# Question 7f402

Sep 10, 2015

The ratio between mass and volume or between volume and mass.

#### Explanation:

Density is essentially the ratio between the mass of a substance and the volume it occupies.

This automatically implies that density is usually given per unit of volume, since, for example, it's easier to say that a substance has a density of

$\rho = 10 \text{g"/"mL"" }$ than $\text{ "rho = "20 g"/"2 mL}$

even though those two ratios are equivalent.

So, let's say that you have a substance that has a density of $10 \text{g"/"mL}$. If you start with a mass of $\text{20 g}$, then you can find out what volume it occupies by using

20color(red)(cancel(color(black)("g"))) * "1 mL"/(10color(red)(cancel(color(black)("g")))) = "2 mL"

Notice that the conversion factor uses the fact that you have $\text{10 g}$ for every $\text{1 mL}$ of substance, which is another way of saying that you get $\text{1 mL}$ for every $\text{10 g}$ of substance.

Likewise, if you start with a volume of $\text{50 mL}$, you can find out its mass by using

50color(red)(cancel(color(black)("mL"))) * "10 g"/(1color(red)(cancel(color(black)("mL")))) = "500 g"

If you have different units for mass and/or volume, then other conversion factors will be needed as well.

For example, let's say that I have $\text{15 kg}$ of the substance, and I want to find out what volume it occupies. I have to first convert kilograms to grams by using

15color(red)(cancel(color(black)("kg"))) * "1000g"/(1color(red)(cancel(color(black)("kg")))) = "1500g"

Now I can find out the volume

1500color(red)(cancel(color(black)("g"))) * "1 mL"/(10color(red)(cancel(color(black)("g")))) = "150 mL"#

As a conclusion, always look for the units to match and be prepared to use the ratio between the mass and the volume, which is density, or the ratio between the volume and the mass, when needed.