Math Is Fun Forum

simone

Member

Registered: 2007-10-05

Posts: 19

urgent help please - venn diagrams - logic

Please anyone I need help. The book I'm practicing on gives B) as the correct answer to this problem, I cannot figure it out! To me it's C) with a value of 27. What am I doing wrong?

Of 75 houses 48 have a patio. How many have a swimming pool?

1) 38 houses have a patio but not a swimming pool

2) houses with both patio and swimming pool = houses without them both

A) Statement 1) alone is sufficient
B) Statement 2) alone is sufficient
C) Statements 1) and 2) together are sufficient
D) Each statement alone is sufficient
E) Statement 1) and 2) together are not sufficient

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: urgent help please - venn diagrams - logic

Separate them into 4 groups:

a: patio and pool
b: patio and no pool
c: no patio and pool
d: no patio and no pool

We are given that a + b + c + d = 75 and that a + b = 48.  By (1), b = 38.  So  we can conclude:

a = 10
c  + d = 27

Yet I do no see any more restrictions.  So what you try now is to see if you can come up with multiple solutions.

a = 10
b = 38
c = 27 - d

Now lets pick 2 values for d.  I'll go with 10 and 20.

a = 10, b = 38, c = 17, d = 10

Total houses: 10 + 38 + 17 + 10 = 75
Number with patio: 10 + 38 = 48
Number with patio and no pool: 38

a = 10, b = 38, c = 7, d = 20

Total houses: 10 + 38 + 7 + 20 = 75
Number with patio: 10 + 38 = 48
Number with patio and no pool: 38

So unless I'm missing something, there are multiple solutions when just given the information from (1).

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,423

Re: urgent help please - venn diagrams - logic

48 houses have a patio, 38 houses have a patio but not a swimming pool. Hence, 10 houses have both a patio and a swimming pool.

From the question, it can be inferred that 10 houses have neither a patio nor a swimming pool. (It is stated in the question that houses with both patio and swimming pool = houses without them both).

Hence, of the 75 houses, 10 have neither a patio nor a swimming pool.

Therefore, 65 houses have either a patio or a swimming pool or both of them.

The formula for finding the number of elements of a set is given by
n(AUB)=n(A)+n(B)-n(AπB) where 'U' denotes intersection and 'π' denotes intersection. n stands for the numberof elements in the set.

In the given question,
n(AUB)=65, n(A)=48, n(B)=? and n(AπB)=10.

Substituting the values if the formula,
65=48+n(B)-10
65+10=48+n(B), 75=48+n(B).
n(B)=75-48=27.

We have made use of both the statements (1) and (2) in deducing the solution.

Hence, you're solution is correct, simone.
The question is incomplete unless it is stated in the question that each house either has a patio or a swimming pool or both of them. If it had stated so, the 2nd statement is redundant.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: urgent help please - venn diagrams - logic

With only statement #1, you can have any number from 10 to 37 swimming pools, so it is not clearly defined.

However, with a little twist in the beauty of the English language interpretation of #2,
you can get an answer.  But first you must take a big leap of faith in the meaning of
"houses without them both".  For this, to get a solution, you can finagle this to mean:
"houses with only one of the two". 
So then 32 houses with both = 16 with one plus 16 of the other.
And 32 + 16 = 48 with a patio.
That is the only thing I can think of.
It is a problem with the English writing I believe.

Last edited by John E. Franklin (2007-10-12 13:53:56)

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