If x, y, and k are positive numbers such that (x/(x+y))(10)

[BTGmoderatorAT]

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?

A. 10
B. 12
C. 15
D. 18
E. 30

Is there a strategic approach to this question? Can any experts help?

[GMATGuruNY]

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?
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 B: 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.

[elias.latour.apex]

It's not necessary to do a lot of complicated math if you simply realize that for it to come out as an integer, x+y will probably need to be 5. If x+y = 3, 4, 6 or 7, you're not going to get integers out. 8 might work if x=2 and y=6, but we'll deal with that after checking 5.

Since x is less than y our choices are:

(1,4)
(2,3)

In the first case, we get (1/5)*10+(4/5)*20 = 2 + 16 = 18. Success. There's no need to check further.

Too many people see a problem and think "I know the formula for that." Then they fall into a default solution method (DSM) that is complicated, time consuming, and error prone.

[Scott@TargetTestPrep]

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?

A. 10
B. 12
C. 15
D. 18
E. 30

We are given:

(x/(x+y))(10) + (y/(x+y))(20) = k

[10x/(x+y)] + [20y/(x+y)] = k

We can combine the two fractions on the left side of the equation because they have the same denominator, (x + y).

[(10x + 20y)/(x+y)] = k

We see that we have a weighted average equation in which x items have an average of 10, and another y items have an average of 20 and a weighted average of k. In this case, the value of k must be between 10 and 20. However, since x is less than y, the weighted average (or k) must be closer to 20 than to 10. Thus k must be 18.

Alternate Solution:

(x/(x+y))(10) + (y/(x+y))(20) = k

[10x/(x+y)] + [20y/(x+y)] = k

We can combine the two fractions on the left side of the equation because they have the same denominator, (x + y).

[(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

Since y > x, x/y is less than 1. Since both x and y are positive, x/y is also positive. Trying each answer choice for k, we see that k = 10, 12 and 15 produce a value of x/y that is greater than or equal to 1 while k = 30 produces a negative value of x/y. k = 18 is the only value that results in a positive proper fraction.

Answer: D

[[email protected]]

Hi ardz24,

We're told that X, Y and K are POSITIVE and that X < Y. We're also told that 10X/(X+Y) + 20Y/(X+Y) = K. We're asked what COULD be the value of K, which means that there's more than one possible answer. Since the answers are NUMBERS, one of the them MUST be a possible answer. You can take advantage of the answer choices and some math "logic" to get to the correct answer. Let's TEST THE ANSWERS....

We can manipulate the given equation into:

10X + 20Y = K(X+Y)

Since all of the variables are POSITIVE, K MUST be between 10 and 20. Here's the proof:

IF.....K=10, then the equation becomes...
10X + 20Y = 10X + 10Y
20Y = 10Y
Since Y is positive, 20Y = 10Y is NOT possible.
Eliminate Answer A

In that same way, K can't be 20 (or greater) because the end equation would be an impossibility.

With the remaining 3 answers, we can TEST the possibilities...

IF...K = 12, then the equation becomes...
10X + 20Y = 12X + 12Y
8Y = 2X
4Y = X
In this scenario, X > Y which is the OPPOSITE of what we were told. This is NOT the answer.
Eliminate B.

IF....K=15, then the equation becomes...
10X + 20Y = 15X + 15Y
5Y = 5X
Y = X
In this scenario, X = Y, which is NOT a match for what we were told. This is NOT the answer.
Eliminate C.

IF....K=18, then the equation becomes....
10X + 20Y = 18X + 18Y
2Y = 8X
Y = 4X
Here, X < Y, which IS a match for what we were told. This IS a POSSIBLE answer.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich