Convert 4.7 mg into kg.

Step-by-step solution

1. 1

   Write down the two prefix relationships separately. The prefix milli means one thousandth and the prefix kilo means one thousand:

   $1\ \mathrm{mg}=10^{-3}\ \mathrm{g},\qquad 1\ \mathrm{g}=10^{-3}\ \mathrm{kg}$

   Going from milligrams to kilograms crosses two prefix steps, which is why a single factor of $1000$ is not enough.

2. 2

   Chain them into one conversion factor. Multiplying the two relationships:

   $1\ \mathrm{mg}=10^{-3}\times 10^{-3}\ \mathrm{kg}=10^{-6}\ \mathrm{kg}$

   So one milligram is a millionth of a kilogram.

3. 3

   Set up dimensional analysis so the units cancel visibly. Multiply the given quantity by the factor written as a fraction with mg on the bottom:

   $4.7\ \mathrm{mg}\times\frac{10^{-6}\ \mathrm{kg}}{1\ \mathrm{mg}}$

   Writing it this way is a built-in check: if the unit you want does not survive the cancellation, the factor was upside down.

4. 4

   Cancel and evaluate. The mg units divide out, leaving

   $4.7\times 10^{-6}\ \mathrm{kg}=0.0000047\ \mathrm{kg}$

5. 5

   Sanity-check the direction of the change. A kilogram is a much bigger unit than a milligram, so the same mass must be a much smaller number of kilograms — and $4.7\times 10^{-6}$ is indeed far smaller than $4.7$. If you had got $4.7\times 10^{6}$, the conversion factor was inverted.