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Hello, Everyone.
My name is Ms. Ku and I'm really happy that you're joining me today because it's one of my favourites again, compound measures.
Compound measures is one of my favourites because it's the real life application of mathematics.
I really hope you enjoy the lesson.
Let's make a start.
Hi everyone and welcome to today's lesson on combining densities under the unit Compound Measures.
By the end of the lesson you'll be able to combine compound measures calculating overall density.
So let's have a look at the keyword, density.
Now the density of a substance is measured as the substance's mass per unit of volume, and we'll be looking at this a lot in our lesson.
Today's lesson will be broken into two parts, combining densities first and then solving problems involving densities.
Let's make a start combining densities.
Now we combine densities every day.
For example, when you mix one fluid with another fluid of a different density, you're combining densities.
Making orange juice with concentrate and water is combining densities.
So when mixing the density of one fluid that is significantly bigger than another fluid, what do you think happens? Well, the fluid separates with the most dense fluid at the bottom.
When combining densities, we work out the overall density of the combined fluid by calculating the overall mass and the overall volume.
For example, 300 centimetres cubed of liquid A with a density of 1.
8 grammes per centimetres cubed and 150 centimetres cubed of liquid B with a density of 1.
2 grammes per centimetres cubed are mixed together to give liquid C.
What's the overall density? Well, to do this, let's have a look at liquid A first.
We're going to work out the mass and volume of liquid A.
Given we know the density is 1.
8 grammes per centimetres cubed, I've popped it into my ratio table.
Now remember the question states I've got 300 centimetres cubed of our liquid, so I'm going to simply multiply by 300, thus giving me a mass of 540 grammes per 300 centimetres cubed of our liquid A.
So now I have a mass and a volume of our liquid.
A.
Let's have a look at our liquid B.
We know the density is 1.
2 grammes per centimetres cubed.
Well for liquid B, we're told it's 150 centimetres cubed.
So I'm going to simply multiply by 150, thus giving me 180 grammes for the 150 centimetres cubed of our liquid B.
So now we have a mass and a volume of our liquid B.
Now we can work out the total mass and the total volume of our liquid C.
Well, it's the sum of the masses of liquid A and liquid B, which is 720 grammes and it's the sum of the volumes of liquid A and liquid B, which is 450 centimetres cubed.
Remember the definition of density, it's per unit of volume, so that means I need to divide by 450 to give me one centimetres cubed.
Divided by 450, I now have 1.
6 grammes per one centimetres cubed, which is a density for liquid C to be 1.
6 grammes per centimetres cubed.
Now it's time for a check.
Alex mixes 30 centimetres cubed of apple concentrate with 240 centimetres cubed of water.
Now the density of the apple concentrate is 1.
2 grammes per centimetres cubed and the density of water is 1.
00 grammes per centimetres cubed.
And we're asked to work out the density of Alex's apple drink and give our answer correct to two decimal places.
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 of all, let's have a look at our apple concentrate.
First we note is 1.
2 grammes per centimetres cubed, so I've popped it into my ratio table.
We also know we have 30 centimetres cubed of our apple concentrate, so I'm going to multiply by 30, giving me a mass of 36 grammes per 30 centimetres cubed.
Now let's have a look at our water.
Well, we know it's one gramme per centimetres cubed but we have 240 centimetres cubed of our water.
So multiplying by 240 means I have 240 grammes per 240 centimetres cubed.
Now we can identify our total mass and our total volume.
The total mass of the apple concentrate and the water is 276 grammes and the total volume of our apple concentrate and our water is 270 centimetres cubed.
Remember, density is per unit of volume, so I need to divide by 270 to give me one centimetres cubed, thus working out to be 1.
02 grammes per one centimetres cubed, which is a density of the entire apple drink to be 1.
02 grammes per centimetres cubed to two decimal places.
Really well done if you got this one right.
We can also combine metals with different densities too and when combining metals, an alloy is created and creating alloys allows manufacturers to create new alloys with specific properties.
For example, by adding chromium to steel, they can create stainless steel which is resistant to rust and erosion.
So let's have a look at a manufacturer who mixes 150 grammes of lead and 340 grammes of tin to make an alloy.
Now lead has density of 11.
34 grammes per centimetres cubed and tin has a density of 7.
31 grammes per centimetres cubed.
And we're asked to work out the density of the entire alloy giving our answer to one decimal place.
So let's have a look at the lead first.
We know lead has a density of 11.
34 grammes per centimetres cubed.
So let's work out the mass and the volume of the lead we have.
Well, the question states I have 150 grammes of lead.
So to convert our 11.
34 grammes to 150 grammes, the multiplier would be 150 over 11.
34.
This would then give me a volume of 13.
23 centimetres cubed.
So now I know my lead has a mass of 150 grammes and a volume of 13.
