Why is kg/m³ to g/cm³1 to 1000?

Question

I understand that changing the divisor multiplies the result by that, but why doesn't changing the numerator cancel that out? I found out somewhere else since posting, is there a way to delete this?

Answer 1

There are 100 cm per m, so there are 100^3 = 1000000 cc per cubic metre, not 1000.

Answer 2

Here is a simple manipulation$$\mathrm{\frac{kg}{m^3}=\frac{1000~grams}{100\times100\times100~cm^3}=\frac{1}{1000}g/cm^3}$$

Answer 3

I always like to think of it this way. We know that $1m = 100 cm$ and $1kg = 1000g$. Then we just replace: $$ \mathrm{\frac{1kg}{1m^3} = \frac{(1kg)}{(1m)^3} = \frac{1000g}{(100cm)^3} = \frac{1000g}{100^3 cm^3} = \frac{1000g}{1,000,000 cm^3} = \frac{1}{1000} \frac{1g}{1cm^3}} $$

Answer 4

Imagine a one-meter solid cube made of some uniform material. That is, you have a block of this material in the shape of a cube, $1$ meter long along each edge.

Now imagine cutting this large cube into one-centimeter cubes. First you cut it into $100$ slabs $1$ meter square and $1$ cm thick. Next you draw a $100\times 100$ grid of $1$-centimeter squares on each slab and cut the slab in $100\times100=10000$ cubes. There were $100$ slabs so now you have $100\times10000=1000000$ cubes each $1$ cm on each edge.

Suppose the material has density $1\ \mathrm g/\mathrm{cm}^3$ so each little cube weighs $1$ gram. How many kilograms does the pile of $1000000$ cubes weigh? That’s the weight of the original cube. How many kilograms per cubic meter is that?