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Hi there.

My name's Ms. Lambert.

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 Abstract Direct Proportion, and that's within the unit direct and the inverse proportion.

By the end of this lesson, you'll be able to identify, write, and solve direct proportion questions involving algebra.

Some key words and terms we'll be using in today's lesson are direct proportion.

A reciprocal.

Two variables are in direct proportion if they have a constant multiplicative relationship.

The reciprocal is the multiplicative inverse of any non-zero number.

Any non-zero number multiplied by its reciprocal is equal to one.

So for example, 2/3 multiplied by 3/2, the reciprocal of 2/3 is equal to one.

Two multiplied by three is six, three multiplied by two is six, six over six we know is equal to one.

Today's lesson is split into two separate learning cycles.

In the first one, we will look at identifying and solving direct proportion problems, and in the second learning cycle, we will look and we will concentrate on solving abstract direct proportion problems. Let's get going with that first one then, identifying and solving direct proportion problems. Here we go.

Which of the following tables show a directly proportional relationship? And you must give a reason for all of your choices.

You're really good at direct proportion so I know you'll be able to answer this question without me giving you any more input.

So you're gonna pause the video, make a decision on which show directly proportional relationship and which don't.

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

Great work.

Let's see if you've got those right, which I'm sure you have.

The first one is correct.

It shows a directly proportional relationship.

Let's take a look at why.

The multiplicative relationship as we move horizontally is multiplied by two, a half multiplied by two is one, and three multiplied by two is six.

We could also consider the vertical relationship.

Half multiplied by six is three, one multiplied by six is six.

Clearly this shows that these two variables have a directly proportional relationship.

And the second one, that does show a directly proportional relationship.

My multiplicative relationship horizontally, it's multiplied by 1.

1, and vertically, it's multiplied by two.

The third one does not show a directly proportional relationship.

If we look at our two multipliers, three multiplied by four over three equals four, and seven multiplied by eight over seven equals eight.

This shows that it is not a constant multiplicative relationship, and therefore, these are not directly proportional to each other.

We could also consider again the vertical movement.

We can clearly see that these are not the same.

Fourth one does not show a directly proportional relationship, and again, we can take a look at those multipliers, and we can clearly see that they are not the same, and the bottom middle one does show a directly proportional relationship.

Eight divided by two is four, 16 divided by two is eight, and eight multiplied by two is 16, and four multiplied by eight is two, and the final one also shows a directly proportional relationship.

With our horizontal multiplier multiplied by 0.

75, and our vertical divide by four.

Alex says, "Hold on! I thought that if variables have a directly proportional relationship, they have a constant multiplicative relationship.

But we are dividing here, not multiplying." I'd like you to explain to Alex why dividing by two is a multiplicative relationship.

Pause the video, write down your explanation, or verbalise your explanation, and then come back and let me know what you've got.

You can pause the video now.

Okay, what did you write? Superb.

We know that dividing by a number is the same as multiplying by the reciprocal.

What is the reciprocal of two? Yeah, you've got it.

It's a half.

The reciprocal of two is a half.

Two multiplied by a half is one.

We could have written that divide by two as multiplied by one half.

Now I'd like you to have a think about this.

True or false, the sum of a number and it's reciprocal is one.

Make your decision and then when you've done that, you can come back.

What did you decide? Did you decide true or false? Of course you said false, and the reason it's false is because actually B, a half multiplied by two is one.

It is the product of those two numbers.

The product of the number and its reciprocal is one.

Here I would like you please to replace each of these divisions with its equivalent multiplication.

Pause the video, and when you've got your three answers, pop back.

Great work, let's check.

Divide by four, we could have write as multiplied by a quarter.

Divided by a third, we could write as multiplied by three, and divide by 0.

75, so a little bit more challenge in this one.

I would probably have written 0.

75 as a fraction, so three quarters, and then I can see my reciprocal would be 4/3, so multiply by 4/3.

Well done, I know you've got all three of those right.

