How many moles of methane are produced when 82.6 moles of carbon dioxide gas react with excess hydrogen gas?

First you need a balanced chemical reaction. Always write out the equation and balance it for problems like this.

CO2 + 4H2 - > CH4 + 2H2O

You want to find the moles of CH4 given 82.6 moles of CO2.

Take your given moles of CO2 and multiply it by the stoichiometric ratio of CO2 and CH4.

The stoichiometric ratio can be obtained by using the coefficients for CO2 and CH4.

The ratio is: 1 mole of CH4 over 1 mole of CO2.

82.6 moles of CO2 times 1 mole of CH4 over 1 mole of CO2.

Units cancel out giving you 82.6 moles of CH4.