Each person attending a fund-raising party for a certain club was charged the same admission fee. How many people attended the party?
1) If the admission fee had been $0.75 less and 100 more people had attended, the club would have received the same amount in admission fees.
2) If the admission fee had been $1.50 more and 100 fewer people had attended, the club would have received the same amount in admission fees.
Let F = admission fee
Let N = number of attendees
So, the TOTAL revenue = FN
Target question: What is the value of N?
Statement 1: If the admission fee had been $0.75 less and 100 more people had attended, the club would have received the same amount in admission fees.
In other words, if the fee had been (F - 0.75) and the number of attendees had been (N + 100) the total revenue would still be FN
We can write: (F - 0.75)(N + 100) = FN
Expand: FN + 100F - 0.75N - 75 = FN
Simplify:
100F - 0.75N - 75 = 0
Since we cannot solve this equation for N, we cannot answer the
target question with certainty.
So statement 1 is NOT SUFFICIENT
Statement 2: If the admission fee had been $1.50 more and 100 fewer people had attended, the club would have received the same amount in admission fees.
In other words, if the fee had been (F + 1.50) and the number of attendees had been (N - 100) the total revenue would still be FN
We can write: (F + 1.50)(N - 100) = FN
Expand: FN - 100F + 1.5N - 150 = FN
Simplify:
-100F + 1.5N - 150 = 0
Since we cannot solve this equation for N, we cannot answer the
target question with certainty.
So statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1 tells us that
100F - 0.75N - 75 = 0
Statement 2 tells us that
-100F + 1.5N - 150 = 0
Here we have two different linear equations involving two variables.
Since we COULD solve this system for N, we could answer the
target question with certainty.
So, the combined statements are SUFFICIENT
ASIDE: We don't need to actually solve the system. We need only recognize that we have SUFFICIENT information to do so.
Answer =
C
Cheers,
Brent