All of the numbers in step one, two, and three are in direct proportion.

Complete the missing entries.

So we need to fill in the missing boxes.

We will start by looking for the multiplicative relationship between set one and set two.

Which row of the table can we use to do this? The third row is the only row we can choose to find that relationship.

Why is the third row the only one that we can use? And the reason for that is it's the only row where we know the values for both set one and two.

The top row we only know value for set one, the second row we only know the value for set two, but the third row, we know the values of set one and set two.

What is the multiplicative relationship between five and 20? Yeah, that's right, multiply by four.

Can we now complete set one? Yes, we can.

To go from set one to set two, I multiply by four.

So to go from set two to set one, what am I going to do? That's right, I'm gonna divide by four, or you might have put multiplied by the reciprocal of four, which is one quarter.

18 divided by four is 4.

5, and then 88 divided by four is 22.

Now we will find the relationship between which two sets? We can either find the relationship between sets one and three, or sets two and three.

What is the multiplicative relationship between eight and 12? That's multiplied by 1.

Remember, if you're not sure, we can do 12 divided by eight, and that gives us 1.

I'm going to find the relationship between set one and three.

I've already done that, haven't I? And then I'm going to check my answers though 'cause I do love to check my answers, if it's possible, using the relationship between sets two and three.

Set one to set three, I multiply by 1.

What value is going to go in here? Well, it's going to be 4.

5 multiplied by 1.

5, which is 6.

75.

And in here we're gonna do five multiplied by 1.

5, which is 7.

And then the final value in set three, I'm going to do six multiplied by 1.

5, which is nine.

Now we can find the rest of set two.

So we'd already worked out that very original multiplier that we'd worked out was five multiplied by four is 20.

So now I can find my missing values in set two.

Eight multiplied by four is 32, and six multiplied by four is 24.

Now we're going to check by finding the relationship between set two and three.

What is the relationship between 32 and 12? And that's multiplied by three over eight.

You may have written that as a decimal, which is fine, but I've decided here to write it as a fraction.

Let's check them.

32 multiplied by three over eight is 12.

18 multiplied by three over eight is 6.

75, 20 multiplied by three over eight is 7.

5, 88 multiplied by three over eight is 33, and 24 multiplied by three over eight is nine.

We've double checked now.

We can be pretty certain that those are the correct answers.

The number of sweets produced by a machine and the number of bags of sweets produced.

Do the number of sweets and the number of bags share a directly proportional relationship? What do you think? Yes, the number of bags is directly proportional to the number of sweets, assuming the same number of sweets is contained in each bag.

As the number of sweets increases, the number of bags increases proportionally.

So for example, double the number of sweets would mean double the number of bags.

The table below shows the number of sweets, S, and the number of bags of sweets, B, produced by a machine.

We need to complete the table.

Alex says, "If we find the multiplicative relationship between bags and sweets, this will tell us how many sweets are in one bag.

This would also be the gradient if we were to draw the relationship graphically." Do you agree with Alex? And of course you said yes because Alex is right.

Once we know the multiplicative relationship, we can find any number of sweets or bags given one piece of information.

So if I know the number of sweets, I can find the number of bags, and if I know the number of bags, I can find the number of sweets.

My multiplicative relationship using the 250 and the 14,000 is multiplied by 56.

How did I get that? I did 14,000 divided by 250.

I know to go from bags to sweets, I multiply by 56.

So therefore, to go from sweets to bags, I must divide by 56.

We would do the inverse.

196,000 divided by 56 is 3,500, and 15,000 multiplied by 56 is 840,000.

I've completed the table.

Your turn now.

The number of cans, C, produced by machine is directly proportional to the number of boxes, B, filled.

If the machine produces 9,000 cans and this fills 500 boxes, which of the following are also true? Pause the video, work out what the relationship is between the number of boxes and the number of cans, cans and boxes, and then you'll be able to decide which of the following are also true.

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

How did you get on? Brilliant.

Well done.

Let's have a look then.