23 centimetres cubed.
Now let's have a look at our tin.
The tin has a density of 7.
31 grammes per centimetres cubed.
So I've popped it into my ratio table and let's work out the mass and the volume of our tin.
Well, we know there are 340 grammes of our tin, so that means we multiply by 340 over 7.
31, giving me a volume of 46.
51 centimetres cubed.
In other words, the tin has a mass of 340 grammes per 46.
51 centimetres cubed.
From here we can work out the total mass and the total volume of our alloy.
By summing the total mass we have 490 grammes.
By summing the total volume we have 59.
74 centimetres cubed.
Remember, density is per unit of volume, so that means we divide by 59.
74, thus giving me a mass of 8.
2 grammes per one centimetres cubed, which is then a density of our alloy to be 8.
2 grammes per centimetres cubed.
Now it's time for your check.
A 300 gramme brass block is made of copper and zinc.
200 grammes of copper is used.
Now copper has a density of 8.
96 grammes per centimetres cubed and zinc has a density of 7.
14 grammes per centimetres cubed.
And the question wants you to work out the density of the entire brass block to one decimal place.
See if you can give it a go.
Press pause if you need more time.
Great work, everybody.
So let's see how you got on.
Let's have a look at our copper first.
We know copper has a density of 8.
96 grammes per centimetres cubed, but we also know there's 200 grammes of copper.
So, the multiplier would be 200 over 8.
96 giving me a volume of 22.
3 centimetres cubed.
Looking at our zinc, we know the density is 7.
14 grammes per centimetres cubed and to make the 300 gramme brass block, we need 100 grammes of zinc.
So therefore, we have to multiply by 100 over 7.
14 to give me a volume of 14.
00 centimetres cubed for our zinc.
Therefore, we know that the total mass would be 300 grammes and the total volume would be 36.
3 centimetres cubed.
Remember, density is per unit of volume, so we need to divide by 36.
3 to give me a mass of 8.
26 grammes per one centimetres cubed, which gives the density of brass to be 8.
26 grammes per centimetres cubed.
Really well done if you got this.
Referring to the previous question, it could also be represented as this, whereby the ratio tables or other working out helps to centralise the information.
For example, you can see from our previous question where there was 200 grammes of copper, so therefore we knew there had to be 100 grammes of zinc in order for there to be 300 grammes of brass.
We also knew the density of copper was 8.
96 grammes per centimetres cubed and the density of zinc was 7.
14 grammes per centimetres cubed.
Notice how the same information is represented in this table.
Now from here we can use our ratio tables as before.
Working out the volume of the copper using the density, we know this is 22.
3 centimetres cubed.
We know the volume of the zinc as was 14.
00 centimetres cubed.
That means we could work out the volume of the brass to be 36.
3 centimetres cubed.
From here, we can use another ratio table just like before to identify the overall density of the brass to be 8.
26 grammes per centimetres cubed.
So notice how we've used the same information but we simply centralised it into a nice table.
Now, it's time for a check.
Rose gold is made by combining pure gold and copper.
Fill in the table and show you're working out, giving your answer to two decimal places where appropriate.
See if you can give it a go.
Press pause if you need more time.
Great work.
If you've chosen to use ratio tables, then we know the mass of the gold is 19.
3 grammes per centimetres cubed but we have 90 grammes of our gold, which works out to give a volume of 4.
66 centimetres cubed.
So I've inserted it into my table.
Looking at copper, we know it had to be 30 grammes of copper as the overall mass of the rose gold was 120 grammes.
Given that we have the mass is 30 grammes and the volume is 3.
3 grammes per centimetres cubed, we can work up the density because the density is per one centimetres cubed.
So we're going to divide by 3.
35 giving me 8.
96 grammes per centimetres cubed, which is my density of the copper.
Now we have all the information we need to work out the density of rose gold.
Let's work out the total volume, the total volume would be 8.
01 centimetres cubed and we know the mass to be 120 grammes.
From here, we can work out our density by working out one centimetres cubed, which is 14.
98 grammes.
So that means we know the density of our rose gold is 14.
98 grammes per centimetres cubed.
Great work, everybody.
So now it's time for your task and we need to work out the combined density of the following.
Give your answer to one decimal place where appropriate.
Press pause as you'll need more time.
Well done.
Let's move on to question two.
The density of orange juice is 1.
5 grammes per centimetres cubed and the density of syrup is 1.
8 grammes per centimetres cubed.
The density of carbonated water is note, 0.
99 grammes per centimetres cubed.
Now, if 60 centimetres cubed of orange juice are mixed with 90 centimetres cubed of syrup and 300 centimetres cubed of carbonated water to make a drink, can you work out the density of our drink? Take your time and press pause if you need.
Great work.
Let's move on to question three.
When a high percentage of silver is mixed with copper, it's called sterling silver.
