## What you need to know

### Density Mass Volume

Mass is a measure of how heavy something is and is usually measured in grams (g) or kilograms (kg). Volume is a measure of how much space something takes up and is usually measured in cubic centimetres (\text{cm}^3) or cubic metres (\text{m}^3). Density is a measure of how tightly the mass of an object is packed into the space it takes up, so we calculate this by dividing its mass by its volume.

\text{density } = \dfrac{\text{mass}}{\text{volume}}

Having a good understanding of the following topics will help with Density, Mass and Volume

### Density Mass Volume Formula Triangle

On top of calculating density, you will also be expected to be able to rearrange this formula and use it to find mass (when given the density and volume) and volume (when given the density and mass). From now on, let d be density, V be volume, and m be mass. Then, our original equation looks like:

d = \dfrac{m}{V}

If we multiply both sides of the equation by V, then swap the left-hand side and right-hand side, we get

m = d \times V

So, we see that we can calculate mass by multiplying the density by the volume. Furthermore, if we then divide both sides by d and swap the left-hand side and right-hand side, we get

V = \dfrac{m}{d}

Therefore, we can calculate volume by dividing the mass by the density. It’s good practice rearranging this formula to get it in the form that you want, but a quick way to remember how to calculate one of these values using the other two is to refer to the triangle below.

### Example 1: Calculating Density

An object has a mass of 570g and a volume of 2,280 \text{cm}^3. Calculate its density.

As stated, density is equal to the mass divided by the volume, so this will be our calculation, but what will the unit of our answer be? In the question, the mass is given in grams and the volume in cubic centimetres, so the density should be in grams per cubic metre, which is denoted either like \text{gm}^{-3} or \text{g/m}^3.

\text{Density } = \dfrac{570}{2,280} = 0.25\text{ g/m}^3

This is the unit of density you’ll see most often but not always, so watch out for the units given in the question. In general, the units will also be in the form “[units of mass] per [units of volume]”. The formula triangle below shows a way to determine the formula by covering up the letter you want to calculate.

### Example 2: Calculating Mass

A cat has volume 0.004 \text{ m}^3 and density 980 \text{ kg/m}^3. Calculate the mass of the cat.

We’re looking for mass, so, constructing the triangle and covering up the m, we get

Therefore, to calculate the mass we must multiply the density by the volume. So

\text{mass } = 0.004 \times 980 = 3.92\text{ kg}

**Note:**we know that the units are kilograms by looking at the units in the question.

### Example 3: Calculating Volume

A bottle of water has a density 1000 \text{ kg/m}^3 and mass 0.5 \text{ kg}. Calculate the volume of the water bottle giving your answer in litres.

We’re looking for volume, so, constructing the triangle and covering up the V,

we get volume = 0.0005 \text{ m}^3 which we multiply by 1000 to get into litres, giving us the final answer to be 0.5L

### Example Questions

1) A bottle is filled with 2 kg of olive oil which has a density of 0.925 \text{ g/cm}^3. What is the volume of the olive oil to the nearest cm³?

We are calculating the volume, so by covering up the V we can see from the triangle above that we have to divide m by d.

Before we can do this, however, we have to make sure that we have the correct units. The mass is in kilograms, but the density is in *grams* per cubic centimetre. This means that we have to first convert the kilograms into grams before proceeding.

2\text{ kg} = 2000 \text{ g}

Therefore, the volume of the olive oil can be calculated as follows:

\dfrac{2000}{0.925} = 2,162\text{ cm}^3.

2) A cube with side length 7m has a density of 10,800,000 kg/m³. Work out the mass of the cube.

We are calculating the mass, so by covering up the m we can see from the triangle above that we have to multiply d by V.

However, we don’t know the volume, but we do know that the shape is a cube with a side length of 7m, so the volume of the cube is:

7 \times 7 \times 7 = 343\text{ m}^3.

