This problem is tough. The only good way to solve this problem is to use a fraction.

First balance the carbon atoms by multiplying CO₂ by 2.

C₂H₆   +   O₂   ®    2 CO₂   +   H₂O

Next multiply the water by 3 to balance the hydrogen atoms.

C₂H₆   +    O₂   ®    2 CO₂   +   3 H₂O

Now you have 7 oxygen atoms on the right and 2 on the left. If you multiply the O₂ by the fraction 7/2, that will give you the required 7 oxygen atoms since O₂ is divided by 2.

C₂H₆   +   7/2 O₂   ®    2 CO₂   +   3 H₂O

Unfortunately you are not supposed to use fractions in balanced chemical equations and in order to get rid of the 2 in the denominator of the 7/2, you must multiply everything by 2 and then you are finally done...whew!

2 C₂H₆   +   7 O₂   ®    4 CO₂   +   6 H₂O