Hey Sandeep,
Good questions - let me tackle that first one, which I think is a pretty good example of GMAT strategy. I did this in less than two minutes using this strategy:
1) Simplify the algebra to get in a position to better solve for k:
x+y is a common denominator on the left, so you really have:
(10x + 20y)/(x+y) = k
Multiply out the denominator to get:
10x + 20y = kx + ky
Now, there are three variables and only one equation, so you can't really solve for k algebraically, so I:
2) Plug in easy-to use numbers to see if you can find a range of values (since they ask "what could be" and not "what is" the value of k, pinning k into a range is probably the best you can do):
Since x < y, let's call them 1 and 2:
10(1) + 20(2) = 1k + 2k
50 = 3k
k = 17.3333
Doing another one, let's try x = 2 and y = 5 to see if different numbers give us something wildly different:
10(2) + 20(5) = 2k + 5k
120 = 7k
k = something just over 17
Since both of our values are in that just-over-17 range, choice D, 18, is looking like a pretty good bet. Maybe try 1-2 more to see if you can get lower than 17 (because 15 looms somewhat close):
Call x 2 and y 3:
10(2) + 20(3) = 2k + 3k
80 = 5k
k = 20
Well, it seems like we can definitely get above the 17 range as high as 20, and now our range of potential values is 17-20, so I'm ready to call it and say that 18 is possible. If we use nonintegers - and we're allowed to...the question doesn't prohibit it - we would be able to tweak our current estimates to get to 18, but with any combinations we've tried already we haven't been able to near 15. The correct answer has to be 18.
A few strategic takeaways here:
1) Cleaning up the algebra is helpful - in most questions that feature variables in a denominator, your goal at some point will be to multiply out that denominator.
2) Because they ask "which COULD BE the value of k", you don't have to solve directly for it if you can use easy-to-plug numbers to establish a range of potential values. You don't need to know exactly what k IS, but rather just what k might look like. Since this question doesn't involve integers specifically, it's not a divisibility problem (so you don't have to worry that 18 might not share necessary factors), so an estimate is probably the most efficient way to tackle this one.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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