Now that we know the volume, we can multiply it by the density in order to calculate the mass:

343 \times 10,800,000 = 3,704,400,000 \text{ kg}

3) A concrete block weighing 2460kg has a volume of 1.2m³. What is the density of the block?

To calculate the answer here, we need to recall the formula:

\text{ density} = \text{ mass} \div \text{ volume}

In this question, the mass is 2460kg and the volume is 1.2m³, so we simply need to substitute these values into the formula accordingly, as follows:

\text{ density} = 2460 \text{ kg} \div 1.2 \text {m³} = 2050 \text{ kg/m³}

4) Metal A has a density of 5g/cm³ and metal B has a density of 3g/cm³.

1200g of metal A and 600g of metal B are melted and mixed and then recast into a block.

a) What is the volume of the block?

b) What is the density of the block?

a) In order to calculate the overall volume of the block, we need to add the volume of metal A and the volume of metal B. Although we don’t have the volume of either metal, we have been given their masses and densities, so we can calculate the volume of each metal accordingly.

By rearranging the density formula, or using the triangle, we can work out how to calculate the volume:

\text{density} = \text{ mass}\div \text{ volume}

so therefore:

\text{volume} = \text{ mass}\div \text{ density}

The volume of metal A can be calculated as follows:

1200\text{ grams} \div5\text{ g/cm³} = 240\text{ cm³}

The volume of metal B can be calculated as follows:

600\text{ grams} \div3\text{ g/cm³} = 200\text{ cm³}

Therefore, if metal A has a volume of 240cm³and metal B has a volume of 200cm³, then their combined volume is simply:

240\text{ cm³} + 200\text{ cm³} = 440\text{ cm³}

b) As we know from question a), the newly-formed block has a volume of 440cm³.

We know that the mass of metal A was 1200g and the mass of metal B was 600g, so mass of the block is:

1200\text{ g} + 600\text{ g} = 1800\text{ g}

The density of this block can be calculated by dividing the mass by the volume as follows:

1880\text{ g} \div 400\text{ cm³} = 4.7\text{ g/cm³}

5) Metal A has a density of 3.2g/cm³.

Metal B has a density of 5.5g/cm³.

Metal C is made from combining metal A and metal B in a ratio of 3 : 7. To one decimal place, what is the density of metal C if it has a mass of 2500g?

This is quite a challenging question with a lot of calculations going on. Like many maths problems, if you have no idea how you are going to arrive at the final solution, just ask yourself what you can work out from the information given. You never know where it will lead!

Since we have been given the mass of metal C and the ratio of metal A and metal B in metal C, we can therefore calculate the mass of metal A and metal B. If the ratio of metal A to metal B is 3 : 7, that means that \frac{3}{10} of the mass of metal C comes from metal A and the remaining \frac{7}{10} is metal B. (We are dealing in tenths here since the sum of the ratio is 10.)

The mass of metal A can be calculated as follows:

2500\text{ g} \times \dfrac{3}{10} = 750\text{ g}

The mass of metal B can be calculated as follows:

2500\text{ g} \times \dfrac{7}{10} = 1750\text{ g}

We now know both the mass and density of both metals A and B, meaning we can work out their respective volumes.

Since

\text{density} = \text{ mass} \div \text {volume}

then

\text{volume} = \text{ mass} \div \text {density}

The volume of metal A can be calculated as follows:

750\text{ g} \div 3.2\text{ g/cm³} = 234.375\text{ cm³}

The volume of metal A can be calculated as follows:

1750\text{ g} \div 5.5\text{ g/cm³} = 318.18\text{ cm³}

If metal A has a volume of 234.375cm³ and metal B has a volume of 318.18cm³, then their combined volume is the volume of metal C.

Volume of metal C = 234.375 + 318.18 = 552.5568….\text{ cm³}

We now know both the mass and volume of metal C, so we are now able to calculate its density.

Density of metal C = 2500\text { g} \div 552.5568\text{ cm³} = 4.5\text{ g/cm³}

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