefully, you decided C.

C was the correct answer.

Use this information to find the value of b.

So now I'd like you to create an equation and solve it to find the value of b.

Pause the video and when you've got your value of b, you can come back.

And you should have got that b was 20.

So that was option A.

And that's how we got it.

So we've got five b that comes from b, b, and three b, and then we've got 20 which comes from five add 15.

And then we solve that equation.

We get b equals 20.

We now know that b is 20 and we've got the information about the bags.

Your job now is to find the number of beads in each of the bags A, B, and C.

Pause the video and when you've worked out how many there are in each one, don't forget to check that it totals 120.

I'll be here waiting when you get back.

You can pause the video now.

And what did you come up with? So B was 20.

We knew that.

A had five more than bag B.

So A had five more.

Five more than 20 is 25.

And C had three times the number in A, so three multiplied by 25, which is 75.

And if we add those up, we get 120.

Now I'd like you to find the ratio of the beads in bag A to bag B to bag C in its simplest form.

This time, no options.

You're just going to work that out for yourself.

Pause the video, and then when you come back, we'll check.

Super.

What did you get? Well, the original ratio would have been 25 to 20 to 75.

All of those are divisible by five.

So we end up with five, four, and 15.

This means that the ratio of bag A to bag B to bag C is five to four to 15.

And really and truthfully, I should have rewritten that again with those colons in between to show that it was a ratio.

I'm sure you did that, even though I forgot.

The ratio of red to yellow sweets in a bag is four to five.

The ratio of hearts to stars in the bag is three to two.

We need to work out the probability of picking out a red heart.

So we've got red and yellow and I've drawn a bar model to represent that.

And then we've got hearts and stars again.

A bar model to represent that.

Notice that the bars are exactly the same length.

The total of the bars is the same length because the number of sweets in the bag doesn't change between the red and yellow and the hearts and stars.

We want the probability of picking a red heart.

And it might be easier to think of this as what fraction of the sweets are red? And we can see that there are four red ones out of a total possible nine.

Now we're looking for hearts.

What fraction of the sweets are hearts? There are three hearts out of the total possible five.

We now need to know then what four-ninths of three-fifths is.

What can we switch that of for? Yeah, that's right, a multiplication symbol, can't we? So this becomes four-ninths multiplied by three-fifths.

So we're going to multiply across our numerators and across our denominators to give us 12 over 45.

The probability, therefore, of picking a red heart is four-fifteenths, because I've simplified that and you can see how I've simplified that.

I've written 12 is three multiplied by four and 45 is three multiplied by 15.

But we know that three over three is equal to one, which is where we get our simplified answer of four-fifteenths.

Your turn.

Have a go at this one.

When you're ready, come back.

Let's check in with those answers then.

What fraction of the sweets are red? That's five-eighths.

What fraction of the sweets are stars? And that is four-elevenths.

So a red star is going to be each of those two probabilities multiplied together, which gives us 20 over 88, which simplifies to five over 22.

And I've shown you there how I've done that simplification if you need that.

Laura is making a dessert in these glasses.

The ratio of jelly to cream to empty space is three to two to one.

How much cream is in the dessert? Give your answer in terms of pi.

So this is a question that we need to be answering.

Here's Laura's working.

And here's Jun's working.

What I'd like you to do now is to decide what is the same and what is different about those two methods? So pause the video and then when you've made your decision of what's the same, what's different, you can come back.

And what did you come up with? Well, this is what I decided.

They both used a ratio of three to two to one.

Well, they have to, don't they? Because we're looking at the ratio of three to two to one from the question.

But Laura finds the total volume first.

So if we look, Laura has found the total volume of the glass and then she's divided that into the ratio of three to two to one of jelly to cream to empty space.

Whereas Jun finds the height of the cream and then finds the volume of just the cream.

So he uses the ratio of three to two to one to find the height of the cream, which is four, and then uses that in our volume-cylinder formula, pi r-squared h.

Notice they both end up with exactly the same answer of 100 pi centimetres cubed, although Laura hasn't quite squeezed the centimetres cubed into her one.

The ratio of gravel to water in this fish tank is one to five.

What volume of water is in the tank? Pause the video.

I will allow you to use a calculator for this one because we get some big numbers, and then when you're ready, come back to check.

And you came up with which of those answers? It was answer A.

Now you can have a go at task B.

So we've been through an example of each of these.

So if you need to at any point, go back and re-watch that example if you get stuck at all.

So pause the video, give this question a go, and then when you're ready, come back.

Great work.

Now let's go question two and question three.

How'd you get on? Quite challenging, weren't they? Particularly that last one I think.

The answer to one, we should have ended up with that b was 20, which then meant bag A had 11, bag B had 20, and bag C had 33 beads, giving a ratio of 11 to 20 to 33, which actually couldn't be simplified.

Question two.

You should have ended up with five over 11.

That was the fraction of the sweets that were yellow.

Two over seven.

That was the fraction of the sweets that were stars.

Multiply those two together, you get 10 over 77.

And question three, you should end up with 1,500 pi.

Remember, you may have found the total volume and then used a ratio or used the ratio to find the height that was jelly.

And that's what I did here.

I found the height that was jelly and then used that to find the volume of jelly.

Summarising our learning today then.

We've looked at ratios and they can be used in many, many different situations.

And during this lesson, we have looked at points on a line segment, including points given as coordinates.

And we've also looked at ratios when we've involved algebra, probability, and volume.

Superb work today.

Well done.

I'm glad that you decided to join me and stick with me right through to the end.

Take care of yourself and hopefully I'll see you again really soon.

Goodbye.