Lesson video

Hello, everyone.

My name is Ms. Ku, and today, we're looking at compound measures.

A fantastic topic, looking at pressure, speed, density, volume, lots of real life applications.

I hope you really enjoy the lesson.

Let's make a start.

Hi, everyone, and welcome to today's lesson on converting between other compound measures, and it's under the unit compound measures.

By the end of the lesson, you'll be able to convert between other compound measures.

Let's have a look at some keywords starting with pressure.

And pressure is the perpendicular force applied to the surface of an object per unit area over which that force is distributed.

We'll also be looking at density.

The density of a substance is measured as the substances mass per unit of volume.

We'll look at that in our lesson.

Today's lesson will be broken into two parts.

Firstly, we'll be reviewing metric conversions, and then moving onto conversions involving density and pressure.

So let's make a start.

Conversion between units of measure is an example of variables, which are directly proportional.

For example, converting between metric units, or converting between units of time, or converting between different currencies, or converting between metric and imperial units.

So let's start by looking at a 1 metre square.

You'll see on the left hand side a 1 metre by 1 metre, and the square on the right hand side is 100 centimetres by 100 centimetres.

So the area of the left square is 1 metre multiply by 1 metre, which is 1 metre squared.

And working out the area of the square on the right, it's 100 centimetres multiplied by 100 centimetres, which is 10,000 centimetres squared.

Now, are these exactly the same squares? Want you to have a little think.

Well, yes they are, they're exactly the same.

So therefore, they must have the same area.

And Aisha recognises that that means 1 metre squared is exactly the same as 10,000 centimetre squared.

And Jacob says, "That makes sense as both lengths and widths have been multiplied by 100." Aisha recognises that this means that the overall effect on the area is multiplying by 10,000.

So in other words, we've just worked out 1 metre squared is equal to 10,000 centimetre squared.

I want you to have a little think now.

What do you think 1 centimetre squared is in millimetre squared? And if you want, you can use this little square to help you.

See if you can give it a go.

Press pause if you need a bit of time.

Well done.

Well, hopefully, you've spotted that 1 centimetre by 1 centimetre is exactly the same as 10 millimetres by 10 millimetres.

Therefore, we've identified that 1 centimetre squared is exactly the same as 100 millimetres squared.

Well done if you got this.

Now it's time for a quick check.

What is 1 metre squared in millimetres squared? Same again, if you want to use the square on the right to help you, please do.

Press pause if you need more time.

Well done, let's see how you got on.

Well, hopefully, you've spotted that 1 metre by 1 metre is exactly the same as 1,000 millimetres by 1,000 millimetres, which gives an area of 1,000,000 millimetres squared.

So that means 1 metre squared is exactly the same as 1,000,000 millimetres squared.

Well done if you got this.

So in summary, we know 1 metre squared is equal to 10,000 centimetres squared.

And we know 1 metre squared is equal to 1,000,000 millimetres squared.

You also discovered that one centimetre squared is the same as 100 millimetres squared.

Sometimes sketching the squares can help you work out the relationship between these measures.

Alternatively, recognising the scaling multiplier of the length and then squaring this will give the equivalent area in the required units.

Now, it's time for another check.

Let's use our knowledge on the fact that 1 metre squared is equivalent to 10,000 centimetre squared.

I want you to convert 542,800 centimetres squared into metres squared.

Press pause if you need more time.

Well done, let's see how you got on.

Well, hopefully, you've spotted it's C.

It's 54.

28 metres squared, and the reason because we divide by our 10,000, thus converting our centimetre squared into metre squared.

Well done if you got this.

Now, let's work out the volume of these identical cubes using metres cubed and centimetres cubed.

You can see the cube on the left is a 1 metre by 1 metre by 1 metre, and the cube on the right is 100 centimetre by 100 centimetre by 100 centimetre.

Given that the cube on the left is in metres, we can work out the volume quite easily.

1 by 1 by 1 gives us a volume of 1 metres cubed.

And the cube on the right, same again, we can work at the volume quite easily.

