confused on question DS #70 in the OG12

the problem says"...EACH of these n friends persuaded n more friends...". Doesn't that mean each one of those first n people got n new people which should give a total of n x n x n x...... n times? But the solution says that those n guys got n more guys which is n x n. this is not correct if EACH of those n guys got another n guys. And the solution won't be possible in that case. I feel the language of this question is misleading. otherwise, each statement in itself is sufficient to answer.

bynddrvn wrote:I understand how you could answer this question with both statement 1 & 2, but not each one by themselves.

"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 amount donated was $120,000."

What I don't understand is how you can answer this question, without knowing how many people each n original people pulled in? Couldn't each original n person pull in 5 people, 16 people? From the first statement we can figure out that each of the original n people are 1/16th of the total, but we don't know what the original number of people are, correct? From the second statement, we know the upward bound and can easily answer the question. Am I missing something simple?

It states clearly in the stem "each of these n friends persuaded n more people..".

Here's a fundamental rule for math: no matter how many times a variable appears in a problem, it will always have the same value.

So, when we break down the stem we see that:

round 1: n donors
round 2: n*n donors (since each of the n donors recruits n more donors)

So, total number of donors is n + n^2 and total money raised is 500(n + n^2).

(1) If the first n donors donated 1/16 of the total, we know that:

part/whole = 1/16

n/(n + n^2) = 1/16

16n = n + n^2
15n - n^2 = 0
n(15 - n) = 0

so n=0 or n=15

Now, one could argue that it's possible for there to be 0 people; however, on the GMAT when we speak about objects, we can assume that n does not equal 0.

(If you look at the OG explanation, it actually says "Assuming n>0" without any further elucidation.)

Therefore, n=15... sufficient.

(2) We can use our total value equation to solve with this information:

500(n + n^2) = 120000
n + n^2 = 120000/500
n^2 + n = 240
n^2 + n - 240 = 0

we want two numbers that multiply to 240 and are 1 apart:

(n+16)(n-15) = 0

n = -16 or n = 15; we can't have a negative number of donors, so n must be 15... sufficient.

Of course, we really didn't need (or want) to do all that math - as soon as we saw that each equation would yield only one positive solution, we knew that each was sufficient alone.

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As an aside, this is DS#70 from OG12 (at least my copy).

Ok, thank you. For some reason, I kept assuming there were two variables, not one.

FYI, I corrected the Q# so others can find this answer if they need it.