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3 positive numbers a, b and c satisfy a/2b-c=2b/3a+c=a/b . What is a/b?

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by Max@Math Revolution » Tue Mar 03, 2020 1:19 am

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[GMAT math practice question] 3.3

3 positive numbers a, b and c satisfy a/2b-c=2b/3a+c=a/b . What is a/b?

A. 1/3
B. 2/3
C. 1
D. 4/3
E. 5/3
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Re: 3 positive numbers a, b and c satisfy a/2b-c=2b/3a+c=a/b . What is a/b?

by Max@Math Revolution » Thu Mar 05, 2020 2:18 am
=>

Assume a/2b-c=2b/3a+c=a/b=k.

Then we have a = k(2b - c), 2b = k(3a + c).
When we add those equations, we have
a + 2b =k(2b-c) + k(3a+c)
a + 2b = 2bk – ck + 3ak + ck
a + 2b = 2bk + 3ak
a + 2b = k(2b + 3a)
or a/b + 2 = k(3a/b + 2) (dividing by b).
Since (a/b) = k , we have
k + 2 = k(3k + 2)
k + 2 = 3k2 + 2k
3k^2 + 2k – k - 2 = 0
3k^2 + k - 2 = 0
or (k + 1)(3k - 2) = 0.
Then k = -1 or k = 2/3
Then k = a/b = 2/3 since a and b are positive numbers.

Therefore, B is the answer.
Answer: B
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