Princess Sarah and Becky are peeling potatoes. Given the same number $P$ of potatoes, Princess Sarah can finish peeling them alone $2$ hours and $36$ minutes before Becky can, alone. If Princess Sarah is given $P + \frac{P}{2}$ potatoes while Becky is given $\frac{9P}{10}$ potatoes, they finish at the same time if they both work alone. Both girls work at constant rates. If Becky helps Princess Sarah peel $2015$ potatoes, how many minutes does Princess Sarah save?
I first set up these equations
$\frac{Potato}{T_{Becky}}$ = Rate of Becky
$\frac{Potato}{T_{Becky}-156}$ = Rate of Princess Sarah
$\frac{\frac{9Potato}{10}}{T}$ = Rate of Becky
$\frac{\frac{3Potato}{2}}{T}$ = Rate of Princess Sarah
I was then able to solve for $T_{Becky}$ which is $390$ mins. Which means $T_{Sarah} = 234$ mins. As a result, peeling potatoes when Becky helps Sarah takes $\frac{585}{4}$ mins. while Sarah takes $234$ mins. and their difference is $\frac{351}{4}$ mins., which is far from the supposedly correct answer $26$ mins. Are there any mistakes in my solution?
You have found the time saved by Sarah if Becky helps her peel $P$ potatoes. As you have found, Becky peels $P$ potatoes in $390$ minutes; Sarah peels $P$ potatoes in $\frac 3 5$ of this time, which is $234$ minutes; working together they will peel $P$ potatoes in $234 \times \frac 5 8$ minutes, and so they save $234 \times \frac 3 8 = \frac {351} 4$ minutes compared to Sarah working alone.
What you do not know is how much time is saved when peeling $2015$ potatoes. And since you have four equations with five unknowns, I don't see how you can solve this problem with the information that is given.