Elementary Algebra - Distribute value amongst a group of people I appreciate your help with this question. I'm ready to respond to any question you might have to help solve this problem. If you have an alternate solution it will be accepted as an answer (if explained). If you spot all the errors (in case 1 or more exist), it will be accepted as an answer. If you confirm that this correct after checking it I will up vote your answer.
A company of total value $v$ is distributed amongst 5 people (I refer to their ownership amounts as $x$,$k$,$e$,$m$,$d$ and $s$) according to the following percentages (shares) of the total amount value $v$:
$x_0=\frac{4}{13}v$
$k_0=\frac{4}{13}v$
$e_0=\frac{2}{13}v$
$m_0=\frac{2}{13}v$
$s_0=\frac{1}{13}v$
Person $x$ decided to leave the company and asked that his share to be distributed amongst the remaining people plus 1 new person $d$. The distribution rule is such that person $s$ would take half of ANY of the persons $k$ or $e$ or $m$ or $d$.
The question:
For each person involved, find the share (percentage of the total amount) under the above mentioned rules.
Note: This is a real-life problem and not a homework - There are no word tricks here or puzzles. Your verification or correction is very much appreciated. Also, if you have a simpler approach, it would be appreciated. Please explain each step clearly if you would, so that I can understand - Thx.
My answer:
Step1: Calculate each individual share of x's share as follows:
share for x will be distributed as:
$x_0=k+e+m++d+s$
given that $k=e=m=d$ and that $s=0.5k$
$x_0=4k+0.5k$
$k=\frac{2x_0}{9}$
Since x's share was $x_0=\frac{4}{13}v$
$k=\frac {2}{9} * \frac{4v}{13}$
$k=\frac {8v}{13*9}$
$k=\frac {8v}{117}$ and the same value applies to $m$,$e$,$d$ ----- (EQ 1)
$s=\frac {4}{13} * \frac{v}{9}=\frac{4v}{117}$
Step 2: Calculate each individual absolute (final) share value:
$x's shares are to introduced individually and person d's shares to be calculated.
$k=\frac{4v}{13} + \frac {8v}{117} = \frac{572v}{1521}$
$e=\frac{2v}{13} + \frac {8v}{117} = \frac{338v}{1521}$
$m=\frac{2v}{13} + \frac {8v}{117} =\frac{338v}{1521}$
$s=\frac{v}{13} + \frac{4v}{117} = \frac{169v}{1521}$
The share for $d$ would be calculated as: total percentage value ($v$) - sum of all share percentages above.
$d =  1-(\frac{572v}{1521} + \frac{338v}{1521} + \frac{338v}{1521}+\frac{169v}{1521})$
$d=\frac{104v}{1521}$
Same result for $d$ could be obtained from (EQ 1):
$d=\frac {8v}{117}$
What do you think? Is this correct? Thx.
Edit:
I have corrected the statement and renamed initial shares with sub zero to  have more accurate formulation.
 A: Yes your answer is correct. The solution is as follows:
A company with $v$ shares of 5 people stands as
$$v=\frac{4}{13}v(x's share) + \frac{4}{13}v(k's share) + \frac{2}{13}v(e's share)+\frac{2}{13}v(m's share)+\frac{1}{13}v(s' share)$$
If 'x' leaves the company and distributes his share amongst k,e,m,s and a new member d such that everyone gets double the shares that s gets then we can say that everyone gets two-ninths of x's share while  s gets one-ninths.
Taking x's share and dividing it into ninths we get:
$$\frac{4}{13}v\div9 = \frac{4}{9*13}=\frac{4}{117}v$$
Everyone except for s gets$\frac{8}{117}v$ extra shares while s gets $\frac{4}{117}v$ extra shares.   So the company shares stand distributed as follows:
$$v=\frac{8}{117}v[d's \ share] + (\frac{4}{13}v+\frac{8}{117})[k's\  \ share] + (\frac{2}{13}v+\frac{8}{117}v)[e\ 's \ share]+(\frac{2}{13}v+\frac{8}{117}v)[m's\  share]+(\frac{1}{13}v+\frac{4}{117}v)[s'\   share]$$
A: Your maths is correct, but you have added unecessary work with:
$$k=\frac{4v}{13} + \frac {8v}{117} = \frac{572v}{1521}$$
$117=13\times9$, and so you do not need to take the denominator to $1521=117\times13$, you only need to use $117$, so:
$$k=\frac{4v}{13} + \frac {8v}{117} = \frac{36v}{117} + \frac {8v}{117} = \frac{44v}{117}$$
and continue as previously.
