Reducing Gift Contribution: $G and $n Factors - Problem #56 (April 22nd, 2013)

[Jameson]

A group of 10 people plan to contribute equally to pay for a friend's gift that costs $G$ dollars. If $n$ additional people wish to contribute to pay for the gift, by how many dollars will the required contribution be reduced per person? Express your answer in terms of $G$ and $n$.
--------------------

 

[Jameson]

Congratulations to the following members for their correct solutions:

1) MarkFL
2) anemone
3) Sudharaka
4) Reckoner

Solution (from MarkFL):

Spoiler

Let the original contribution $C(n)$ be:

$\displaystyle C(10)=\frac{G}{10}$

Now, when $n$ additional contributors are added from an original 10 contributors, we find:

$\displaystyle \Delta C=C(10+n)-C(10)=\frac{G}{10+n}-\frac{G}{10}=G\left(\frac{1}{10+n}-\frac{1}{10} \right)=-\frac{Gn}{10(10+n)}$

Hence, we find the decrease in the contribution required from each person is:

$\displaystyle \frac{Gn}{10(10+n)}$