Bunuel wrote:
Tough and Tricky questions: Word Problems.
The Farmer in the Deli sandwich shop sells two kinds of sandwich: tuna melts and veggie melts. Each customer buys exactly one sandwich. If there were 300 customers yesterday, what fraction of veggie melts sold yesterday were bought by female customers?
(1) \(\frac{1}{2}\) of all sandwiches sold yesterday were tuna melts, and \(\frac{1}{3}\) of all customers yesterday were male.
(2) Yesterday, twice as many tuna melts were bought by females as there were veggie melts bought by males.
Kudos for a correct solution. Official Solution:The Farmer in the Deli sandwich shop sells two kinds of sandwich: tuna melts and veggie melts. Each customer buys exactly one sandwich. If there were 300 customers yesterday, what fraction of veggie melts sold yesterday were bought by female customers?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 (\(\frac{1}{2}\) of 300) V 150 Total 100 200 300
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:
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 \(x\):
\(2x + (150 - x) = 200\)
\(x + 150 = 200\)
\(x = 50\)
We see that \(\frac{100}{150}\), or \(\frac{2}{3}\), 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.
Answer: C.
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