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x, y and z satisfy xy/x+y = 1/3, yz/y+z = 1/5, and xz/x+z =

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x, y and z satisfy xy/x+y = 1/3, yz/y+z = 1/5, and xz/x+z =

by Max@Math Revolution » Tue Dec 17, 2019 11:40 pm
[GMAT math practice question]

x, y and z satisfy xy/x+y = 1/3, yz/y+z = 1/5, and xz/x+z = 1/6. What is x + y + z?

A. 7/4
B. 5/4
C. 3/4
D. 7/5
E. 3/5
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by Max@Math Revolution » Thu Dec 19, 2019 10:46 pm
=>

Since (xy)/(x+y) = 1/3, we have (x + y) / xy = 3, x/xy + y/xy = 3, or 1/x + 1/y = 3.
Since (yz)/(y+z) = 1/5, we have (y + z) / yz = 5, y/yz + z/yz = 5, or 1/y + 1/z = 5.
Since (xz)/(x+z) = 1/6, we have (x + z) / xz = 6, x/xz + z/xz = 6 or 1/z + 1/x = 6.

If we add 3 equations, then we get
1/x + 1/y + 1/y + 1/z + 1/z + 1/x = 3 + 5 + 6
2/x + 2/y + 2/z = 14
2(1/x + 1/y + 1/z) = 14
Then, we have 1/x + 1/y + 1/z = 7, by dividing both sides by 2.

So, we have (1/x + 1/y + 1/z) - (1/y + 1/z) = 7 - 5, 1/x + 1/y + 1/z - 1/y - 1/z = 2, 1/x = 2, 1 = 2x, or x = 1/2.

We have (1/x + 1/y + 1/z) - (1/x + 1/z) = 7 - 6, 1/x + 1/y + 1/z - 1/x - 1/z = 1, 1/y = 1, 1 = 1y, or y = 1.

We have (1/x + 1/y + 1/z) - (1/x + 1/y) = 7 - 3, 1/x + 1/y + 1/z - 1/x - 1/y = 4, 1/z = 4, 1 = 4z, or z = 1/4.

Then, we have 1/x + 1/y + 1/z = 1/2 + 1 + 1/4 = 2/4 + 4/4 + 1/4 = 7/4.

Therefore, A is the answer.
Answer: A
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