24. Saba, Rayan and Derin are cooperating to complete a task. They each work at a constant rate independent of whoever else is working on the task. When all three work together, it takes 5 minutes to complete the task. When Saba is working with Derin, the task takes 7 minutes to complete. When Rayan is working with Derin, the task takes 15 minutes to complete.
How many minutes does it take for Derin to complete the task on his own?
A 21
B 28
C 35
D 48
E 105
Solution
E

Suppose that it takes Saba, Rayan and Derin, working on their own, $s, r$ and $d$ minutes, respectively, to complete the task.
Then in 1 minute Saba completes $\frac{1}{s}$ of the task, Rayan completes $\frac{1}{r}$ of the task, and Derin completes $\frac{1}{d}$ of the task. Hence when all three are working together in 1 minute they complete $\frac{1}{s}+\frac{1}{r}+\frac{1}{d}$ of the task. Since it takes them 5 minutes to complete the task when they all work together,
$$
\frac{1}{s}+\frac{1}{r}+\frac{1}{d}=\frac{1}{5}
$$

Similarly, as it takes Saba working with Derin 7 minutes to complete the task,
$$
\frac{1}{s}+\frac{1}{d}=\frac{1}{7}
$$

Likewise, as it takes Rayan working with Derin 15 minutes to complete the task,
$$
\frac{1}{r}+\frac{1}{d}=\frac{1}{15} .
$$

We therefore have
$$
\begin{aligned}
\frac{1}{d} & =\left(\frac{1}{s}+\frac{1}{d}\right)+\left(\frac{1}{r}+\frac{1}{d}\right)-\left(\frac{1}{s}+\frac{1}{r}+\frac{1}{d}\right) \\
& =\frac{1}{7}+\frac{1}{15}-\frac{1}{5} \\
& =\frac{15}{105}+\frac{7}{105}-\frac{21}{105} \\
& =\frac{1}{105}
\end{aligned}
$$

Therefore it takes Derin 105 minutes to complete the task on his own.