Sets

IMO B

This problem can be solve using a Venn diagram.

Let's start rephrasing the question
P = number of houses that have a patio
S = number of houses that have a swimming pool
P&S = number of houses that have a swimming pool and a patio
NP&S = number of houses that have neither a swimming pool nor a patio

75 = P + S - P&S + NP&S and we know that P =48 so
75 = 48 + S - P&S + NP&S ---> S = 27 + P&S - NP&S

Now take a look at the statements
1) 38 houses have a patio but no swimming pool
this means that 38 houses only have patio so
P = 38 + P&S ---> 48 = 38 + P&S ---> P&S = 10

can you get the value of S in this equation S = 27 + P&S - NP&S, using that P&S = 10? No insufficient.

2) Number of houses that have a swimming patio and a swimming pool is equal to the number of houses that have neither pool nor patio
P&S = NP&S
S = 27 + P&S - NP&S ---> S = 27 sufficient

Claret wrote:Of 75 houses, 48 have a patio. How many houses have a swimming pool?
a. 38 houses have a patio but no swimming pool
b. Number of houses that have a swimming patio and a swimming pool is equal to the number of houses that have neither pool nor patio

Mike's solution is great!

I just wanted to add the general 2-set equation which can be very useful on this type of question:

True # of Objects = (# in group 1) + (# in group 2) + (# in neither group) - (# in both groups)

or

True # = G1 + G2 + neither - both

As Mike applied it to this question:

75 = 48 + pool + neither - both

(1) tells us that "both" = 10 (if 48 have a patio, and 38 have only a patio, then 48-38=10 have both).

So: 75 = 48 + pool + neither - 10

Still 2 variables, can't solve... insufficient.

(2) neither = both, so

75 = 48 + pool + neither - neither

75 = 48 + pool... we can certainly solve for "pool".. sufficient.

(2) is sufficient, (1) isn't... choose (B).

thanks mikeCoolBoy and Stuart!

will keep the formula in mind

Actually without knowing any formula ---- one can answer the question using logic


There are 75 houses , and there are two characteristics -- Patio and swimming pool so what can happen ?

We know from the given that only 48 have a patio what can we derive from this ............ the 48 might or might not have SP , but we don't know how many houses are with out both ........... so lets look at the statements

Statement I :38 houses have a patio but no swimming pool

This doesn't give the with out figure so insufficient

Statement II : Number of houses that have a swimming patio and a swimming pool is equal to the number of houses that have neither pool nor patio

Aha ! this give the neither case so its sufficient so ans = B