Work out the missing information giving answers to one decimal place where appropriate.
Press pause if you need more time.
Great work.
Let's move on to these answers.
Well, here are our answers.
Please press pause if you need more time to mark.
Well done.
For question two, here's the working out, so you should have had a density for the entire drink to be 1.
22 grammes per centimetres cubed.
Well done.
For question three, here are our missing values.
Massive well done if you filled out our table.
Fantastic work, everybody.
Let's move on to solving problems involving densities.
Some problem solving questions involve converting between units of measure.
These units can be units of mass, units of volume, units of capacity.
In either case, it's important to know the equivalent units of measure so to convert.
For example, here we have a table showing the measurements for metal A and metal B and they are mixed to give metal C.
What do you think you need to do first? Have a little look.
Press pause if you need more time.
Well, hopefully you've spotted we have different units here.
For example, the density is grammes per centimetres cubed and the mass is in kilogrammes.
We also have a volume in metres cubed.
So we have different units.
So let's convert all to the same units.
Converting grammes to centimetres cubed is going to be more efficient.
So let's have a look at metal A.
We know it's 3,000 kilogrammes per 1.
5 metres cubed.
So to convert it into centimetres cubed to remember that conversion, one metres cubed is equivalent to 1 million centimetres cubed.
So therefore we know 1.
5 metres cubed is 1,500,000 centimetres cubed.
Now we need to convert our kilogrammes into grammes.
So we know 3,000 grammes is exactly the same as 3 million grammes.
Now we have mass in grammes and volume in centimetres cubed.
So let's work up the density grammes per centimetres cubed.
Remember the density is per unit of volume, so I'm going to divide by 1,500,000 giving me one centimetres cubed, which is two grammes.
So now we have our density of metal A in grammes per centimetres cubed.
So now what I'd like you to do is work out the missing information and give the density of metal C in grammes per centimetres cubed.
See if you can give it a go.
Press pause if you need more time.
Well done.
So let's look at metal B where we know the density of metal B is four grammes per one centimetres cubed, but we have 60 kilogrammes.
So I'm gonna convert this into grammes.
60 kilogrammes is 60,000 grammes.
So the multiplier would have to be 15,000, which means the volume must be 15,000 centimetres cubed.
So now we know the volume of metal B is 15,000 centimetres cubed, we can convert everything into the relevant units, which means we have metal C is 3,060,000 grammes.
We know the volume of metal C is 1,515,000 centimetres cubed giving a density for metal C to be 2.
02 grammes per centimetres cubed.
Well done if you got this.
Great work, everybody.
So now it's time for your task.
For question one, here we have a table showing the measurements for metal A and metal B.
They're to give metal C.
I want you to fill in the table and give your answers to two decimal places where appropriate.
Press pause as you'll need more time.
Well done.
Let's move on to question two.
360 grammes of liquid A with a density of 1.
2 grammes per millilitre is mixed with 300 grammes of liquid B with a density of 1.
5 grammes per millilitre.
Work the overall density of the mixture.
Press pause as you'll need more time.
Great work.
Let's move on to question three.
A solid is made from a cylinder and cone.
Now the density of the cylinder is two grammes per centimetres cubed and the density of the cone is 4.
8 grammes per centimetres cubed.
And we're asked to work out the average density of the solid, giving your answer to three significant figures.
I've given you a helpful formula here to work out the volume of a cone.
It's 1/3 pi r squared times the height.
See if you can give it a go.
Press pause if you need more time.
Great work.
Let's see how you got on.
Well, for question one, you should have got these values in our table.
Please press pause if you need more time to fill in.
Fantastic work.
Let's move on to question two.
So for question two, I've just centralised all our results into this table, so hopefully you can work out.
But the density of our overall mixture is 1.
32 grammes per centimetres cubed.
Really well done if you've worked this one out.
For question three, well, we had to work out the volume of the cylinder and the volume of the cone using our knowledge on these formulas.
The volume of the cylinder can be found by pi multiplied by our radius squared, multiplied by the height, which is our 4,050 pi centimetres cubed.
The mass of the cylinder is therefore 8,100 pi grammes.
The volume of the cone is found by 1/3 pi times 15 squared, times 20, which gives us 1,500 pi centimetres cubed.
This then gives us a mass of 7,200 pi grammes.
Therefore, the overall density of our solid to three significant figures is 2.
76 grammes per centimetres cubed.
Really well done if you got this.
Great work, everybody.
So in summary, combining densities is so important and it's everywhere when making drinks, combining foods, and even in the alloys and take for granted every day.
When combining densities, finding the individual measures allows you to work out the overall density and ratio tables allow us to efficiently calculate this.
Sometimes for more complex problems, we also need to use knowledge on unit conversions.
Great work, everybody.
It was wonderful learning with you.
