A problem by Koustav Mondal
Level
pending
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?
18
10
15
30
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1 solution
(10x + 20y)/(x+y) = k
10x + 20y = kx + ky
20y - ky = kx - 10x
y(20 - k) = x(k - 10)
(20 - k)/(k - 10) = x/y
and since 0 < x < y, then 0 < x/y < 1, and it must be that 0 < (20 - k) / (k - 10) < 1. From the answer choices we can be sure k - 10 isn't negative, so we can multiply through this inequality by k-10 to find that 0 < 20 - k < k - 10, or that 15 < k < 20.