Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:
In this overlapping sets problem, there are two kinds of sandwiches (tuna melts and veggie melts, abbreviated T and V). There are also two kinds of customers: male and female. Since each customer buys exactly one sandwich, customers and sandwiches are interchangeable. Thus, we can set up one table to keep track of both type of sandwich and type of customer, as follows:
[] M F Total
T
V
Total 300
We are looking for the ratio of two numbers on this chart: veggie melts bought by females and the total number of veggie melts.
Statement (1): INSUFFICIENT. We can fill in the chart’s total row and total columns, but the four cells in the upper left remain unknown.
[] M F Total
T 150 (1/2 of 300)
V 150
Total 100 200 300
(1/3 of 300)
Thus, we cannot figure out the needed ratio.
Statement (2): INSUFFICIENT. We can use the relationship between “female tuna melts” and “male veggie melts,” introducing a variable as follows:
[] M F Total
T 2x
V x
Total 300
However, without more information, we cannot find the needed ratio.
Statements (1) and (2) together: SUFFICIENT. Combining the tables above, we get the following:
[] M F Total
T 2x 150
V x 150
Total 100 200 300
We can fill in the remaining cells with expressions — for instance, “female veggie melts” can be written as 150 – x, since the veggie row must sum to 150. Now we can add up the female column and solve for x:
2x + (150 – x) = 200
x + 150 = 200
x = 50
Thus, the completed chart looks like this:
[] M F Total
T 50 100 150
V 50 100 150
Total 100 200 300
We see that 100/150, or 2/3, of the veggie melts sold yesterday were bought by female customers.
Note that we could have addressed this problem without knowing the total number of customers (300). We are only looking for a ratio between two numbers on the chart.
The correct answer is (C): Both statements together are sufficient, but neither statement alone is sufficient.