How many moles of hydrogen gas would be needed to react with excess carbon dioxide to produce 53.6 moles of water vapor?

You first need to write your chemical equation:

CO2 + 4 H2 → CH4 + 2 H2O (balanced equation)

4 moles of hydrogen gas will produce 2 moles of water vapor since carbon dioxide is in excess.

ratio is then 4 H2 / 2 H2O

If you produce 30.6 moles of water vapor, how many moles of hydrogen gas is needed:
let's solve with ratio 4 H2 / 2 H2O = x H2 / 30.6 H2O

Therefore you need twice as many moles of hydrogen gas: 61.2 moles of hydrogen gas is needed.

Hope it helps.