Algebra question (men and women in choir)

[aishwaryasingh0311]

If there are x men and y women in a choir, and there are z more men than there are women in that choir, what is z?

(1) x2 - 2xy + y2 - 9 = 0

(2) x2 + 2xy + y2 - 225 = 0

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.


Now the ans is a)statement one is sufficient

Can you please explain me how is the second equation not sufficient?

In the explanation it is written that when we solve the second equation it could be x+y=15 or -15
But it is very obvious that the no. of men and women in a choir cannot be-(minus)
So according to that we have to chose a positive no. only, which is 15 and which qualifies the second option.

Please help...

[campb441]

First you should make the equation when comparing men and women in the choir.

x = men
y = women
and there are z more men than women
Therefore:

x = y + z or you can rewrite this as x - z = y

1) Sufficient

x^2 - 2xy + y^2 - 9 = 0
x^2 - 2xy + y^2 = 9 additive inverse
(x - y)(x - y) = 9
(x - y)^2 = 9 this creates a perfect square
x - y = 3 square root each side
x -(x - z) = 3 substitute for y from the equation at the top
z = 3 solve for z

2) NOT sufficient

Start by repeating the same steps but you'll see after you substitute for y you won't cancel out your x variables like you did above. This is all because of the + 2xy in this problem instead of the - 2xy like the problem above.

x^2 + 2xy + y^2 - 225 = 0
x^2 + 2xy + y^2 = 225 additive inverse
(x + y)(x + y) = 225
(x + y)^2 = 225 perfect square
(x + y) = 15 square root each side
x + (x - z) = 15 substitute for y
2x - z = 15 ... not solvable

[jnelson0612]

Wow, very nice! Thanks campb!

[thulsy]

I have a question regarding this problem: Why can't z be negative?
i.e. why can't we say "there are -3 more men than there are women in that choir"? I thought that was equivalent to "there are 3 more women than there are men in that choir". Just as we can say that "the population increase is -2.5%" or "the population increases by -2.5%".
I mean, here the figure can be negative, meaning the population actually decreases. So here I suppose that a negative integer (-3) associated with "more" is supposed to mean the corresponding positive integer (3) associated with "less". I'm not a native speaker of English, and it seems like I'm missing here. Please kindly explain. Thanks.

[CurtisVincentBorns]

I solved the problem pretty similar to campb but it seems more simplistic to me. Please correct me if I'm wrong...

The question states that "z more men than there are women in that choir"

So if:

x=men
y=women
and there are z more men than woman.

The equation be written y+z=x
which then can be rewritten as z=x-y

So then the rephrasing of the question should be "what is x-y?"

1) sufficient
x^2 - 2xy + y^2 - 9 = 0
x^2 - 2xy + y^2 = 9 additive inverse
(x - y)(x - y) = 9
(x - y)^2 = 9 this creates a perfect square
x - y = 3 square root each side

***can we not just stop here?
we now have x-y = 3 = z

2) NOT sufficient

x^2 + 2xy + y^2 - 225 = 0
x^2 + 2xy + y^2 = 225 additive inverse
(x + y)(x + y) = 225
(x + y)^2 = 225 perfect square
(x + y) = 15 square root each side

***this gives us x+y, which is not what we were looking for

[tim]

thulsy, i agree with you. the problem should stipulate that z is a positive integer..

[amw1984]

Is there a hint or something which might help me know to move the 9 and 225 to the other side of the equation?

[RonPurewal]

Is there a hint or something which might help me know to move the 9 and 225 to the other side of the equation?

in terms of "hints", you can notice that 9 and 225 are perfect squares.
if you also happen to recognize these two factoring patterns...
x^2 + 2xy + y^2 = (x + y)^2
x^2 - 2xy + y^2 = (x - y)^2
... then you're good to go.

also, strictly speaking, you don't have to move those terms to the other side.
you could also factor
x^2 + 2xy + y^2 - 225
= (x + y)^2 - 225
= (x + y)^2 - 15^2
= (x + y + 15)(x + y - 15)
and likewise with the other one.

[RonPurewal]

I have a question regarding this problem: Why can't z be negative?
i.e. why can't we say "there are -3 more men than there are women in that choir"? I thought that was equivalent to "there are 3 more women than there are men in that choir". Just as we can say that "the population increase is -2.5%" or "the population increases by -2.5%".

one VERY important thing to realize here is that you should never, ever, EVER entertain thoughts like this one.

the gmat will NEVER use "tricky" phrasing in math problems.
if you are thinking of a "tricky" interpretation of some statement, STOP thinking about that interpretation right now.

i.e., if it says "x is y more than z", then, yes, ladies and gentlemen, x is actually more than z. (i.e., you don't have to think about the cases in which y is negative.)
if it says "x increases by p%", then, yes, x actually increases. i.e., p won't be negative.
etc.

usually, they will go out of their way to specify these things (almost to the extent of being annoying with their over-specificity). but, even if they don't, always go with the common-sense interpretation of the words.

[RonPurewal]

Many other procedures can easily apply to solve the problem.

Please try to post things that add value to the discussion. Thanks.