100 centimetres multiplied by 100 centimetres multiplied by 100 centimetres gives me 1 million centimetres cubed.

Now, given that we know 1 metre is exactly the same as 100 centimetres, therefore, we know 1 metre cubed is exactly the same as 1 million centimetres cubed.

Let's see if you can use this knowledge, and work out what is 1 centimetres cubed in millimetres cubed.

Use these cubes to help if you need, and the second question ask you to identify what is 1 metre cubed in millimetres cubed? Same again, you can use these cubes to help.

See if you can give it a go.

Press pause one more time.

So let's see how you got on.

Well, the first question asks you to convert 1 centimetre cubed into millimetres cubed.

To do this, let's have a look at the conversion of centimetres into millimetres.

A 1 centimetre by 1 centimetre by 1 centimetre cube is the same as a 10 millimetre by 10 millimetre by 10 millimetre cube.

This meaning 1 centimetre cubed is exactly the same as 1,000 millimetres cubed.

Really well done if you got this.

The second question wants you to identify what is 1 metre cubed in millimetres cubed? Well, let's convert our metres into millimetres.

A 1 metre by 1 metre by 1 metre is exactly the same as 1,000 millimetres by 1,000 millimetres by 1,000 millimetres.

This means that 1 metre cubed is exactly the same as 1 billion millimetres cubed.

Massive well done if you go this one.

So in summary, we know 1 metre cubed is exactly the same as 1,000,000 centimetres cubed.

1 metre cubed is exactly the same as 1 billion millimetres cubed, and we know 1 centimetres cubed is the same as 1,000 millimetres cubed.

Sometimes these are tricky to remember, but sketching the cubes can help you work out the relationship between these measures.

Alternatively, recognising the scaling multiplier of the length and cubing this will give the equivalent volume in the required units.

So now, we know 1 metre cubed is equivalent to 1,000,000 centimetres cubed.

What we're going to do is convert 653,000 centimetres cubed into metres cubed.

Which one do you think it would be? Press pause if you need more time.

Well done, let's see how you got on.

Well, it is C, and the reason why is because we have 653,000 divided by 1,000,000, and this gives us 0.

653 metres cubed.

Very well done if you got this.

Great work, everybody.

Now, it's time for your task.

I want you to fill in the missing values for question one in the required units of measure.

See if you can give it a go.

Press pause if you need more time.

Great work.

Question 2 wants you to calculate the area of this triangle, and give your answer to two decimal places and the units I want is metres squared.

See if you can it a go.

Press pause if you need more time.

Great work.

Let's move on to question 3.

Question 3 wants you to calculate the volume of this cuboid, and give your answer in centimetres cubed.

See if you can give it a go.

Press pause if you need more time.

Well done, let's move on to these answers.

Here are answers to question 1 and 2.

Press pause if you need more time to mark.

Well done, let's move on to question 3.

This is our answer, 504,000 centimetres cubed.

In a couple of different ways to work this one out, you could have either converted each length into centimetres and work out the volume from there.

Or you could have written your answer in metres cubed, and then converted it into centimetres cubed.

Couple of different ways there.

Well done if you got this answer.

Great work, everybody.

So now let's have a look at conversions involving density and pressure.

Aisha has two cubes of different metals.

Cube A has a length of 15 millimetres by 15 millimetres by 15 millimetres, and a mass of 27 grammes.

Cube B has a mass of 54 grammes and has length 3 centimetres by 3 centimetres by 3 centimetres.

Aisha says, "I can convert 15 millimetres to 1.

5 centimetres, and then everything is just doubled.

So that means the density would be doubled too." Let's have a look to see if Aisha is correct.

Well, first things first, we can start by converting to the same units.

You might notice cube A is given in millimetres, and cube B is given in centimetres.

So converting to the same units makes things a little bit more easier when comparing.

Then let's insert our information into a ratio table, starting with cube A.

Cube A has a mass of 27 grammes.

Now, let's have a look at the volume.

Well, to work out the volume, it'd be 1.

5 cubed centimetres cubed.

