Reading through the question, we see that we're dealing with a group of houses where some have a swimming pool and some have a patio. Scanning over the statements, we see that some houses have
only a pool, some houses have
only a patio, some have
neither, and some have
both. Almost anytime we see the word "both" in GMAT Quant questions, we're dealing with an
overlapping sets problem - we are looking at two criteria (here, having a pool and having a patio) and where they overlap (here, having "both" a pool and a patio).
Overlapping sets problems have a lot of information, so it's really easy to get lost in them. A good trick is to
use a visual representation to keep track of what you know:
- For two overlapping criteria, use a table, where each axis represents one criterion.
- For three overlapping criteria, use a venn diagram, where each circle represents a criterion.
Here, we have two overlapping sets, so we're going to use a table. We'll go ahead and fill in only what was stated directly in the question. We want to find the total number of houses that have a Pool, so we'll represent that in our table as x:
Because of the way we've set the table up, the two numbers in each row should add up to the total at the end of the row and the two numbers in each columns should add up to the total at the bottom of the column. This means that
if we have at least two of the three values in each row or column, we should be able to solve for the third. Looking at our table, we see that our total row along the bottom has two values. If there are 75 houses in total and 48 of those houses have patios, 75 - 48 = 27 of those houses must
not have patios. We can go ahead and fill that information in our table:
Doesn't seem like we can get much more out of our table at this point, so we'll move on to our Statements.
Statement 1
To start we'll fill in the information directly given in the statement:
We see that our first column has two values, so we should be able to solve for the third. If there are 58 houses with patios and 38 of those houses do
not have pools, 48 - 38 = 10 of those houses must have pools:
Looking at the row and the column that contain x, we see that we only have one number value for each, meaning that we can't solve for x.
Statement 1 is insufficient.
Statement 2
This statement doesn't give us any concrete numbers to work with, but it does tell us that two of our values (houses with
both pools and patios and houses with
neither pools nor patios) are equal to each other. When we know that the same number shows up in two places, but we don't know what that number is, it's a good idea to represent that number with a variable - if we represent both values as, say, n, we know that they are the same number and can combine or eliminate them down the line:
Now we're getting somewhere! We don't have two number values in any row or column, but we can use both the top row and the second column to represent No Patio/Pool with variables: if there are x total houses with pools and n of those houses have patios, x-n must
not have patios,
and if there are 27 total houses that do
not have patios, and n of those houses do
not have pools, 27-n must have pools:
Since the number of houses with no patio and a pool equals both x-n and 27-n, we can set the two equal to each other to solve for x:
x-n=27-n
x=27
We were able to determine that 27 houses have pools, which means that
Statement 2 is sufficient. The correct answer is B: Statement 2 alone is sufficient to answer the question.
We actually featured this problem recently on the
PrepScholar GMAT blog as one of the
5 Hardest Data Sufficiency Questions. I recommend checking out the article for more strategies and trends we can take away from this and other 700+ level problems!
