Bunuel wrote:
x, y and z satisfy \(\frac{xy}{x+y} = \frac{1}{3}, \frac{yz}{y+z} = \frac{1}{5}\), and \(\frac{xz}{x+z} = \frac{1}{6}\). What is \(x + y + z\)?
A. \(\frac{7}{4}\)
B. \(\frac{5}{4}\)
C. \(\frac{3}{4}\)
D. \(\frac{7}{5}\)
E. \(\frac{3}{5}\)
I know I can do the algebra, but it sure looks like it's going to be somewhere between annoying and frustrating, and it's probably going to take longer than I'd like to devote to a single question unless necessary. If I think like a test-writer, I might be able to get to the correct answer without that damage to my brain and the clock.
Look at the answer choices. None is greater than 2. It's possible that one of the variables will be negative, but it seems unlikely. If all three are positive, let's see what happens if we make one of them equal to 1.
If x = 1:
From the xy equation, we'd have y = 0.5. From the xz equation, we'd have z = 0.2. Plugging those into the yz equation yields 1/7, not 1/5, so x can't be 1.
If y = 1:
From the xy equation, we'd have x = 0.5. From the yz equation, we'd have z = 0.25. Plugging those into the xz equation yields 1/6, which is exactly what we want. Well, wouldn't you know...by thinking about how GMAT questions are built, we avoided the algebra and got the answer pretty quickly.
x + y + z = 0.5 + 1 + 0.25 = 1.75 = 7/4
Answer choice A.