Question

Guests at a conference ate a total of fifteen chicken samosas. Each guest who was neither a student nor a vegetarian ate exactly one chicken samosa. No chicken samosa was eaten by any guest who was a student, a vegetarian, or both. If half of the guests were vegetarians, how many guests attended the conference?

(1) The vegetarians attended the conference at a rate of 2 students to every 3 non-students, half the rate for non-vegetarians.

(2) 30% of the guests were vegetarian non-students.

• A. If Statement (1) alone is sufficient, but statement (2) alone is not sufficient. Or If Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
  
• B. If BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• C. If Each statement ALONE is sufficient
• D. If Statements (1) and (2) TOGETHER are NOT sufficient.

Answer 1

The correct option is A

If Statement (1) alone is sufficient, but statement (2) alone is not sufficient. Or If Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

Use a 2 set matrix to solve this problem.

First set is vegetarians vs. non-vegetarians; second set is students vs. non-students.

$\begin{array}{cccc}& \text{VEGETARIAN}& \text{NON-VEGETARIAN}& \text{TOTAL}\\ \text{STUDENT}& & & \\ \text{NON-STUDENT}& & 15& \\ \text{TOTAL}& X& X& ?\end{array}$

We are told that each non-vegetarian non-student ate exactly one of the 15 chicken samosas, and that nobody else ate any of the 15 chicken samosas. This means that there were exactly 15 people in the non-vegetarian non-student category. We are also told that the total number of vegetarians was equal to the total number of non-vegetarians; we represent this by putting the same variable in both boxes of the chart.The question is asking us how many people attended the conference;

Statement(2) INSUFFICIENT: This statement gives us information only about the cell labeled "vegetarian non-student"; further it only tells us the number of these guests as a percentage of the total guests. The 30% figure does not allow us to calculate the actual number of any of the categories.

Statement (1) SUFFICIENT: This statement provides two pieces of information. First, the vegetarians attended in the ratio, of 2:3 students to non-students.

We're also told that this 2:3 rate is half the rate for non-vegetarians; i.e. the rate for non-vegetarians is 4:3 We can represent the actual numbers of non-vegetarians as 4a and 3a and add this to the chart below. Since we know that there were 15 non-vegetarian nonstudents,

we know the missing common multiple, a is $\frac{15}{3}=5$. Therefore, there were (4)(5) = 20 non-vegetarian students and 20 + 15 = 35 total non vegetarians (see the chart below). Since the same number of vegetarians and non-vegetarians attended the conference, there were also 35 vegetarians, for a total of 70 guests.
$\begin{array}{cccc}& \text{VEGETARIAN}& \text{NON-VEGETARIAN}& \text{TOTAL}\\ \text{STUDENT}& & \text{4a or 20}& \\ \text{NON-STUDENT}& & \text{3a or 15}& \\ \text{TOTAL}& \text{x or 35}& \text{x or 35}& \text{? or 70}\end{array}$