A baker wants to use all his flour to make bread. Working alone, he could finish the flour in 6 hours. With an assistant, the same could be done in 2 hours. The baker started working alone and, after some time, tired, he called his assistant and thus, after 150 minutes the flour ran out. For how many minutes did the baker work alone?
(a) 30
(b) 35
(c) 40
(d) 45
(e) 50
The correct option is (d).

Let $x$ be the amount of flour, in kilograms, that the baker has. Working alone, he would use $x / 6$ kilograms of flour per hour. Working with his assistant, they would use $x / 2$ kilograms of flour per hour. Let $t$ be the time, in hours, that the baker worked alone. Since the flour runs out in 150 minutes, or 2.5 hours (2 hours and 30 minutes), the time he worked with his assistant was $2.5 - t$ hours. Therefore, the amount of flour used during the time the baker worked alone was $(x / 6) \times t$ and the amount used during the time the baker worked with his assistant was $(x / 2) \times (2.5 - t)$. Since

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-051.jpg?height=166&width=1130&top_left_y=1639&top_left_x=543)

we have $x = \frac{1}{6} x t + \frac{1}{2} x (2.5 - t)$. The initial amount $x$ of flour the baker had was non-zero. Therefore, we can divide both sides by $x$, yielding $1 = \frac{1}{6} t + \frac{1}{2} (2.5 - t)$, and thus $t = 0.75$ hours, or the baker worked alone for 45 minutes.