Each person attending a fund-raising party for a certain club was

Each person attending a fund raising party for a certain club was charged the same admission fee. How many people attended the party?

Let the original admission fee be \(x\) and the original # of people be \(n\). Question: \(n=?\)

(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 the admission fee

\(xn=(x-0.75)(n+100)\) --> \(100x-0.75n=75\). Two unknowns, one equation, not sufficient to calculate \(n\).

(2 ) If the admission fee had been 1.50$ less and 100 fewer people had attended, the club would have received the same amount in the admission fee

\(xn=(x-1.5)(n-100)\) --> \(100x-1.5n=-150\). Two unknowns, one equation, not sufficient to calculate \(n\).

(1)+(2) \(100x-0.75n=75\) and \(100x-1.5n=-150\) --> we have two distinct linear equation with two unknowns, hence we can solve for \(n\). Sufficient.

Answer: C.
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TomB, how can you assign a random value (100) to xy and claim that x can be solved??

Let a = admission fee and n = number of people attended

(1) If the admission fee had been $0.75 less and 100 more people attended, the club would have received the same amount in admission fees.

an = (a - 0.75)(n + 100)
an = an - 0.75n + 100a -75
0.75n - 100a = -75

Insufficient

(2) If the admission fee had been $1.50 more and 100 fewer people had atteneded, the club would have received the same amount in admission fees.

an = (a + 1.50)(n - 100)
an = an + 1.5n - 100a - 150
1.5n - 100a = 150

Insufficent

Subtrace (1) from (2)
1.5n - 100a = 150
0.75n - 100a = -75

0.75n = 225 >> n = 300 & a can be solved

let assume x=admin fee,y=no.of. people
x*y=100.
1) (x-.75)* (y+100)= 100 (statement 1 mentioned same amount,so i assume the amount as 100)
from question stem x*y=100
so x=100/y, sufficient to know x value.

2) (x+1.50) * (y-100) =100
x*y =100
x=100/y,sufficient to know x value.

Bunnel, plz explain why should we solve the problem like this.

Each person attending a party was charged the same admission fee. How many people attended the party?

(1) If the fee had been 0.75$ less and 100 more people had attended, the club would have made the same amount of money.
(2) If the fee had been 1.50$ more and 100 fewer people had attended, the club would have made the same amount of money.

I have a specific question regarding this problem :

Obviously Statement 1 and 2 are insufficient alone, however once we find out equation 1: xy = (x-3/4)(y+100) from statement 1 and xy = (x+3/2)(y-100) from statement 2, why can't we simply equate them?
(x-3/4)(y+100) = (x+3/2)(y-100)

I'd appreciate any help! Thanks

Well, actually you can: (x-3/4)(y+100) = (x+3/2)(y-100) --> 9y=800x+300.

From xy = (x-3/4)(y+100) we'll have that 3y=400x-300, so 9y=1200x-900. Equating again: 800x+300=1200x-900 --> x=3 --> y=300.

Though you can directly simplify xy = (x-3/4)(y+100) to get 3y=400x-300 and xy = (x+3/2)(y-100) to get 3y=200x+300 --> 400x-300=200x+300 --> x=3 --> y=300.

Hope it's clear.
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Let Original admission fee= x
Original number of people = n
1 xn =(x-.75)(n+100)
=>100x - .75 n = 75 ---1

Not sufficient .

2.The second statement should have been
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 fee
xn= (x+1.5)(n-100)
=> 100x - 1.5 n = -150 ---2

Not sufficient .
Combining 1 and 2 , we get 2 distinct linear equations with 2 unknowns
n=300
x=3

Answer C

If the admission fee had been 1.50$ less and 100 fewer people had attended, the club would have received the same amount in admission fee Then,
xn= (x-1.5)(n-100)
=> 100x +1.5n = 150 --3

On solving 1 and 3 , we get
x=1
n=33.33


We can even see that statement 2 has an issue by analyzing that for a product xn to remain same if one of the values increase , the other should decrease . The product won't be same if both the values decrease or increase .
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Let attend fee be x, number of person be y:
Form 1, (x-0.75)(y+100)=xy----100x-0.75y-75=0
From 2, (x-1.5)(y-100)=xy ----100x-1.5y-150=0
Combine 1 and 2, we can get specific value of x and y.
Answer is C

Bunuel As mentioned above, the original prompt is written incorrectly:

2) should be "If the admission fee had been $1.50 less more and 100 fewer people had attended, the club would have received the same amount in admission fees."

Although, I guess in this case it doesn't cause any harm.

Below is the correct equation from the GmatPrep test:
1) fp = (f-.75)(p+100)
fp = fp + 100f - .75p - 75
0 = 100f - .75p - 75
2 variables, 1 equation, insufficient

2)fp = (f+1.5)(p-100)
fp = fp - 100f + 1.5p - 150
0 = -100f + 1.5p - 150
2 variables, 1 equation, insufficient

3) 2 variables, 2 equations, sufficient. In the 2 equations the signs are reversed relative to f,p so we know they are not the same equation.

combining:
0 = 100f - .75p - 75
0 = -100f + 1.5p - 150
+_________________
0 = 0 + .75p - 225
225 = .75p
p = 300

300f = (f-.75)(300+100)
300f = 400f - 300
300 = 100f
f = 3

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