Stoichiometry of incomplete combustion
Complicated little rascal!
What we know: 
1) The N2 is inert. 
2) The oxygen is in excess. 
3) 90% of the 7.8 mole % CH4 reacts; this is 7.02 mole % CH4. 
4) This gives 8 moles CO2 per mole of CO, or 88.89% CO2 plus 11.11% CO. 

The resultant output contains CO2: 7.02 mole % of the original CH4 x 88.89% = 6.24 mole % CO2.

The resultant output contains CO: 7.02 mole % of the original CH4 x 11.11% = 0.78 mole % CO.

The resultant output also contains 10% unreacted CH4. The 90% conversion refers to the starting material, not to the products. (Yield % would refer to one product.)

So the stoichiometry is 80% of the methane goes to CO2, 10% goes to CO, and 10% is unreacted.