Question

In a laboratory, you determine that the density of a certain solid is 5.23×10−6kg/mm3. Convert this density into kilograms per cubic meters.

Answer 1

To do this, you must know the conversion factor between meters and millimeters.
1 m = 1000 mm
Divide the above equation by 1 m, and you get:

\(\dfrac{1~m}{1~m} = \dfrac{1000~mm}{1~m}\)


or

\(\dfrac{1~m}{1000~mm} = 1\)

Since 1000 mm over 1 m equals 1, you can multiply any number by it without changing its value, just like you can multiply any number by 1 without changing its value.

Now look at your problem:

\(5.23 \times 10^{-6} \dfrac{kg}{mm^3} \)

\(=5.23 \times 10^{-6} \dfrac{kg}{mm^3} \times \dfrac{1000~mm}{1~m} \times \dfrac{1000~mm}{1~m} \times \dfrac{1000~mm}{1~m}\)

\(=5.23 \times 10^{-6} \dfrac{kg}{mm^3} \times \dfrac{(1000~mm)^3}{(1~m)^3} \)

\(=5.23 \times 10^{-6} \dfrac{kg}{mm^3} \times \dfrac{(10^3~mm)^3}{1~m^3} \)

\(=5.23 \times 10^{-6} \dfrac{kg}{\cancel{mm^3}} \times \dfrac{10^9~\cancel{mm^3}}{1~m^3} \)

\(=5.23 \times 10^{-6} \times 10^9 \dfrac{kg}{m^3} \)

\(=5.23 \times 10^3 \dfrac{kg}{m^3} \)