Lesson video

Hi everyone, my name is Ms Ku and I'm really happy that you're joining me today.

Today we're going to look at some beautiful representations of direct and inverse proportion.

So let's make a start.

Hello everyone.

Welcome to this lesson on abstract inverse proportion under the unit direct and inverse proportion.

And by the end of the lesson, you'll be able to identify, write, and solve inverse proportion questions involving algebra.

Now let's have a look at some keywords.

Two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other.

Today's lesson will be broken into two parts.

Firstly, we'll be identifying and solving inverse proportions, and then we'll be solving abstract inverse proportion problems. So let's make a start, identifying and solving inverse proportions.

Now, when two variables are inversely proportional, there is a constant multiplicative relationship between one variable and the reciprocal of the other.

Ratio tables are an excellent approach to show inverse proportion.

For example, robots each work at an exact same rate when stacking shelves.

and it takes four robots 12 hours to stack a wall of shelves.

How many hours would it take eight robots to stack the shelves.

Now Laura says it'll take 24 hours because I'm going to use ratio table.

I doubled the four robots and doubled the 12 hours.

And Sophia says, that does not make sense, Laura.

Now can you explain why Laura's answer of 24 hours is incorrect? Have a little think.

Well, simply because the more robots used, the fewer the hours it will take.

So therefore it cannot be 24 hours.

This means the number of robots and the number of hours are inversely proportional.

As they are inversely proportional, there is a constant multiplicative relationship between one variable and the reciprocal of the other.

Therefore, when we multiply the number of robots by two, we multiply the hours by the reciprocal, which is a half, so that means it will take eight robots six hours to stack a wall of shelves.

Now recognising inversely proportional variables is important, and this can be easily done by understanding the context of the question.

In other words, ask yourself, as one variable doubles, does the other half? If so, then we know we multiply one variable by the constant and the other variable by its reciprocal.

We can also show how dividing a number is the same as multiplying by the reciprocal.

Therefore inverse proportion can also be shown using division.

It's important to note either approach can be used.

So let's have a look at a quick check.

Which of the following are examples of inverse proportion? And I want you to explain how you know.

Do you think it's A, 10 hoses have a constant rate of flow of water and it takes five hoses two days to fill a pool.

How many days do you think it will take 10 hoses.

Do you think that's inverse proportion or direct proportion? B: one pound is equivalent to $1,30.

How much will 10 pounds be equivalent to? Do you think that's inverse proportion or direct proportion? C: 10 jars hold five kilogrammes marmalade.

How many jars would hold two kilogrammes of marmalade.

Inverse proportion or direct proportion? And D in a factory, 10 machines take five days to produce a lorry load of items. How long will it take two machines.

Have a little think, and press pause if you need more time.

Well done.

Let's see how you got on.

Well, the first one is most certainly inverse proportion.

As the more hoses that will pump water, the less time it will take to fill the pool.

B is directly proportional, as the more pounds, the more dollars.

Exchange rates are always directly proportional.

C, this is directly proportional to the more marmalade, the more jars needed.

And D, this is inversely proportional.

As the machines used, the fewer days it will take to fill a lorry load.

Alternatively, the fewer machines used, the more days it will take to fill a lorry load.

Well done if you got these right.

Now, let's have a look at these inverse proportion scenarios and I want you to work out the answers to these.

We have exactly the same questions as before.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, using a wonderful ratio table, I've put in five hoses takes two days.

Recognising it's inversely proportional.

I know if I multiply the five hoses by two, therefore I multiply the two days by the reciprocal of two, which is one half.

Or alternatively you can say divide by two.

Therefore it takes 10 hoses one day to fill the pool.

Let's have a look at B.

While using our wonderful ratio table again, you can see 10 machines, it takes five days.

What do I do to 10 to make two? Well I divide by five.

This is the same as multiplying by one fifth.

Either one is fine.

From here, well, I know they are inversely proportional, so that means I multiply the number of days by the reciprocal of one fifth, which is five, thus giving me 25 days.

So it takes two machines 25 days to fill a lorry.

Really well done if you've got this.

Let's have a look at another check.

Label which ratio table shows direct proportion, inverse proportion or neither? And same again, I want you to explain how you know.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well for A, this shows direct proportion given the multipliers of four.

For B, we're subtracting.

There is no multiplicative relationship.

For C, this shows inverse proportion, given we have multipliers of seven and one seventh.

And for D, this also shows inverse proportion given we have a multiplier and divisors of three.

You might also say a multiplier of three and it's reciprocal one third.

Well done if you got this.

Variables showing inverse proportion can be represented in different ways.

Regardless how the variables are given, understanding the inverse relationship allows you to apply the correct multipliers.

For example, P and Q are inversely proportional and we are asked to fill in the table.

So given the fact that we know they are inversely proportional, we can recognise that multiplicative relationship between the multipliers and the reciprocal.

Then we can multiply our 1000 by one quarter.