Working out 1.

5 cubes gives me 3.

375 centimetres cubed.

Now remember, density always wants per unit of volume.

So I'm going to divide our mass and our volume by 3.

375 giving me 1 centimetre cubed.

In other words, per unit of volume.

So dividing 27 by 3.

375 means we have 8 grammes per 1 centimetre cubed.

In other words, we have our density, 8 grammes per centimetres cubed.

Now, let's have a look at cube B.

For cube B, we know the mass is 54 grammes.

Let's work out that volume.

It's in centimetres, so it's simply 3 cubed centimetres cubed.

Working this out gives me 27 centimetres cubed.

Remember again, the definition of density wants per unit of volume.

So I want to find out what the mass is per 1 centimetres cubed.

So I'm going to divide by 27.

Dividing everything by 27 gives me 2 grammes per 1 centimetre cubed.

In other words, I know my density.

The density of cube B is 2 grammes per centimetres cubed.

So therefore, I know the density of cube B is a quarter of the density of cube A.

The method we looked at was converting to the same units first, allowing us to compare densities more easily.

But now, let's have a look at Izzy's approach.

But Izzy says she prefers to work out the densities in the given units, and then convert.

So let's have a look how Izzy tackles this question.

First of all, she looks at the mass 27 grammes, and then she works out the volume in millimetres cubed.

So she's worked out the volume to be 3,375 millimetres cubed.

Just like before, she's worked out the mass per 1 millimetres cubed, and then she's converted it into centimetres cubed.

But she has made an error.

Can you spot where Izzy's error is? See if you can give it a go.

Well done.

Well, hopefully, you've spotted, she's made a mistake converting millimetres cubed into centimetres cubed.

Remember, there's 1,000 millimetres cubed in 1 centimetres cubed.

So that means the density of cube A should be 8 grammes per centimetres cubed, which we looked at before.

Therefore, when comparing or converting compound measures when lengths are given, you can either convert the lengths into the desired units first, and then calculate the density, or you can calculate the density in the given unit, and convert into the desired units after.

Which do you prefer? Now, it's time for a quick check.

Here, we have two questions.

Here is a cube of red brass and the mass is 560 grammes, and we're asked to work out the density of the cube in kilogrammes per metres cubed.

The second question says here's a block of nickel in the shape of a triangular prism, and it weighs 178 grammes.

We're asked to work out the density of the cube in grammes per centimetres cubed.

So you can give it a go.

Press pause if you need more time.

Well done.

So let's see how you got on.

Well, for the first question, I'm going to convert our 560 grammes into kilogrammes.

Given the fact that the question wants it in kilogrammes per metres cubed.

Then I'm gonna convert my length of four centimetres into metres.

Same again, given the question wanted it to be kilogrammes per metres cubed.

From here, I'm gonna put it into my ratio table.

I know the mass is 0.

56 kilogrammes, and the volume of our cube can be found by 0.

04 cubed, which works out to be 0.

000064 metres cubed.

So I have 0.

56 kilogrammes per 0.

000064 metres cubed.

But remember, density always wants per unit of volume.

So I'm going to divide both my mass and my volume by 0.

000064 to give me one metre cubed and a mass of 8,750 kilogrammes.

In other words, my density is 8,750 kilogrammes per metres cubed.

Really well done if you got this.

Well done, let's have a look at the second question.

The question wants us to look at grammes per centimetres cubed.

So converting to the relevant units, I'm going to convert all of my units into centimetres.

From here, I can work out the volume of our triangular prism to be 2 multiply by our 1.

25, all divided by 2 multiplied by that 1.

6.

Therefore, putting it into a ratio table, I know the mass of 178 grammes is given per 2 centimetres cubed.

But remember, density is measured per unit of volume.

So I need to divide everything by 2 to give me 89 grammes per 1 centimetres cubed, which is a density of 89 grammes per centimetres cubed.

Really well done if you got this.

The importance of unit conversion applies with any compound measures including pressure.

For example, here's a square based pyramid resting on a shelf, and the weight is given as five newtons.

