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

By the end of this lesson, you will 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 feel 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 and 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 point A, B, C, and D are in order on a straight line.

The ratio of AB to BD is 3:5, and the ratio of AC to CD is 11:5.

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 representation to help us solve this problem.

Now we know that the ratio AB to BD is 3:5.

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

We also know that the ratio of AC to CD is 11:5, 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 (indistinct) 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.

3 X 2 is 6 and 5 X 2 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 gonna 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 we 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 6:5:5.

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, with 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 6:2:7.

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 are 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 3:8.

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

The total length of the line 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 gonna multiply my top ratio by two.

This will create an equivalent ratio where the total is 22, 3 X 2 is 6 and 8 X 2 is 16.

Just that quick double check.

Is the sum of 16 and 6, 22? It is.

Now to find the question mark, I'm going to take the length of BD and I'm gonna subtract the length of CD, given me 16 - 7, which is 9.

The ratio AB to BC to CD is 6:9:7.

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 'cause we've got some coordinates.

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

The coordinates of A are 7,4.

The coordinates of B are 1,2 given that the AB to BC is 1:2.

So the ratio of AB to BC is 1:2.

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 7,4.

Now point B has the coordinates of 1,3, 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 gonna be important in a moment when we are deciding whether we're gonna be adding or subtracting our values to find the value of the coordinates of C.

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

So I've just labelled that XY for the XY coordinates, at the moment.

We know that the ratio AB to BC is 1:2.

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 1:2.

So therefore we're going to take that six and two and we're gonna 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 'cause now we need to find point C.

So we're gonna 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 moving to the left, I'm moving toward my negative number, so I'm gonna be subtracting 12.

So I'm gonna do one subtract the 12, which gives me -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 2 - 4, which is -2.

The coordinates of point C are -11,-2.

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 are ready to move on, we're going to now with a different question.

Okay, 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 2,-9 and B is -2,-1.

The ratio this time of AB to BC is 4:3.

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 2,9.

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.

Doesn't matter in his 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 'cause we've got the ratio AB to BC is 4:3.

Now we're looking for the change in X.

The A coordinate X was 2 and in B was -2.

So the change in X, 2 to -2 is a change of four.

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

A's Y coordinate was -9 and B was -1.

That's a change of eight.

The difference between those two is eight.

Now last time because we had a 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 4:3.

So we're going to draw myself a little ratio table, AB to BC is 4:3 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.

4 X 0.

75 is 3 and 8 X 0.

75 is 6.

And I 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 Y coordinate of point C, starting at B, my X coordinate is -2 and we can see that three is to the left of that.

So I'm going to take my -2 and I'm going to subtract that three, giving me -5.

Now let's take a look at the Y coordinate.

The Y coordinate of B is -1 and we can see that the six is going upwards.

So therefore we're going to add that six, a -1 + 6 is 5.

This means that the coordinates of the point C are -5,5.

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 rewatch 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 the answer.

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

Great work.

Shall we check the answer? Yeah, of course, come on.

We have A to B was a change in 2.

0 to 2 is 2 and then -8 to -7 is a change of 1.

Here it was a unit ratio so I could just multiply those values by 3 and giving me a change in X of 6 and a change in Y of 3.

I'm now going to go 6 to the right of 2, which gives me 8 and I'm going to go 3 above -7, which is -4.

The coordinates of point C are 8,-4.

Laura is making a dessert in these glasses.

The ratio of jelly to cream to empty space is 3:2:1.

How much cream is in the dessert? And we need to give our answer in terms of pi.

Laura decides she's going to answer the question like that and Jun decides who's gonna answer the question this way.

What is the same and what is different about the two methods? Pause the video, makes some decisions about what's the same and what is different.

And then when you are ready, come back and we'll see if you agree with me.

What did you decide? Well they both used the ratio of 3:2:1.

They'd have to, wouldn't they? 'Cause otherwise they'd end up with the wrong answer 'cause we know that's the ratio of jelly to cream to empty space.

But Laura finds the total volume first.

We can see that Laura has found the volume of the glass is 300 pi cubic centimetres and then she's used that ratio of 3:2:1 to find the volume of it that is cream.

Whereas Jun finds the height of the cream first.

So we can see he's got the ratio of 3:2:1 and where he knows the total height of the glass is 12 and so finds the height of the cream is four centimetres and then works out.

Notice they end up with exactly the same answer.

I wonder which method you prefer.

Notice here also Laura should really have concluded with her answer the units of centimetres cubed.

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

The ratio of gravel to water in this fish tank is 1:5 What volume of water is in the tank? Pause the video and then when you've got your answer, come back.

Of course here you may use a calculator 'cause it's fairly big numbers, Superb work.

What did you come up with? You should have come up with A.

Now you are 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 are 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 And now onto question five.

Pause the video when you've got the answer, come back and we'll check your answers for the whole of task, and good luck.

Okay, let's check those answers.

Question number one, the ratio is 3:8:4.

Question two, the ratio is 6:7:8.

Question three, the coordinates of point C were -9,-8.

