# 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

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?

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!

• If I am understanding this right: $m$ and $s$ want to divide both earnings and expenses according to their share of ownership? If that is the case, then you want the end result to be that: $m$ earned $\frac{5}{9}$'s of $T$ and spent $\frac{5}{9}$'s of $E$ on expenses, and $s$ earned $\frac{4}{9}$'s of $T$ and spent $\frac{4}{9}$'s of $E$. Is that correct? Apr 11, 2016 at 20:54
• @JustinBenfield, you are correct, the main confusion comes from how to treat the down payment that (s) has made which =$B$ and is part of $E$. Apr 11, 2016 at 21:32
• There is a case breakdown depending on whether or not $B>\frac{5}{9}E$. If $B\leq \frac{5}{9}E$ then, it's simply a matter of subtracting $\frac{5}{9}E-B=s$'s end of year expenses. Apr 11, 2016 at 21:37
• For the case of $B>\frac{5}{9}E$, $s$ has effectively paid more than his/her fair share of the expenses. So $m$ needs to pay $s$ the remaining amount of his $\frac{4}{9}E$ that exceeds their end of year expenses. Apr 11, 2016 at 21:41
• @JustinBenfield, while the current case is the simple case, your point is quite valid and It would be nice to elaborate. I have made an edit saying that the current problem is focused on the case where $B<T$ and $B<=E$. Apr 12, 2016 at 21:37

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 $T-E+B$

• Thank you for your answer. It looks like you consider solution 2 is correct. Do you think that solution 1 is wrong? Apr 9, 2016 at 23:37
• I expected when I add the 2 equations to get $T-E$, that is the difference between the total income and total expenses, but I am not getting that. Apr 10, 2016 at 0:40
• I made an edit, added a new solution, kindly check, thx. Apr 10, 2016 at 1:04
• Given that I'm not a native English speaker I may misunderstood. But it seems to me that the B payment must be divided between m and s based on their shares Apr 10, 2016 at 9:47
• It seems to me that B is a part of E so after the down payment the income from the house is T-(E-B) =T-E+B, let's call this value V. They have to divide V in relation to the shares, so 5V/9 to s and 4V/9 to m. But s payed B all by himself while B (as every income and cost of the house) should have been divided in relation to the share. So m has to give his share of B which is 4B/9 to s and you get exactly the second solution. Apr 10, 2016 at 10:21

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).

• Thx. for your post, do you agree that Solution 2 is correct for $B\leq\frac{5}{9}E$? Apr 13, 2016 at 23:02
• Solution 2 is correct, the reason you're not getting $T-E$ when you add the two equations is that you haven't modeled the fact that $s$ is already out $B$ dollars for the down payment (and so they are sitting at $-B$ dollars net before the end of year income and expenses are handled). Apr 13, 2016 at 23:06
• Thank you for your explanation. Apr 13, 2016 at 23:12