The question wants us to work out the pressure exerted on the shelf by the pyramid in newton's per metre squared.

So let's have a look at those units.

It's required to write our pressure as newton's per metres squared, but our pyramid is given in centimetres.

So all I need to do now is to convert our centimetres into metres as you can see here.

From here, I can use my ratio table again.

The force is five newtons.

The base of our pyramid is a square, so working out the area of that square is 0.

25 squared, which works out to be 0.

0625 metres squared.

In other words, we have a force of 5 newtons per 0.

0625 metres squared.

But remember, pressure wants it per unit of area.

So I'm going to divide by our 0.

0625, which gives me 1 metre squared and a force of 18 newtons.

So this means, I've got 18 newtons per 1 metre squared, which is a pressure of 18 newtons per metre squared.

Again, Izzy says she prefers to work on the pressure in the units given, and then convert.

So that means, Izzy would work out what the area would be in centimetres squared, which is 625 centimetres squared.

Then she works out the pressure as newton's per centimetre squared, and then she recognises 10,000 centimetre squared is 1 metre squared, and then convert our newton's per centimetre squared into newton's per metre squared.

Thus, giving us exactly the same answer as before, 18 newtons per metre squared.

So therefore, when lengths are given in compound measures in questions, you can either convert the lengths into the desired units first, and then calculate the pressure, or you can calculate the pressure in the given units, and then convert into the desired units after.

You'll notice both of them give exactly the same answer.

For which do you prefer? Great work, everybody.

Let's have a look at a check.

A cylindrical bucket with radius 12 centimetres is filled with sand and it sits on a table.

It has a weight of 120 newtons.

Now, we're asked to work up the pressure exerted on the table by the bucket in newton's per metre squared, and we're asked to give our answer to three significant figures.

Quick question, take your time, and press pause.

Well done, let's see how you got on.

Well, first of all, let's calculate the area in contact with the table in metres squared.

We know that the radius of our circle is 12 centimetres, so converting into metres gives us 0.

12 metres.

Now, we also know the formula for the area of a circle.

It's pi multiply by a radius squared.

So therefore, substituting our radius in metres gives us an area of 0.

0144 pi metres squared.

Notice how I've kept it in terms of pi just for that accuracy.

From here we can insert into our ratio table.

We know 120 newtons is applied to an area of 0.

0144 pi metres squared.

But remember pressure wants it per unit of area.

So we need to divide by 0.

0144 pi in order to give me per metre squared.

That means I also divide the force of 120 by 0.

0144 pi giving me 2652.

58 et cetera newtons.

So therefore, I know the pressure in three significant figures is 2,650 newtons per metre squared.

Really well done If you've got this.

Great work everybody.

However, sometimes the question will not give the lengths.

So we must use our knowledge on area and volume conversions to work out densities and all pressures in the same units.

For example, this question gives us two cubes and asks which is denser, 240 kilogrammes per metres cubed or 2.

4 grammes per centimetres cubed? Well to work this out we must convert to the same units.

So you can convert kilogrammes per metres cubed to grammes per centimetres cubed or vice versa.

For me, let's have a look at converting kilogrammes per metres cubed into grammes per metres cubed.

Using a ratio table again, I'm going to show 240 kilogrammes per metres cubed.

Now to convert it into grammes per centimetres cubed, let's have a look at our conversion.

We know 1 million centimetres cubed is equal to one metres cubed.

So that means dividing by 1,000,000 means one centimetre cubed gives me a mass of 0.

00024 kilogrammes.

Now from here I need to convert our kilogrammes to grammes.

So to do that we simply multiply by 1000.

So multiplying by 1000 gives me 0.

24 grammes per centimetres cubed.

So now I know 240 kilogrammes per metre cubed is exactly the same as 0.

24 grammes per centimetres cubed.

Well in comparison that means we know the 2.

4 grammes per centimetres cubed is more dense.

Alternatively, let's have a look at converting our grammes per centimetres cubed into kilogrammes per metres cubed.

Very much the same, we're gonna put it into a ratio table, 2.

