Data Sufficiency

Hello everyone,

So for the problem below, the detailed solution literally takes a whole page in the GMAT book.
Is there a fast way to answer this question? if so, how did you come up with it?

I dont know if the solution in the GMAT book can be found in < 2 mins

Question is below (i also attached a photo of it):

If x and y are positive integers, what is the value of √x + √y (radical of x + radical of y)
(1) x + y = 15
(2) √xy = 6 (radical of (x times y) = 6

LGmat wrote:If x and y are positive integers, what is the value of √x + √y (radical of x + radical of y)
(1) x + y = 15
(2) √xy = 6 (radical of (x times y)) = 6

GMAT quant is not really a math test but rather a test of hacking skills and vision.

With that in mind, here's one way to get to the answer to that question.

For Statement 1 I would plug in some numbers to confirm what immediately seems to be the case, that there are multiple possibilities.

11 + 4 = 15 So √x + √y = approximately 3.xx + 2 = 5.xx (something greater than 5)

1 + 14 = 15 So √x + √y = approximately 1 + 3.xx = 4.xx (something less than 5)

So, yes, there are multiple possible answers and Statement 1 is insufficient.

For Statement 2 we could do something similar.

We could use x = 1 and y = 36. So √x + √y = 1 + 6 = 7

Or we could use x = 4 and y = 9. So √x + √y = 2 + 3 = 5

So Statement 2 is insufficient.

Now comes the key part. We have to determine whether we can find the answer by combining the statements, and to do that you have to see something specific. What you need to see is the following.

(√x + √y)² = x + 2√xy + y

That's the same as (x + y) + (2√xy).

We know from Statement 1 that x + y = 15, and from Statement 2 that √xy = 6.

So (x + y) + (2√xy) = 15 + 12 = 27, and we have our answer. Since (√x + √y)² = 27, √x + √y = √27.

You actually don't need to calculate all the way to √27, by the way. Once you notice that (√x + √y)² = x + 2√xy + y, and realize that the two statements give you the values of x + y and 2√xy, you are done.

So combined the statements provide sufficient information and the correct answer is C.

That explanation may seem long, but it's full of explanatory stuff, and actually, getting the answer takes just quickly plugging in some numbers or otherwise determining that neither statement is sufficient alone and then seeing that (√x + √y)² = x + 2√xy + y.

Marty Murray wrote: That explanation may seem long, but it's full of explanatory stuff, and actually, getting the answer takes just quickly plugging in some numbers or otherwise determining that neither statement is sufficient alone and then seeing that (√x + √y)² = x + 2√xy + y.

Thanks Marty, I now understand!

LGmat wrote:If x and y are positive integers, what is the value of √x + √y?
(1) x + y = 15
(2) √xy = 6

Statement 1: x+y = 15
If x=1 and y=14, then √x + √y = √1 + √14.
If x=2 and y=13, then √x + √y = √2 + √13.
Since √x + √y can be different values, INSUFFICIENT.

Statement 2: √xy = 6
Since x and y are POSITIVE, we can safely square both sides:
(√xy)² = 6²
xy = 36.
If x=1 and y=36, then √x + √y = √1 + √36 = 7.
If x=2 and y=18, then √x + √y = √2 + √18.
Since √x + √y can be different values, INSUFFFICIENT.

Statements combined: xy = 36 and x+y = 15.
Thus, x and y must be positive integers with a PRODUCT OF 36 and a SUM OF 15.
Factor pairs of 36:
1*36
2*18
3*12
4*9
6*6.
Only the option in red yields a sum of 15.
If x=3 and y=12, then √x + √y = √3 + √12.
If x=12 and y=3, then √x + √y = √12 + √3.
Since the value of √x + √y is THE SAME in each case, SUFFICIENT.

The correct answer is C.

0 + 15 = 15 So √x + √y = approximately 0 + 3.xx = 3.xx (something less than 4)

Careful, Marty.
Since x and y must be POSITIVE INTEGERS, this case is not valid.

Ok, I changed the numbers so that they are now all positive integers.

Funny thing, maybe I did what's a typical GMAT quant mistake generating thing to do. I am not sure, but maybe I was so happy about noticing that I could use (√x + √y)² = x + 2√xy + y to find the answer, that I distracted myself from keeping in mind a key detail, that x and y have to be positive integers.

Also, this question freaked me out a little, in that partly since I barely had the positive integer detail on my radar screen, I didn't see the hack Mitch outlined and I got the impression that a test taker would be stuck unless he or she saw the quadratic factors approach that I used.

I guess there is always more than one way though, and, LGmat, there you have it. This question is totally hackable via multiple paths.

LGmat wrote:
If x and y are positive integers, what is the value of √x + √y (radical of x + radical of y)
(1) x + y = 15
(2) √xy = 6 (radical of (x times y) = 6

We know that
x + y + 2√x√y = (√x + √y)^2

Statement 1 gives us (x + y)
and Statement 2 gives us √xy

So, we need both the statements to find the value of √x + √y

Hence the answer is C.

Please note that we need to solve the whole question. We just need to find that the data given is sufficient or not.