## Mary persuaded $$n$$ friends to donate $$\500$$ each to her election campaign, and then each of these $$n$$ friends

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### Mary persuaded $$n$$ friends to donate $$\500$$ each to her election campaign, and then each of these $$n$$ friends

by Vincen » Thu Apr 15, 2021 10:41 am

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## Global Stats

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 $$\dfrac1{16}$$ of the total amount donated.
(2) The total amount donated was $$\120,000.$$

Source: GMAT Prep

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### Re: Mary persuaded $$n$$ friends to donate $$\500$$ each to her election campaign, and then each of these $$n$$ friends

by [email protected] » Fri Apr 16, 2021 5:28 am

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## Global Stats

Vincen wrote:
Thu Apr 15, 2021 10:41 am
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 $$\dfrac1{16}$$ of the total amount donated.
(2) The total amount donated was $$\120,000.$$

Source: GMAT Prep
Target question: What was the value of n?

When I scan the two statements, it seems that statement 2 is easier, so I'll start with that one first...

Statement 2: The total amount donated was \$120,000
Let's summarize the given information....

First round: n friends donate 500 dollars.
This gives us a total of 500n dollars in this round

Second round: n friends persuade n friends each to donate
So, each of the n friends gets n more people to donate.
The total number of donors in this round = n²
This gives us a total of 500(n²) dollars in this round

TOTAL DONATIONS = 500n dollars + 500(n²) dollars
We can rewrite this: 500n² + 500n dollars

So, statement 2 tells us that 500n² + 500n = 120,000
This is a quadratic equation, so let's set it equal to zero to get: 500n² + 500n - 120,000 = 0
Factor out the 500 to get: 500(n² + n - 240) = 0
Factor more to get: 500(n + 16)(n - 15) = 0
So, EITHER n = -16 OR n = 15
Since n cannot be negative, it must be the case that n = 15
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Statement 1: The first n people donated 1/16 of the total amount donated.
First round donations = 500n
TOTAL donations = 500n² + 500n
So, we can write: 500n = (1/16)[500n² + 500n]
Multiply both sides by 16 to get: 8000n = 500n² + 500n
Set this quadratic equation equal to zero to get: 500n² - 7500n = 0
Factor to get: 500n(n - 15) = 0
Do, EITHER n = 0 OR n = 15
Since n cannot be zero, it must be the case that n = 15
Since we can answer the target question with certainty, statement 1 is SUFFICIENT