How many liters of hydrogen, H2, are needed to react with 10 liters of nitrogen gas in the reaction forming ammonia?
You are on the right track. Dimensional Analysis is your friend:

$$\frac{10\,\text{L  $\ce{N2}$}}{1} \cdot \frac{1\,\text{mol  $\ce{N2}$}}{22.4 \,\text{L  $\ce{N2}$}}\cdot \frac{3\,\text{mol  $\ce{H2}$}}{1\,\text {mol  $\ce{N2}$}}\cdot \frac{22.4\,\text{L  $\ce{H2}$}}{1\,\text {mol  $\ce{H2}$}}=\,?\,\, \text{L $\ce{H2}$}$$

If you do this, you will realize that the quantities cancel exactly and give you a nice clean number.