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This problem trains for: SAT-I, AMC-8, GMAT.

Two thirds of my yearly mileage has been used to go to work and return home. However, on the way back, I always stop for shopping, making my return trip longer by 15% than the trip to work. What fraction of my yearly mileage is allocated for shopping?

Denote the total mileage by x. Denote the distance from home to work by y. The distance from work to home is:

The mileage used to go and return from work can be written in two ways: as a fraction of the total mileage and as depending on y:

Solve for y:

The mileage used to make the detour for shopping is:

and it can be expressed as a fraction of the total mileage:

Notice how the computations have been optimized by not computing all the intermediate results, but leaving them until the end result, when more simplifications were available.