Mary persuaded n friends to donate $500 each to her election

I have understood the approach GT took to solve the problem its very similar to mine... but i cannot make out how can the first stmt give a solution for n as 0 or a -ve value.

Could someone explain this?

It cannot give negative solution for \(n\), though it can give \(n=0\) as a solution. See below:

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?

# of people donated at the firs stage - \(n\), amount donated - \(500n\);
# of people donated at the second - \(n^2\), amount donated - \(500n^2\);
Total amount donated - \(500n+500n^2\)
Little assumption here: \(n>0\).

(1) The first n people donated 1/16 of the total amount donated --> \(500n=\frac{1}{16}(500n+500n^2)\) --> \(n=15\) (we can rule out \(n=0\), which is also a solution of this equation). Sufficient.

(2) The total amount donated was $120,000 --> \(500n+500n^2=120,000\) --> \(n=15\). Sufficient.

Answer: D.

Hope it's clear.

Man this question makes me mad that i got it wrong initially and it took me a while to figure it out..

OK so (1)

n/(n + n^2) = 1/16
16n = n^2 + n
n^2 -15n = 0
n(n-15) = 0
But n cant really be zero
Sufficient

(2)
(n + n ^2)* 500 = 120,000
n + n^2 = 240
n^2 + n -240=0
(n +16) (n-15) = 0
But n cant really be -16
Sufficient

Answer is D

I couldnt figure out the way to factor n^2 +n -240 = 0 for a long time
I guess my real issue was trying to solve it.. once i constructed the quadratic i shouldve just moved on with life!!!

How to solve equation like 500n^2 + 500n = 120,000

This translates to quadratic equation n^2 + n = 240

Should one use formula of \sqrt{b^2 - 4ac}

You can solve it using the formula for quadratics, though it's better to use another approach:

\(500n+500n^2=120,000\) --> \(n+n^2=240\) --> \(n(n+1)=240\). Since \(n\) is an integer then we have that the product of two consecutive integers is 240, now it's easy to find that \(n=15\).

Hope it's clear.

Hey all,

I think I don't get the wording: If EACH of the n friends persuaded n people, wouldn't it be n^n??

Wouldn't really change the outcome, but I'd like to know it exactly....

Thanks!

I still am confused with the phrase...
"then each of these n friends persuaded n more people"
How could this be \(n*n\) why not \(n+n\)

If 3 people persuade 3 more people then the total would become 3+3=6 right.??
Pls clarify

Each of these n friends persuaded n more people, not that n people together persuaded n more people.

Hope it's clear.

Can someone please explain the logic in putting the equation in part (1) = 500n? In my own working i came up with the RHS, but in my mind that should be put equal to the total amount donated. I am not sure how this is equal to 500n (which in turn is equal to the total number of 1st tier friends who donated).
I am sure its simple and I am just missing a logical step. The rest of Brunels/OG's solution is crystal clear.

Amount donated by the first n people = \(500n\);
Total amount donated = \(500n+500n^2\).

(1) says that the first n people donated 1/16 of the total amount donated, thus \(500n=\frac{1}{16}(500n+500n^2)\) --> \(16(500n)=500n+500n^2\).

Hope it's clear.

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:

Cheers,
Brent

[/quote]

You can solve it using the formula for quadratics, though it's better to use another approach:

\(500n+500n^2=120,000\) --> \(n+n^2=240\) --> \(n(n+1)=240\). Since \(n\) is an integer then we have that the product of two consecutive integers is 240, now it's easy to find that \(n=15\).

Hope it's clear.[/quote]

Within context of GMAT DS question, the moment I manage to set up such relationship n(n+1) = 240, will it be safe to say there is 1 solution for n without trying to find a pair of factors that fit? This would save some time. Whenever I get to this point, I always try to find a pair just to make sure it will not be the case of a) having no solution for n or b) having 2 solutions for n.

n(n + 1) = (positive number) will always have two solutions, one negative and one positive but not always these solutions will be integers.

For example:

n(n + 1) = 2 --> n = -2 or n = 1;

n(n + 1) = 2 --> \(n = -\frac{1}{2}-\frac{\sqrt{13}}{2}\) or \(n = -\frac{1}{2}+\frac{\sqrt{13}}{2}\)

\({\rm{Total}}\,\, = \,\,500 \cdot n + 500 \cdot n \cdot n\,\,\,\,\,\,\left[ \$ \right]\)

\(? = n\)

\(\left( 1 \right)\,\,\,500 \cdot n = {1 \over {16}} \cdot 500 \cdot n \cdot \left( {1 + n} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,\,\left( {500\,n} \right)\,\,\,\left[ {\,n\, \ne \,0\,} \right]} \,\,\,1 = {1 \over {16}} \cdot \left( {1 + n} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,n\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)

\(\left( 2 \right)\,\,\,500 \cdot n\left( {1 + n} \right) = 120000\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,500} \,\,\,\,n\left( {1 + n} \right) = 240\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,n\,\, > 0\,\,\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)

\(\left( * \right)\,\,15 \cdot 16 = 240\,\,\, \Rightarrow \,\,\,\left\{ \matrix{\\
\,n\left( {n + 1} \right) < 240\,\,{\rm{for}}\,\,0 < n < 15 \hfill \cr \\
\,n\left( {n + 1} \right) > 240\,\,{\rm{for}}\,\,n \ge 16 \hfill \cr} \right.\,\,\,\,\,\,\left( {{\rm{Now}}\,\,{\rm{rethink}}\,\,{\rm{without}}\,\,{\rm{knowing}}\,\,{\rm{that}}\,\,n = 15...} \right)\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Dear Bunuel , In this case the question asked that each person persuaded other n person to donate and no one has donated more than once. In this case it also possible that the few persons who were persuaded ,did not donate at all then can we confidently say that the total amount donated was 500n+ 500n^2 ? Please clarify

VeritasKarishma
chetan2u

Here persuaded means convince, so you have to take that all did what they were asked to do.

Nice one. took me a while, but then once you've figured out the pattern, it's easy.

So, if marry had N people donating for her campaign, and then those N people found another N people (each) to donate. how many people are now involved?

Thought bubble:- say, she had 4 people initially and then these 4 found another 4 each, now we have 16 more people with us apart from the first 4. Or maybe she had 3 people initially and then these 3 found another 3 each making the total at 9 more. see the pattern?

if you have n people initially, you will get n^2 in the next phase of the question. That's it.

(1) The first n people donated 1/16 of the total amount donated

\(n(500)\)= \({n(500) + n^2(500)/16}\) --- Single variable equation, will get a definite answer.

A. suffices.

(2) The total amount donated was $120,000.

\(n(500) + n^2 (500)= 12* 10^4\) ---Single variable, we'll get a definite answer.

B. suffices.

D. is the answer.