Munkhjin and Dorjoo can solve all homework problems together in 6 hours. Munkhjin can solve all problems alone in 15 hours. How many hours does it take Dorjoo to solve all the problems alone?

1. Problem Description

Munkhjin and Dorjoo can solve all homework problems together in 6 hours. Munkhjin can solve all problems alone in 15 hours. How many hours does it take Dorjoo to solve all the problems alone?

2. Solution Steps

Let $T_M$ be the time Munkhjin takes to solve all the problems alone, and $T_D$ be the time Dorjoo takes to solve all the problems alone. Let $T_{M+D}$ be the time they take to solve the problems together.

We are given that $T_{M+D} = 6$ hours and $T_M = 15$ hours. We want to find $T_D$.

Let $R_M$ be the rate at which Munkhjin solves the problems, and $R_D$ be the rate at which Dorjoo solves the problems.

$R_M = \frac{1}{T_M}$ and $R_D = \frac{1}{T_D}$.

When they work together, their rates add up. So, $R_{M+D} = R_M + R_D$.

Also, $R_{M+D} = \frac{1}{T_{M+D}}$.

Therefore, $\frac{1}{T_{M+D}} = \frac{1}{T_M} + \frac{1}{T_D}$.

Plugging in the given values:

$\frac{1}{6} = \frac{1}{15} + \frac{1}{T_D}$.

We want to solve for $T_D$.

$\frac{1}{T_D} = \frac{1}{6} - \frac{1}{15}$.

$\frac{1}{T_D} = \frac{5}{30} - \frac{2}{30}$.

$\frac{1}{T_D} = \frac{3}{30}$.

$\frac{1}{T_D} = \frac{1}{10}$.

$T_D = 10$.

3. Final Answer

10