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OG12 PS148

by alex.gellatly » Mon Sep 03, 2012 8:28 pm
OK I know this question has been posted several times on this site... but I still can't seem to understand the best approach for solving it.

If x, y, and k are positive numbers such that (x/x+y)(10)+(y/x+y)(20)=k and if x<y, which of the following could be the value of k?

10
12
15
18
30

Much thanks in advanced
A useful website I found that has every quant OG video explanation:

https://www.beatthegmat.com/useful-websi ... tml#475231
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by neelgandham » Mon Sep 03, 2012 9:48 pm
Alex - Here it goes.

(x/x+y)(10) + (y/x+y)(20) = k
(x/x+y)(10) + (y/x+y)(10) + (y/x+y)(10) = k - Splitting (y/x+y)(20) into (y/x+y)(10) + (y/x+y)(10).
[(x+y)/(x+y)]*(10) + (y/x+y)(10) = k - Adding (x/x+y)(10) + (y/x+y)(10), we get [(x+y)/(x+y)]*(10).
10 + (y/x+y)(10) = k - Dividing numerator and denominator with (x+y), [(x+y)/(x+y)]*(10) = 10.
A)10
10 + (y/x+y)(10) = k = 10, then
10 + (y/x+y)(10) = 10
(y/(x+y))(10) = 0
y = 0. BUT y is a positive number (From the question). So, answer choice A is incorrect.
B)12
10 + (y/x+y)(10) = k = 12, then
(y/(x+y))*(10) = 2
10*y = 2*(x+y)
10y = 2x + 2y
8y = 2x
x = 4y. Since x and y are positive numbers and x = 4y, x is greater than y. BUT x<y(from the question).
So, answer choice B is incorrect.
C)15

10 + (y/x+y)(10) = k = 15, then
(y/(x+y))*(10) = 5
10*y = 5*(x+y)
10y = 5x + 5y
5y = 5x
y = x.
BUT x<y(From the question). So, answer choice C is incorrect.
D)18

10 + (y/x+y)(10) = k = 18, then
(y/(x+y))*(10) = 8
10*y = 8*(x+y)
10y = 8x + 8y
2y = 8x
y = 4x. x<y(From the question). So, answer choice D is correct.
E)30
10 + (y/x+y)(10) = k = 30, then
(y/(x+y))*(10) = 20
10*y = 20*(x+y)
10y = 20x + 20y
-10y = 20x
y = -2x.
If x is positive, then y is negative and If x is negative, then y is positive. BUT x, y, and k are positive numbers(From the question). So, answer choice D is incorrect.
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by Anurag@Gurome » Tue Sep 04, 2012 12:08 am
alex.gellatly wrote:OK I know this question has been posted several times on this site... but I still can't seem to understand the best approach for solving it.

If x, y, and k are positive numbers such that (x/x+y)(10)+(y/x+y)(20)=k and if x<y, which of the following could be the value of k?

10
12
15
18
30

Much thanks in advanced
We have 10x/(x+y) + 10y/(x+y) + 10y/(x+y) = k.
10 + 10y/(x+y) = k.
Now it is given that x < y.
x + y < 2y.
y/(x + y) > 1/2.
10y/(x + y) > 5.
10 + 10y/(x + y) > 15.
Also since both x and y are positive, y/(x+y) < 1.
10 + 10y/(x + y) < 20.
Therefore 15 < k < 20.
The only possible value of k is 18.

The correct answer is D.
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