So that means we can multiply our a hundred by four.

Multiplying by 1000 by one quarter gives me my 250, therefore multiplying my a hundred by four gives me my 400.

Now let's have a look when P is 100.

Well, when P is 100 we multiplied 1000 by one 10th.

So therefore I'm multiplying Q by the reciprocal of one 10th, which is 10, thus giving me 1000.

Now let's have a look at Q.

Well, I know if I multiply a hundred by 50, I get my Q value of 5,000.

So multiplying the P value by the reciprocal of 50, one 50th gives me the P value to be 20.

And lastly, let's find out what we multiply a thousand by to give 1,600.

Well, it's eight fifths.

Multiplying the Q by the reciprocal gives me 62.

5.

We could also check to see if the answers are correct by checking the multipliers with other variables.

For example, let's have a look at the P value 250 to the P value of 100.

Clearly I'm multiplying by two fifths, so therefore I must multiply the corresponding Q value by the reciprocal, which at five over two.

Yes it works.

So I know we're right.

This is a nice little way just to check to see if all the values are correct.

Now it's time for your check.

A and B are inversely proportional and Izzy has filled in the table below but she has made an error.

I want you to find the error and work out the correct value.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well hopefully you've spotted it's right here.

The B value cannot be 0.

15.

And there are lots of different ways to show where she made her mistake, but I'm just choosing the A value of 24 to the A value of 300, and the B value of two to the incorrect B value of 0.

15.

So you can see.

If I were to multiply the 24 by 25 over two, therefore I should multiply the B value by its reciprocal, which is two over 25.

That does not give me 0.

15.

So therefore this is the error.

The correct answer would be 0.

16.

Great work if you've got this, and a huge well done if you showed it in a different way.

Now more complex inverse proportion questions involve more variables to consider.

For example, 12 3D printers can print a doll's house in eight hours.

This means we have 12 machines complete the doll's house in eight hours.

So that means a doll'S house requires 96 machine hours.

So if it takes 96 machine hours to build a doll's house, this means we need 96 machine hours full stop to build a doll's house.

So we can vary the machines and vary the hours as long as we get 96 machine hours.

So using the correct multipliers, given the inversely proportional relationship, I can multiply the number of machines by half, therefore multiplying the hours by two.

I still have 96 machine hours.

Alternatively, I could multiply the 12 machines by a quarter and multiply the hours by four, still giving me 96 machine hours.

So you can see this inversely proportional relationship here.

Now knowing this allows the questions to be more varied.

For example, we still have our 12 3D printers and they can print a doll's house in eight hours.

But after three hours, four of the 12 3D printers break down.

How long will it take for the other eight printers to finish the doll's house? Well first of all, we know we need 96 machine hours to build the doll's house.

So I put this in my ratio table.

This means so far we have 36 machine hours because we had 12 machines working nonstop for three hours, which gave us 36 machine hours.

Now given the fact that we have four which have broken down means we only have eight machines working.

Those eight machines must produce a total of 60 machine hours.

Given that 96, subtract our 36 gives us our 60.

So how do we find the number of hours needed for our remaining eight machines? All we need to do is find out what do we multiply eight by to give our 60? Well it's simply 7.

5, because 7.

5 multiplied by our eight gives us our 60.

This is a great question, adding a little bit more complexity to those inverse proportional relationships.

Well done.

Let's have a look at a check.

It would take seven machines eight hours to build a car.

All of the machines work at the same rate.

After five hours, one of the machines breaks down.

How long does it take for the remaining six machines to build the car? See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well first things first, we know seven machines take eight hours to build a car.

So that means we need 56 machine hours to build the car.

We know seven machines worked for five hours, which gave us a total of 35 machine hours.

Now given the fact that we need 56 machine hours to build the car, I'm going to subtract 35 from our 56 giving me 21 machine hours.

So my six remaining machines have to produce 21 machine hours.

So what's my multiplier? How many hours do I need? It has to be 3.

5 because 3.

5 multiply by my six gives me 21.

Really well done if you got this.

Great work everybody.

So now it's time for your task.

Read the questions carefully and identify which are the direct proportions, inverse proportions or neither.

Press pause if you need more time.

Great work.

Let's move on to question two.

Work out the answers to the following.

Press pause as you'll need more time.

Well done.

Let's move on to question three.

Identify which are direct proportion, inverse proportion or neither, and explain.

Press pause if you need more time.

Well done.

Let's have a look at question four.

The table show two variables P and Q, which are inversely proportional.

I want you to identify the mistake in each one and work out the correct answer.

Give it a go.

Press pause for more time.

Well done.

And lastly, let's have a look at question five.

Great complex question, read it carefully.

Press pause for more time.

Let's move on to these answers.

Question one, you should have had all of these answers.

Press pause if you need more time to mark.

Let's move on to question two.

You should have had these answers.

Press pause if you need more time to mark.

