Data Sufficiency

Mary persuaded n friends to donate 500$ each to her election campaign and then each of these n friends persuaded n more people to donate 500$ each to Mary's campaign. If no one donated more than once and if there were no other donations what was the value of n?

1)the first n people donated 1/16 of the total amount donated
2)the total amt donated was 1,20,000

Mary persuaded 'n' friends to donate 500 dollars each to her election campaign and then each of these n friends persuaded n more people to donate 500 dollars each to Marys campaign. If no one donated more than once if there were no other donations what was the value of n?

1. The first n people donated 1/16 of the total amount donated.
2. The total amount donated was 120,000.

If n=2, and each of these 2 people invite 2 other people, then the total number of ADDITIONAL people invited = 2*2 = 2² = n².
If n=3, and each of these 3 people invite 3 other people, then the total number of ADDITIONAL people invited = 3*3 = 3² = n².
Thus, the total number of ADDITIONAL people invited = n².

Since each person donates $500:
Total donated by the first n people = 500n.
Total donated by the n² additional people = 500n².

Statement 1: The first n people donated 1/16 of the total amount donated.
Thus, of every $16 donated, $1 was donated by the first n people, while $15 was donated by the n² additional people.
Thus:
500n / 500n² = 1/15
1/n = 1/15
n=15.
SUFFICIENT.

Statement 2: The total amount donated was 120,000.
Since each person donates $500, the total number of people = 120,000/500 = 240.
Since n=15 in statement 1, let's see whether this value also satisfies statement 2:
If n=15, the total number of people = n + n² = 15 + 225 = 240.
Clearly, no other value of n will yield the required total of 240 people.
Thus, n=15.
SUFFICIENT.

The correct answer is D.

To think about this in a conceptual fashion, think about the '2 layers of friends'. The first layer is the friends that Mary invites, and the second layer is the friends that Mary's friends invites. As 'n' increases, the second layer will increase more rapidly - it is exponential growth.

1) If the second layer is growing more quickly, the proportion of the first layer to the total will get smaller as n grows. If the ratio is constantly changing, there will only be 1 place where the ratio is 1/16 (see the math above for specifics).

2) Same logic here. As 'n' grows, the total amount donated will increase and there will only be one place where the dollar amount is 120,000. Again, see math above.


Try to think about what is happening in the problem more than how to do the math.

D is the answer.

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Mary persuaded n friends to donate 500$ each to her election campaign and then each of these n friends persuaded n more people to donate 500$ each to Mary's campaign. If no one donated more than once and if there were no other donations what was the value of n?

1)the first n people donated 1/16 of the total amount donated
2)the total amt donated was 1,20,000


(Solution)
From the original condition
Mary


sum of initial donation: 500n

sum of second donation: 500n,500n,500n,500n,500n... there are n donations of 500n, thus -> 500n2

Since we need to find out n, we only need 1 equation for the 1 variable. Therefore the answer is D.

If we solve the problem directly using 1) 2) separately, the answer that is sufficient for both conditions are D.

1) 500n=(1/16)(500n+500n2) , n=15, sufficient (o)
2) 500n+500n2=120,000, n=15, sufficient (o)


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