Elementary Algebra problem about calculation of profit First of all, this is an algebra problem and it is not a homework and no word tricks are intended. Here it goes:
Two partners $m$ and $s$ own a rental building.
partner $s$ owns $5/9$ of the shares.
partner $m$ owns $4/9$ of the shares.
An income of $T$ was realized at the end of the year. The partners would split the income and expenses according to their corresponding shares (as shown above).
Maintenance expenses of amount $E$ were paid during the year. To cover this amount, $s$ paid from her pocket a cash amount of B dollars with no interest as a down payment for the amount $E$. The rest of $E$ was to be paid at the year end.
At the end of the year, they wanted to calculate the due amount for each.
I want to find out how much partner (s) get, and partner (m) at the end of the year so there are no owning amounts of either sides to the other.
I tried the following 2 solutions but neither made sense to me. You help is appreciated.
Currently the focus is on the case where $B<T$ and $B<=E$.
Solution 1
Separate the borrowed amount from the expenses and distribute the income then adjust shares as appropriate:
Let $e=E-B$, i.e. $e=$ Expenses - Borrowed amount
Partner s gets: $(5T/9)$ -$(5e/9)$ + $B$, i.e. income share - expense share + down payment amount
Partner m gets: $(4T/9) - (4e/9) - B$, i.e. income share - expense share - down payment amount
Adding the two equations above:
What partners $m$ and $s$ get = $T-e=T-E+B$. Is this correct? Why I am not getting $T-E$ when I add the two equations?
Solution 2
Partner $s$ gets: $(5T/9) - (5E/9) + B$, i.e. income share - expense share + down payment amount
Partner $m$ gets: $(4T/9) - (4E/9)$, i.e. income share - expense share without subtracting the down payment since it is already calculated as part of $E$.
This does not look correct because adding the two equations above:
What partners $m$ and $s$ get = $T-E+B$ is this correct?, why I am not getting $T-E$ when I add the two equations?
EDIT: Added this other solution:
Solution 3
This is similar to Solution 2 but, since $B$ is an amount that has to be paid by m to s, the equations for shares are:
Partner $s$ gets: $(5T/9) - (5E/9) + B$, i.e. income share - expense share + down payment amount
Partner $m$ gets: $(4T/9) - (4E/9) - B$, i.e. income share - expense share - down payment amount that appears in $s$ share above. This way if we add the two equations we get $T-E$ which makes sense. However, it makes me think that m has paid for the down payment twice!
 A: If it wasn't for the down payment the division would be $\frac{5}{9}(T-E)$ for $s$ and $\frac{4}{9}(T-E)$ for $m$. So just add B to the amount due to $s$ (let's say that he take his down payment and then they procede to the division) and you have your solution. 
Note that adding the two seems to give a wrong answer (i. e. More money than the total) because you don't consider that $s$ gave those money before. When the division happen the total is in fact $T-E+B$
A: For the case at hand, where $B\leq\frac{5}{9}E$, $s$ will pay the remaining $\frac{5}{9}E-B$ of her share of the expenses, and $m$ pays his $\frac{4}{9}E$ share. They receive their entitled shares of $T$ as well (order these are done in does not matter mathematically, but might matter in real life).
If however, $B>\frac{5}{9}E$, then $s$ has already payed more than her fair share of the expenses, and is owed $B-\frac{5}{9}E$ from $m$ for the portion of $m$'s expense share that $s$ ended up covering with the down payment. In this scenario, $m$ pays the remaining $E-B$ of the expenses and pays $s$ the $B-\frac{5}{9}E$ amount (the total payout by $m$ across both remaining expenses and recompense for $s$'s previous down payment exceeding $\frac{5}{9}E$ will equal $\frac{4}{9}E$). The income $T$ is again divided according to their entitled shares, and as before, mathematically speaking this can happen before or after expenses are paid (order might matter in real life).
