If x, y and k are positive numbers such that 10*x/(x+y)+20*y/(x+y)=k and if x<y, which of the following could be the value of k?
A. 10
B. 12
C. 15
D. 18
E. 30
Putting the sum over a common denominator, we get:
(10x + 20y) / (x+y) = k.
Let x = the number of $10 shirts purchased at a certain store.
Let y = the number of $20 shirts purchased at a certain store.
Total cost of the $10 shirts = 10x.
Total cost of the $20 shirts = 20y.
Total number of shirts purchased = x+y.
Thus, the AVERAGE cost per shirt is equal to the following:
(10x + 20y) / (x+y).
In the problem above, the value of k is equal to the AVERAGE cost per shirt.
Since each shirt costs either $10 or $20, the average cost per shirt must be BETWEEN 10 and 20.
Since y>x, the number of $20 shirts purchased is GREATER than the number of $10 shirts purchased, with the result that the average cost per shirt must be CLOSER TO 20 than to 10.
Of the answer choices, the only viable option is k=18.
The correct answer is
D.
An alternate approach is to plug in the answers, which represent the value of k.
Let x=1.
When we plug in the correct answer choice for k, x < y.
Answer choice C: k=15
(10*1 + 20y)/(1+y) = 15
10 + 20y = 15 + 15y
5y = 5
y = 1.
Since x=y, eliminate C.
Answer choice D: k=18
(10*1 + 20y)/(1+y) = 18
10 + 20y = 18 + 18y
2y = 8
y = 4.
Since x < y, success!
The correct answer is
D.
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