## Blog

### The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a GMAT question on our blog.  Today, they have followed up with the answer:

The fastest way to solve this problem is to pick smart numbers for the two distances, so that you’ll get integer numbers of gallons. The distance from A to B should be a multiple of 12, and the distance from B to C should be a multiple of 18. With a little trial and error, we can find suitable numbers.

A-B: 72 miles — the car burns 6 gallons.
B-C: 36 miles — the car burns 2 gallons.

In total, the car goes 108 miles on 8 gallons, so the average fuel efficiency is 108/8 =13.5 miles per gallon.

You can check a different set of numbers:
A-B: 36 miles — the car burns 3 gallons.
B-C: 18 miles — the car burns 1 gallon.

In total, the car goes 54 miles on 4 gallons, so again, the average is 13.5 miles per gallon.

This value may not be what you expected: maybe you though that the answer would be 14, which is a weighted average of 12 and 18, weighted 2:1 toward the 12. However, that weighting is faulty. If you want to weight two ratios or rates (such as miles PER gallons), then you must weight by the denominator (gallons), NOT by the numerator (miles). As we saw by picking numbers, the gallons used on each stage of the trip wind up in a 3:1 ratio, and the weighted average of 12 and 18 (weighted 3:1) is 13.5.

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