Convert 4.7 grams into nanograms.

Step-by-step solution

1. 1

   Recall what the prefix nano- means. In SI, nano- is the factor $10^{-9}$, so $1\ \text{ng}=10^{-9}\ \text{g}\qquad\Longleftrightarrow\qquad 1\ \text{g}=10^{9}\ \text{ng}=1{,}000{,}000{,}000\ \text{ng}.$

2. 2

   Decide whether to multiply or divide. A nanogram is much smaller than a gram, so any given mass contains a very large number of nanograms - the numerical value must go up. That means multiplying by $10^{9}$, not dividing. Getting this direction backwards would give $4.7\times 10^{-9}$, which is off by eighteen orders of magnitude.

3. 3

   Set up the conversion as a fraction whose units cancel. $4.7\ \text{g}\times\frac{10^{9}\ \text{ng}}{1\ \text{g}}$ The grams cancel top and bottom, leaving nanograms - a reliable way to confirm the direction before touching the arithmetic.

4. 4

   Do the multiplication. $4.7\times 10^{9}=4{,}700{,}000{,}000\ \text{ng}.$ In scientific notation this is $4.7\times 10^{9}\ \text{ng}$; the coefficient stays between $1$ and $10$, so no renormalising is needed.

5. 5

   Sanity-check by chaining through a familiar unit. $4.7\ \text{g}=4700\ \text{mg}$ (milli- is $10^{-3}$), then $4700\ \text{mg}=4{,}700{,}000\ \mu\text{g}$ (micro- is another $10^{-3}$), then $4{,}700{,}000\ \mu\text{g}=4{,}700{,}000{,}000\ \text{ng}$ (nano- is a third $10^{-3}$). Three factors of $10^{3}$ make $10^{9}$, matching the direct route.