Of the 75 houses in a certain community, 48 have a patio. How many of

Here is my standard approach to solve such problems
P= # of houses with patio only.
Q= # of houses with pool only.
R= # of houses with patio & pool.
S= # of houses with no patio & no pool.

Given P+Q+R+S=75
& P+R=48
What is Q+R? (1)

Solving the first two equations
Q+S+48=75
Q+S=27 (2)

Now let's look at the statements
Statement 1
P=38 unnecessary and insufficient.

Statement 2
R=S
Substituting in (1)
Q+S?
We know form (2) that Q+S=27
hence sufficient.

We have to find the value of X + Y = ?

we know from statement 1 that 10 is the area covered by both and 38 is only patio we have no info about neither not sufficient

Statement 2

We have 48 - x + x + y + x = 75

we can find the value for x + y = 27 sufficient


Good problem

B
--
From B, we have
x = number of houses without P and S = number of houses having P and S

75-x = P + S - x; 75=48+S; S=27

finally can got my computer log in for work... so I can participate on GClub when I have nuffin to do.

Tricky problem. This is why its important to write everything out and not do it in your head.

youl notice we want x+z, which = m.

notice carefully that x+z also =27... its that simple, but so easy to miss... I missed it and said C at first.

B

I dont know how to draw table here... attached a document file with the solution, let me know is that clear for you..

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.

Total = only s + only p + Neither -Both
Now, lets forget about stmt 1 for now.

We have, total =75, neither=both=x, and from the stem we have 48 houses with patio ( these 48 can include houses with Only patio or Both. So we are not sure if we can take s=48 in the above formula.

I may be wrong...not sure.

since only is not given in the question , 48 includes both houses that contains patio with swimming pool and patio without swimming pool.. did you go through the document attached, i think that is very clear...

We have a universal formula: Total= A + B - Both(A&B) + Neither

(1) is obviously insufficient.

(2) is sufficient because from the given conditions we get this: Both(A&B) = Neither = T(Let it be T, to make it simple)
=> 75 = 48 + B - T + T => B = 27.

The best answer is B!

My answer is D.

My Reasoning ;From the main question and also from statement 1-> it is apparent that all the houses have a swimming pool or a Patio. Hence you can answer the question with just A.


Statement B introduces that some houses do no have either. Also you can answer the question with statement B.

Please help me correct my reasoning.

Thanks and kudos to every one

Why it is obvious that all houses there have either a patio or a swimming pool? Why there could not be a house without either of them?
_________________

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

_________________

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

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

Correct Answer B

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.

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.

The easiest way to solve this question is to draw a table.

With the information given in the question stem, the table looks like this:


We want to find x.

Statement (1) is clearly insufficient from filling in the boxes.


Statement (2) seems wordy, but is as simple as the table below:


\((x-y) + y = 27\)

Therefore, \(x = 27\)

Use double-matrix for this Q.

Given: Total = 75. Patio = 48
Need: Pool = ?

1) P but not S = 38 => Not Sufficient
2) Both = None = x
S = Both + Only S => x + (27 - x) = 27
Sufficient

ANSWER: B

Bunuel

How can we preclude that there is no third set we have to consider?
Since the statement does not state that only a pool or a patio are possible, can we always assume that there is no third set involved, when we solve such a question?
Because if there was a third set involved, we must use the formula
Total = A+B+C- [-two Group overlaps] + all three + neither. Then statement 2 would be insufficient, since we dont know the other variables.