Four, the coordinates of C were 15,-4.

And finally question five, our answer was 1,500 pi centimetres cubed.

How did you get on with those? Superb work, well done.

Now we can move on to our second learning cycle and here we're going to be concentrating on ratios within probability.

The ratio of red to yellow sweets in a bag is 4:5.

The ratio of hearts to stars in the bag is 2:3 We need to work out the probability of picking out a red heart.

Here's a bar model to show the proportions of red and yellow sweets.

So in the ratio of 4:5 and here is a bar to show the proportions of hearts and stars, the ratio 3:2, notice the total length of both of my bars is the same because a number of sweets is the same no matter whether I'm comparing red and yellow or hearts and stars.

We are looking for a probability of picking a red heart.

What fraction of the sweets are red? We can see clearly see there are four red parts outta a total possible nine.

So the fraction of sweets that are red is 4/9.

We're also looking for hearts, we're looking for red hearts.

What fraction of the sweets are hearts? And here we can see that three parts are hearts outta a possible total five, which is 3/5.

We now need to find 4/9 of 3/5.

What can we exchange the word of for? That's right, we can exchange it for a multiplication symbol.

We end up with 4/9 X 3/5, which is 12/45.

The probability of picking a red heart is 12/45 and then I've simplified that to 4/15 and I've shown you in that middle step there, that intermediate step how I've simplified that.

Your turn to have a go now.

So pause the video, work out the probability of picking a red star for me and then when you're ready, come back.

Well done on that.

Let's check your answer.

The fraction of sweets that were red were 5/8.

The fraction of sweets that were stars was 4/11.

So the probability that we find a red star if we put dip our hand in the bag is 5/8 X 4/11, which was 20/88 and that simplifies to 5/22.

Well done if you got that right.

The ratio of red to green counters in a bag is 7:2.

A counter is chosen at random, replaced and a second counter is then chosen.

What is the probability that two red counters are chosen? Here is a tree diagram to help us think about this problem in a logical way.

The ratio of red to green is 7:2.

What is the probability that a counter chosen is red? Using that ratio, what is the probability that I choose a red counter? And it might be easier to think of this as what fraction of the counters are red.

So what fraction of the counters are red? Yep, 7/9.

If we were to draw a bar model seven parts would be red and two parts would be yellow.

So the total number of parts is nine.

We can now put that onto our tree diagram.

So red is 7/9.

Because we're replacing the counter, the probability of red remains the same.

So I now also know that red is 7/9.

We want the probability of picking two reds, so that's a red followed by a red.

So that's 7/9 X 7/9, which gives us 49/81.

Your turn now.

I'd like you to have a go at this question, pause the video and then come back when you are ready.

Well done.

Let's go through this one then.

So you should have ended up with the probability of red was 3/8.

The probability is not changing for the second pick because we're replacing the counter.

So my probability is 3/8 X 3/8, which is 9/64.

Is that what you got? Brilliant.

Sofia puts 30 red and green counters in a bag in the ratio of 1:4.

She takes two counters at random.

So this time she's putting her hand in and taking two counters at once.

What is the probability she takes two red counters? We know the ratio is 1:4, which gives us a total of five.

We know that there are 30 counters in total in the bag.

So now we can create our equivalent ratio looking at that multiplicative relationship between five and 30.

And it's what? Yeah, multiplied by six.

1 X 6 is 6, 4 X 6 is 24, always worth doing that little double check, is the total of six and 24, 30? Yes it is.

Here's my tree diagram.

We know the probability of picking a red counter, we're looking at two red counters.

So probability of picking a red counter on the first pick is 6/30.

What about the second pick? Well the second pick this time, there is one less red counter, because we've already picked a red counter that's already in our hand, and there's one less counter in total.

So the probability of picking a red on the second pick is 5/29.

The probability then of picking two reds is the product of those two probabilities, 6/30 X 5/29, which is 30/870 and that simplifies to 1/29.

Notice here the question doesn't ask us to simplify, so it's absolutely fine to leave our answer as 30/870.

Sofia puts some red and green counters in a bag in the ratio of 1:4.

She takes two counters at random.

The probability she takes two red counters is 1/30.

So this time we don't know the total number of counters in the bag, but we do know what the value of the probability is.

How many counters of each colour are in the bag? Here's my tree diagram.

I've decided not to draw the whole thing because I know I'm concentrating just on those two red counters.

What is the probability that the first counter chosen is red? And again, it might be easier to think of this as what fraction of the counters are red.

And that's 1/5.

We can see there's one part red and the total number of parts represented by the ratio is five, so 1/5.

The second pick, we can't simply just reduce the numerator and denominator by one like we did in the previous question because unfortunately we don't know how many counters are in the bag.

We'll let the number of red counters be r.

Using the ratio of 1:4, what is the number of green counters in terms of r? It's 4r.

So if r is representing the one, then four must be 4r.

What is the total number of counters in the bag in terms of r? Well that's the red and the green.

So the red was r, green was 4r.

So the total number of counters in the bag is going to be represented by 5r.

Now we can reduce the number of red counters by one and the total number of counters by one.

