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GMATPrep Question (Test 1)

by Chick » Thu Dec 17, 2009 11:32 pm
Of the 75 houses in a certain community, 48 have a patio. How many of the houses in the community have a swimming pool?

(1) 38 of the houses in the community have a patio but do not have a swimming pool.

(2) The number of houses in the community that have a patio and a swimming pool is equal to the number of houses in the community that have neither a swimming pool nor a patio.

Answer: B

Can someone explain why statement 2 is sufficient on its own? I am getting "C" as the answer - need both stems to answer how many houses have a swimming pool....
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by papgust » Thu Dec 17, 2009 11:51 pm
Take a look at the image. I set up a double matrix.
From statement II, we have 2 x's in the matrix and let y be the question.
Now, swimming pool with NO patio will be (27-x). This means that y = x + (27-x) = 27 (Answers the question). Hence its B.
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by Chick » Fri Dec 18, 2009 12:59 am
Thanks a lot for your well-illustrated explanation!! That makes sense. I use the matrix as well to solve these questions. This is a high level one but at first glance, it looked as though the answer could not be determined without a value for x. I should have written out the equation as you did. Thanks again!
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by Testluv » Fri Dec 18, 2009 1:11 am
Hi Chick,

We can also use the counting equation for two overlapping sets:

True number of objects = (Number in group 1) + (Number in group 2) - (Number in Both) + (Number in Neither group)

We subtract objects that appear in both groups because those objects have been counted twice: once in the group 1 count, and then again in the group 2 count. But, if we were concerned with finding (ie, counting) the true number of objects, we wouldn't want to double count those objects. Likewise, if we were concerned about counting the true number of objects, we would have to add in those objects that we didn't count in the group 1 and group 2 counts--that's why we add in the "number in neither group".

In this question, let number of houses with patios be "p" and the number of houses that have swimming pools be "s". From the stem, we know that the true (or total) number of houses is 75, and that p is 48. Thus, we have:

75 = 48 + s - Both + Neither

And the question is asking us for the value of s.
Can someone explain why statement 2 is sufficient on its own? I am getting "C" as the answer - need both stems to answer how many houses have a swimming pool....
(2) tells us that the number of houses with "Both" is equal to the number of houses with "neither". That is, Both = Neither. Thus, subbing into our equation, "both" and "neither" will cancel each other out, leaving only s, which we would then be able to solve for.
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by Stuart@KaplanGMAT » Fri Dec 18, 2009 1:25 am
Chick wrote:Of the 75 houses in a certain community, 48 have a patio. How many of the houses in the community have a swimming pool?

(1) 38 of the houses in the community have a patio but do not have a swimming pool.

(2) The number of houses in the community that have a patio and a swimming pool is equal to the number of houses in the community that have neither a swimming pool nor a patio.

Answer: B

Can someone explain why statement 2 is sufficient on its own? I am getting "C" as the answer - need both stems to answer how many houses have a swimming pool....
The matrix is a great way to solve these questions, but we can solve even quicker if we know the overlapping sets formula:

Total # of items = # in group 1 + # in group 2 + neither - both

From the original, we have:

75 = 48 + swimming pool + neither - both

(2) tells us that neither = both, or that neither - both = 0. Subbing into the equation we now get:

75 = 48 + swimming pool + 0

which we can certainly solve.
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by Stuart@KaplanGMAT » Fri Dec 18, 2009 1:25 am
Bah, Testluv hit "enter" quicker than I did!
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