A was true, B was true, C was false, and D was true.

If we look, we started off the information given in the question.

9,000 cans gave us 500 boxes.

To go from boxes to cans, I multiply by 18, 360 multiplied by 18 is 6,480, not 5,400.

So we can see that that one is false.

Now you're ready to do the first independent task for today's lesson, which is task A.

Question number one, decide which of the following would share a directly proportional relationship or not.

A, the time taken to get from Lands End to John O'Groats and the speed of a car.

B, the number of tickets sold and the amount of money made.

C, the cost of fuel and the size of the vehicle's fuel tank.

D, the time taken to paint a fence and the number of people.

E, the amount of water in a swimming pool and the rate at which a tap runs.

And F, the number of cookies being made and the amount of butter needed.

Pause the video and decide which of those are directly proportional.

Good luck.

Okay, moving on then to question two and question three.

The following have a directly proportional relationship.

So I'm telling you that they do.

You need to use the information given to answer the questions.

Question two, 36 tickets are sold and the amount of money made is 630 pounds.

How much money is made if 500 tickets are sold? And B, how many tickets are sold if 1,015 Pounds is made? Three, a lorry's fuel tank has a capacity of 325 litres and costs 503 pounds 75 to fill.

A, how much will it cost to fill a tank of 60 litres? And B, what is the capacity of a motorcycle's fuel tank if it costs 31 Pounds to fill? Pause the video, and then come back when you've got your answers.

Of course here you may use a calculator.

And questions number four and five.

I'm not going to read these ones out to you.

Just pause the video and then when you've got your answers, come back and we'll be ready to check questions one, three, to five.

Great work, let's check those answers then.

Question number one, A was not directly proportional, B was, C was, D was not, E was, and F was.

So the ones that were were B, C, E, and F.

Question number two, 36 tickets make 630 Pounds.

So one ticket would be divided by 36, 17.

50, or 17.

5 as I've written it there.

So the money made is equal to those tickets multiplied by 17.

50.

So if I buy 500 tickets, multiplying by 500, giving me 8,750 Pounds for part A, and then B, I would be dividing by 17.

5, which gives me 58 tickets.

So part A, 8,750, and part B 58 tickets.

Question three, we're looking for that multiplicative relationship, and here it's multiplied by 1.

55.

So the cost is equal to the litres multiplied by 1.

55 60 multiplied by 1.

55 or 1.

55 Pound, I should say, is 93 Pounds.

So part A, 93 Pounds, and part B I'm going back the other way.

So I would need to divide by 1.

55, meaning the motorcycle's fuel tank has a capacity of 20 litres.

Question number four, the volume of water in litres is equal to the number of minutes multiplied by 11.

14 multiplied by 11, so after 14 minutes there would be 154 litres of water in the pool.

If there are 2,750 litres of water in the pool, how many minutes did this take? So we'll do 2,750 divided by 11, and that would mean it took 250 minutes.

And finally, question number five.

The relationship butter needed in grammes is the number of cookies multiplied by 12.

How much butter is needed to make 220 cookies? So I multiply 220 by 12, giving me 2,640 grammes, or you may have changed that into kilogrammes, 2.

64 kilogrammes.

And then part B, I do 6,000 'cause we were looking at how much is for six kilogrammes, which is 6,000 grammes.

6,000 divided by 12, we could make 500 cookies.

Let's move on then.

We'll now start looking at what happens.

We've got abstract problems here, so they won't have a context like the ones we've just looked at, but the process is going to be the same.

A is directly proportional to B.

When A equals 18, B equals 126.

Find the value of B when A is 123, and find the value of A when B is 189.

As A and B are directly proportional, they share a multiplicative relationship.

Therefore, we can put the values of A and B into a ratio table.

Here is my ratio table.

We know when A equals 18, B equals 126.

We want to find out the value of B when A is 123, and the value of A when B is 189.

We'd need to look then for our multiplicative relationship.

