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 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 \(\$120,000.\)

Answer: D

Source: GMAT Prep

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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 \(\$120,000.\)

Answer: D

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

Answer: D

Cheers,
Brent
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