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# Of the 75 houses in a certain community, 48 have a patio.

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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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24 Nov 2015, 11:47
Did anyone tried solving this problem with double matrix? One of the guys did, but looks like his excel is incomplete
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Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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06 Mar 2016, 08:23
Steinbeck wrote:
Did anyone tried solving this problem with double matrix? One of the guys did, but looks like his excel is incomplete

From stem , we get 75-48 = 27 houses with no patio
From Stem 2: Let b be the number of houses with a pool , which also happens to be number of houses without a pool.

b + (27-b) = houses with swimming pool only
27 = houses with swimming pool only.

--------------------Patio ------No Patio Total
------------ Swim b---------(27-b)----------(b+27-b)=27
--------No Swim-------------- (b)--------------
-------------------- 48---------(75-48=27)----(75)
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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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01 May 2016, 07:22
4
Keats wrote:
Did anyone tried solving this problem with double matrix? One of the guys did, but looks like his excel is incomplete

using double set matrix and from option B, we can solve as below

S.P = swimming pool
Attachments

double set.png [ 3.74 KiB | Viewed 6655 times ]

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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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01 Aug 2016, 01:23
bmwhype2 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.

I marked answer D thinking A is also sufficient because I didn't consider the case where a house has neither a swimming pool nor a patio. I did see the "neither" case in option B but thought it doesn't have relevance to option A and also in main question there wasn't anything mentioned about "neither" case. How do we come to know whether "neither" case is applicable to a question or not? If they haven't particularly mentioned in the question that 'each house has a patio or a swimming pool' then it is necessary to consider the "neither" case?
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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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05 Aug 2016, 09:36
Interesting sets question. B plays with you and if you setup a matrix table the swimming pool equation will become x - (27 - x) and then the x actually cancel and you get the # of swimming pools right there. Important to fill out all the details that you can.
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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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05 Aug 2016, 15:30
2
bmwhype2 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?

Given the question, we have the following table:

$$\begin{array}{c|cc|c} & \textbf{P} & \textbf{NP} & \\ \hline \textbf{S} & A & B & \color{green}{Z} \\ \textbf{NS} & C & D & \\ \hline & 48 & 27 & 75 \\ \end{array}$$

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

$$\begin{array}{c|cc|c} & \textbf{P} & \textbf{NP} & \\ \hline \textbf{S} & A & B & \color{green}{Z} \\ \textbf{NS} & 38 & D & \\ \hline & 48 & 27 & 75 \\ \end{array}$$

$$A = 10$$

We have no way of determining $$B$$

Insufficient

(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.

$$\begin{array}{c|cc|c} & \textbf{P} & \textbf{NP} & \\ \hline \textbf{S} & X & B & \color{green}{Z} \\ \textbf{NS} & C & X & \\ \hline & 48 & 27 & 75 \\ \end{array}$$

We can see that although we cannot solve the matrix, we can see that a vertical and horizontal equation are equal due to the added symmetry.

$$B + X = 27 \implies Z = 27$$

Sufficient

(B) statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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06 Aug 2016, 04:41
Top Contributor
Let,
# of houses with patio=n(P)
# of houses with Swimming pool=n(S)
# of houses with patio & Swimming pool=n(Sum of two groups overlap)
# of houses with no patio & no Swimming pool=N
So,
48+n(S)-n(Sum of two groups overlap)+N=75

(1) No information about the number of houses with no patio & no Swimming pool(N), Not sufficient

(2) statement said that, n(Sum of two groups overlap)=N,So we can find n(S)=75-48=27 , Sufficient

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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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02 Feb 2017, 16:20
2
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.
Hence B.
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Re: Of the 75 houses in a certain community, 48 have a patio.  [#permalink]

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21 Apr 2018, 04:31
keats wrote:
Did anyone tried solving this problem with double matrix? One of the guys did, but looks like his excel is incomplete

Yes, i was at the beginning a bit confused. But finally got it:

Here is a general formula:

A + B - Both + Neither

from question:
48 + S - Both + Neither = 75

Statement 1
38 have patio but no swimming pool

then formula becomes:
(48-38) + S - Both + Neither = 75
10 + S - Both + Neither = 75 ----> Is this sufficient to find S? No, therefore not sufficient. Why? because you are left with three variables - S,Both and Neither with one equation

Statement 2
Both = Neither

this means either we can replace from formula Both or Neither like this:

from question:
48 + S - Both + Neither = 75

Replacing "Both"
48 + S - Neither + Neither = 75 -----> you can see that Neither cancels out
then you are left with 48 + S = 75, from here you don't need to solve it since you can find the value of S Sufficient

Replacing "Neither"
48 + S - Both + Both = 75 -----> you can see that "Both" cancels out
then you are left with 48 + S = 75 same outcome as above.

My point is either way will eliminate both and neither in the formula.
Re: Of the 75 houses in a certain community, 48 have a patio. &nbs [#permalink] 21 Apr 2018, 04:31

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