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

Question

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 fee (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 fee Image7.JPG

Bunuel · Accepted Answer

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.

FQ · Answer

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

TomB · Answer

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.

mockney · Answer

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

Bunuel · Answer

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.

Skywalker18 · Answer

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 .

anairamitch1804 · Answer

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

energetics · Answer

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

Sneha_0410 · Answer

Hey Bunuel,

I have a basic doubt regarding Data Sufficiency questions. In cases like this, can we directly conclude that a statement is insufficient by applying the rule, "two unknowns and only one equation = not sufficient"?

I'm asking because there may be situations (like the one shown in the attached screenshot) where the required value can still be determined from a single equation. So, is there any general pattern or shortcut that helps us decide whether a statement is sufficient without actually solving it, or should we always work through the statement to check?

Edit: I would appreciate any helpful response to this, MartyMurray, KarishmaB

Show Spoiler

Attachment:

GMAT-Club-Forum-xunazvco.png

https://gmatclub.com/forum/download/file.php?id=89921 GMAT-Club-Forum-xunazvco.png

MartyMurray · Answer

In a way, there is a general pattern. That pattern is that, if there are additional constraints, the information provided by a single equation with two variables may be sufficient for answering the question when combined with those additional constraints.

In the example you provided, the additional constraint that the prices must be integer values allows us to determine the values of the two variables using the single equation.

So, a key question you can ask in evaluating a statement in a Data Sufficiency question is that of whether there are in the scenario any stated or inferable constraints that work with the implied equation to enable answering the question.

egmat · Answer

Hi Sneha_0410,

Great instinct to question the shortcut instead of trusting it blindly. In the thread you're looking at, Bunuel writes "two unknowns, one equation, not sufficient" for each statement, and that conclusion  is  correct here - but the reasoning is not a universal law. It's a heuristic that only holds under specific conditions.

Why it works in  this  problem

Statement  (1)  gives  100x − 0.75n = 75 . Here  x  and  n  are  free  - they can be any real numbers that fit the line, so  n  has infinitely many possible values. That's the real reason it's insufficient, not the raw count of variables vs. equations.

When the shortcut breaks (the exceptions you sensed)

The rule can fail whenever the two variables aren't truly free, or when the question doesn't ask for a single variable. Watch for:

-  Hidden constraints  (integers, positives). Example: if  x, y  are  positive integers  and  2x + 3y = 8 , the  only  solution is  x = 1, y = 2 . One equation, two unknowns - yet fully pinned down.
-  The question asks for a combination, not one variable.  Example:  "What is x + y?"  with the single equation  2x + 2y = 10 . You can't find  x  or  y  alone, but  x + y = 5  is locked.  Sufficient. 
-  Non-linear equations , which can force unique or very limited solutions.

The real takeaway

So don't apply "2 unknowns, 1 equation => insufficient" mechanically. Instead ask two quick questions before deciding:

1.Are the variables genuinely free , or does something (integer/positive/etc.) restrict them?
 2.What exactly is being asked  - one variable, or a combination the equation might already fix?

If the variables are free reals and you need one of them individually, then yes, a single linear equation is  insufficient  - that's why  (1)  and  (2)  each fail here, and combining them gives the second independent equation that finally pins  n = 300 . But when either condition above is in play, always check rather than assume.

Answer: C

KarishmaB · Answer

Also note that 2 equations and 2 unknowns doesn't always mean sufficient. Ensure that the equations are distinct else they could both be describing the same line. 
Essentially, x + y = 4 and 2x + 2y = 8 are the same equation and will not give you a unique value for x and y without any other constraints. Sometimes the same equations are harder to see due to the presence of fractions etc. One must check this.

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

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