4 grammes per one centimetre cubed.

Well, considering we know 1,000,000 centimetres cubed is 1 metres cubed, if we multiply by 1,000,000, this means our 1 metres cubed gives a mass of 2,400,000 grammes.

Then converting our grammes into kilogrammes identifies 2,400 kilogrammes per metres cubed.

So that means our 2.

4 grammes per centimetres cubed is exactly the same as 2,400 kilogrammes per metres cubed.

Once again, that is greater than our 240 kilogrammes per metres cubed.

So that means we know 2.

4 grammes per centimetres cubed is more dense.

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

Here are two solids and their densities, which solid has the greater density? And I want you to show all your working out.

See if you can give it a go.

Press pause one more time.

Well done.

Let's see how you got on.

Well for the sphere I'm going to change our kilogrammes per metres cubed into grammes per millimetres cubed.

You could have changed the cube into kilogrammes per metres cubed.

I've just decided to change our kilogrammes per metres cube into grammes per millimetres cubed.

Now to do this, I'm going to use my ratio table and insert our 1,130 kilogrammes per metres cubed, then converting kilogrammes to grammes.

I have 1,130,000 grammes, which is still our 1 metres cubed.

Knowing that 1 metres cubed is 1 billion millimetres cubed, that means I have these values in my ratio table.

From here, simply dividing by 1 billion gives me my density per millimetres cubed.

Therefore I end up with a density of 0.

00113 grammes per millimetres cubed.

So now we have two densities of the same unit, we're able to compare.

And we can see the cube has a greater density as 0.

001226 grammes per millimetres cubed is greater than our 0.

00113 grammes per millimetres cubed.

Great work if you've got this.

Fantastic work everybody.

So now it's time for your task.

Here are two cubes, which cube has the greatest density and you must show all your working out.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's move on to question 2.

Here are two solids on top of a table, a cube and cylinder.

The cube has a weight of 30 newtons and lengths of 24 centimetres.

The cylinder has a radius of 120 millimetres and a weight of 60 newtons.

Which solid is exerting the greatest pressure.

And you're asked to show all your working out.

So if you can give it a go.

Press pause if we need more time.

Well done.

Let's move on to question 3 and 4.

Question 3 says, "Find the pressure exerted by a force of 500 newton's on an area of 80 centimetres squared." But we want our answer in newton's per metre squared.

And question 4 says, "Find the density of a solid with a volume of 12 metres cubed and a mass of 480 kilogrammes.

But we want our answer in grammes per centimetre cubed.

See if you can give them a go.

Press pause if even more time.

Well done.

Let's move on to question 5.

Question 5 wants you to put these metals in ascending order of density and you must show you're working out.

See if you can give it a go.

Press pause for more time.

Great work everybody.

Let's go through these answers.

Here's my wonderful working out.

Please press pause if you need more time to mark.

Well done.

For question 2 once again, here's all my working out.

Press pause if you need more time to mark.

Well done.

For question 3 and for question 4, here's all the working out.

Press pause if you need more time to mark.

Great work.

And for question 5, here's my working out for brass, thus giving me density of 8.

52 grammes per centimetres cubed.

For the gold, here's my working out to give me a density of 19.

3 grammes per centimetres cubed.

For mercury, here's my working out to give me a density of 13.

53 grammes per centimetres cubed.

For aluminium, well, it's already grammes per centimetres cubed so no working out required there.

And for copper, here's my working out, which gives me a density of 8.

96 grammes per centimetres cubed.

So therefore putting the metals in ascending order of density gives me aluminium, brass, copper, mercury, and then gold.

Really well done if you got this.

Fantastic work everybody.

So in summary, when comparing or converting compound measures when lengths are given, you can either convert the lengths into the desired units first, then calculate the compound measure, or calculate the pressure or density in the given units and then convert into the desired units.

However, sometimes the question will not always give the relevant measures, so we must use our knowledge on area and volume conversions to work out the densities and all pressures into the most appropriate units.

Massive well done everybody.

It was great learning with you.