For question three, here's all my working out and my explanations of whether they are direct, inverse or neither.

Press pause if you need more time to mark.

For question four, hopefully you spotted, these are the mistakes and these are the correct answers.

Really well done if you've got this.

Press pause if you need more time to mark.

Let's move on to that tough question five.

Here's my working out and there's the final answer.

It takes a total of 11 hours to build the computer.

Well done if you got this.

Great work everybody.

So let's move on to the second part of our lesson, solving abstract inverse proportion problems. Now two variables are inversely proportional when there is a constant multiplicative relationship between one variable and the reciprocal of the other.

And a variable can be inversely proportional to the exponent or root of the other variable.

For example, here we have Y is equal to three over X squared.

This means Y is inversely proportional to X squared.

Here we have W is equal to eight over the square root of P.

This means W is inversely proportional to the square root of P.

We can also say X squared is inversely proportional to Y or we could say the square root of P is inversely proportional to W.

Now Laura says, but what about the three and the eight? Now these are the constant, sometimes called the constant of proportionality and they are important, but they remain constant as the variables increase or decrease.

So therefore, when identifying proportional relationships, we only compare the variables, not the constant of proportionality.

For example, P is equal to four over 7Q to the four.

This means P is inversely proportional to Q to the four and it means four sevenths is the constant of proportionality.

Knowing this, let's see if we can move on to a check.

Link the statements to the correct equations.

Not all of them can be paired.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well firstly, hopefully you've spotted four over seven X cubed means that Y is inversely proportional to X cubed.

Next, Y is equal to three over 11X means Y is inversely proportional to the X.

Next we have Y is equal to 10 over the square root of X.

This means Y is inversely proportional to the S square root of X.

Next we have Y is equal to two over X squared.

This means Y is inversely proportional to X squared.

And lastly Y is equal to seven over nine multiplied by the cube root of X.

This means Y is inversely proportional to the cube root of X.

Really well done if you got this.

So you can use a ratio table even when variables are inversely proportional using roots and exponents.

For example, a table of values is formed whereby P is inversely proportional to Q squared and we're asked to work out the missing values.

Now, just like before, we recognise that multiplicative relationship between the multiplier and its reciprocal.

So looking at Q squared, what do I do to want to give four while I multiply by four? Knowing it's inversely proportional means I multiply the Q by one quarter, which gives me 1.

5.

Still referring to the P value six and the Q squared values one, I'm going to multiply one by 25 and six by one over 25 giving me 0.

24.

Same again using the same P value and Q value, I'm multiplying by one hundredths to give me 0.

06 here.

And lastly, what do I multiply the six by to give the 0,015.

Well it's one over 400, so therefore I multiply the Q squared value by 400, thus giving me 400.

Now Laura says, why does the ratio table have P and Q squared? Shouldn't it just be P and Q? Now, ratio tables show the constant multiplicative relationship.

And given that P and Q squared are inversely proportional, the constant multiplicative relationship is between P and Q squared, not between P and Q.

Now it's time for a check.

W and the square root of X are inversely proportional.

I want to fill in the table.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, I'm going to multiply these values by two fifths and five over two.

I'm going to multiply these values by one fifth and five, and multiply these values by one sixth and six, and multiply these values by one 10th and 10.

Massive well done if you got these right? Let's have a look at another check.

Notice how all that done is represent the table in a different way, exactly the same process.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, I've just multiplied these values by 1000 and 1000th.

Multiply these values by 125 and one over 125, and multiply these values by one over 25 and 125.

Multiply these values by one over 1000 and 1000.

Massive well done if you've got this one right.

Great work everybody.

So now it's time for your task.

I want you to fill in the tables given the following inverse proportions.

Take your time.

Press pause if you need.

Well done.

Let's move on to question two.

Fill in the tables given the following inverse proportional relationships.

See if you can give it a go.

Press pause as you will need more time.

Very well done.

Let's move on to question three.

Question three.

Izzy again has made mistakes.

Can you explain her misunderstanding and can you draw and complete the correct table of variables given A and B squared are inversely proportional.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's go through these answers.

Well, you should have got these values for question one A and B.

Press pause if you need.

Well done.

Let's move on to question two.

We should have had these values.

Press pause as you'll need more time to copy this down.

Really, really well done if you've got any of these.

Question three.

Great question.

Izzy should have used a table showing the variables of A and B squared, not A and B.

So correcting the table, we should have had these values and then working out our values for A, we should have had this.

Massive well done if you got this.

Great work, everybody.

So in summary, when two variables are inversely proportional, there is a constant multiplicative relationship between one variable and the reciprocal of the other.

Recognising inverse proportion is so important, and this can be done by understanding the context of the question.

In other words, ask yourself, as one variable doubles, does the other half.

If so, we know we multiply one variable by a constant and the other variable by the reciprocal.

Variables can be inversely proportional to the square cube or the root of another variable.

Really well done today.

It was great learning with you.