What is the multiplicative relationship between 18 and 126? We can multiply by seven, 18 multiplied by seven is 126.

And if you weren't sure, we could do 126 divided by 18.

We now know to go from A to B, we multiply by seven.

123 multiplied by seven is 861.

So what do I do to go from B to A? That's right, I do the inverse.

I divide by seven.

189 divided by seven is 27.

So we've used those two values, and then we found the missing value of B and A.

Y is directly proportional to X.

We've got the same type of problem when X is 65, Y is 78, and then we need to find some values of X and Y given values of X and Y.

Again, we know they're directly proportional so they share that multiplicative relationship so we can use the ratio table X and Y.

When X is 65 Y is 78, and then we need to find Y when X is 42, and we need to find X when Y is 43.

So I'm looking for that multiplicative relationship, so I'm going to do 78 divided by 65, giving me 1.

X multiplied by 1.

2 gives me Y so when X is 42, Y is 50.

And Y then would be yeah, divide by 1.

43.

2 divided by 1.

2 is 36.

G is proportional to H.

When G equals 108, AG equals 48.

So again, we've just changed the letters.

We're still told that it's directly proportional relationship, so we can put it into that ratio table.

We know when G is 108, eight is 48, and then we can fill in the values we know.

Now we can find the missing values in the table, can't we? What's my multiplicative relationship? I'm gonna do 48 divided by 108, which gives me four over nine.

So there I need to use that exact fraction because it's a recurring decimal.

567 multiplied by 4/9 is 252.

And to go back the other way, I'm gonna do the inverse.

I'm gonna divide by 4/9.

20 over three divided by 4/9 is 15.

Right, have a go at this one.

I'd like you please given that A is directly proportional to B to find the missing values in the table below.

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

Of course, you may use the calculator.

Great work.

Multiplicative relationship between A and B.

A multiplied by 0.

6 gives us B, so therefore, B divided by 0.

6 is gonna give us A.

So our missing values when B is 23.

52, A is 39.

When A is 0.

25, B is 0.

15.

When A is 3/16, B is 0.

1125, and when B is 5/8, A is 25/24, and you may have that written as a decimal.

Now we can have a go, or I should say, you can have a go at task B.

I'd like you please to find the odd one out in each row, and that's the one that does not share the same directly proportional relationship as the others.

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

Well done.

Question number two, three, and four.

I'm not going to read those out.

I'd just like you to have a go at those.

Remember if you need to, you could go back and rewatch those example ones that we went through, but I don't think you'll need to.

I think you're more than ready to give these three questions a go.

Here we are then, here are our answers.

So in the first row, it was the one in the fourth column.

All of them were multiplied by 1.

5, but the one in the fourth column was multiplied by 1.

In the second row, it was the one in the first column, in the third row, it was the one in the third column, and in the final, in the last row, it was the one in the final column.

So in the bottom right hand corner.

Question two, you needed to find the multiplicative relationship which was multiplied by 1.

2, going from A to B.

So going from B to A we're divided by 1.

When A is 37, B is 44.

4, and when B is 96, A is 80.

Question three, the relationship between X and Y divide by six going from X to Y.

So from Y to X is multiplied by six.

When X is 15.

6, Y is 2.

6, and when Y is 13.

5, X is 81.

Question number four, the relationship was multiplied by two thirds going from G to H.

So from H to G would be divided by two thirds.

When G is 54, H is 36, and when H is 5/9, G is 5/6.

Now let's summarise what we've done during today's lesson.

Two variables are in direct proportion if they have a constant multiplicative relationship.

Since variables in direct proportion have a constant multiplicative relationship, we can use a ratio table to find variables if we know a pair of values, and there's an example there of one that we went through.

We were looking at A being directly proportional to B, we found that multiplicative relationship going from A to B.

So from B to A will be the inverse, and then we can find those missing values.

Great work today.

Really enjoyed working alongside you, and I know that you're gonna join me really, really soon.

Take care of yourself.

Goodbye.

I've finished the video