What is the number of red counters on the second pick in terms of r? It's r - 1.

r - 1 because there's one less red counter because we already picked a red counter on the first count.

What's the total number of counters on the second pick in terms of r? It's 5r - 1.

We know the total number of counters was 5r.

There's one less because we've already picked that red one, so therefore it's 5r - 1.

Therefore the probability of picking a counter on the second pick is r - 1 over 5r - 1.

What's the probability of picking two red counters in terms of r? We know it's the product of picking a red counter on the first pick and a red counter on the second pick, which is represented here.

We know that that has a numerical value of what? Yeah, it has numerical value of 1/30 because we're told in the question the probability she takes two counters is 1/30.

We can now therefore create an equation.

We then need to solve this equation.

So you should have solved equations like this before.

Let's solve this equation.

So I've just combined my numerator and denominators on the left hand side, then gonna multiply by the left hand denominator giving me r - 1 equals 5/30 and then multiply by 5r - 1.

I'm then going to multiply both sides of my equation by 30 because that we want to eliminate that denominator of 30 on the right hand side, which then simplifies to 30r - 30.

So I've expanded the bracket on the left hand side and I've also expanded the bracket on the right hand side.

Now we can collect together our like terms. We end up with 5r = 25.

And then we're gonna divide both sides of the equation by five to end up with r = 5.

Now there's quite a lot there, so please do pause the video, go back and rewatch that at your own pace if you need to.

We knew that there were red counters was r and green counters was 4r, but we now know that r is five.

This means there are five red counters and 20 green counters.

We'll now have a go at another one of those together.

Sofia puts some red and green counters in the bag and the ratio of 1:3.

She takes two counters at random.

The probability she takes two red counts is 13/220.

How many of each colour are in the bag? So same question as we've just done, but the ratio and the probability are different.

The setup is the same.

What's the probability that the first counter is chosen is red? And again, might be easier to think of it as what fraction of the counters are red? It's 1/4.

We look at the ratio, one part was red outta a total possible four parts in the whole ratio.

So we now know that the probability of picking red on the first one is 1/4.

The second pick, remember we can't simply reduce the numerator, denominator by one because we don't know the total number of counters in the bag.

We will let the number of red counters be r.

Using the ratio 1:3, what is the number of green counters in terms of r? It's 3r.

What's the total number of counters in terms of r? We've got red and green, well red was r, green was 3r.

So the total number of counters is 4r.

Now we can reduce the number of red counters by one and the total number of counters by one.

What is the number of red counters on the second pick in terms of r? r - 1.

What's the total number of counters on the second pick in terms of r? And that's 4r - 1.

We knew the total number of counters was 4r and we are subtracting one because we already have a red counter in our hand.

Therefore the probability of picking a red counter on the second pick is r - 1 over 4r - 1.

What is the probability of picking two red counters in terms of r? And that is finding the product of the two separate probabilities.

What's the numerical value of picking two red counters? Yeah, that's 13/220.

So now we've got our equation, 1/4 multiply by r - 1 over 4r - 1, is 13/220.

We can now solve this equation.

So the first step, combine the first two fractions.

Second step we're gonna multiply by four and then bracket 4r - 1.

When we do that, we end up with r - 1 = 52/220 4r - 1.

Then we're going to eliminate the denominator on the right hand side by multiplying both sides by 220, giving us 220r - 220 = 208r - 52, collect together like terms, giving us 12r = 168 and then divide both sides by 12.

Giving us r is 14.

So the red counters we know is r.

So there are 14 red counters and then green counters with 3r and 3 X 14 is 42.

There are 14 red counters and 42 green counters.

I'd like you now to have a go at this check for understanding.

So if you need to do, like I said before, go back and re-watch those two examples, obviously that's absolutely fine.

But if you feel you are ready now, have a go at this question, pause the video and then when you come back, we will check through the answers for you.

Good luck with this.

Great work on that.

Let's see how you got 'em.

Let the number of red counters be r, sorry, the number of green counters then is 2r.

The total number of counters is 3r.

The probability that the first counter is red is 1/3, that comes from the ratio.

And the second counter is red is r - 1 over 3r - 1.

Now I'd like you to create and solve the equation.

Pause the video and then when you are ready, I'll be here waiting.

So you should have ended up with this.

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

Ultimately, you should have got the answer, r = 5.

Now knowing that r = 5, I'd like you please to complete the question.

We had red counters r, green counters 2r.

We now know that r is 5, this means we know there were five red counters and 10 green counters.

Did you get that right? Of course, you did.

Ready now for your independent task.

Task B, questions one and two.

Pause the video and then when you are ready, come back.

And questions three and four.

While done on those, now let's check those answers.

Question number one, the answer is 10/77.

Question two was 4/121.

Question three was 35/92 and that was in its simplest form, absolutely fine if you've left it as 210/552.

And question number four, you should have ended up with 13 red counters and 65 green counters.

The working out is there, so if you've made an error, pause the video and then take a look and see if you can see where you went wrong.

Summarising our learning today now, 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.