67. A bus that runs once daily between the villages of Alpaca and Down makes only two stops between, at Billow and at Comfort. Today, the bus left Alpaca with some passengers. At Billow, one- half of the passengers got off, and 6 new ones got on. At Comfort, again one-half of the passengers got off, and, this time, 5 new ones got on. At Down, the last 13 passengers on the bus got off. How many passengers were aboard when the bus left Alpaca?
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Let's denote the initial number of passengers on the bus when it left Alpaca as \( x \). 1. After leaving Alpaca, the bus makes its first stop at Billow. Half of the passengers get off: \[ \text{Passengers left} = \frac{x}{2} \] Then, 6 new passengers get on: \[ \text{Passengers at Billow} = \frac{x}{2} + 6 \] 2. The bus then proceeds to Comfort. Again, half of the passengers get off: \[ \text{Passengers left} = \frac{1}{2} \left( \frac{x}{2} + 6 \right) = \frac{x}{4} + 3 \] Then, 5 new passengers get on: \[ \text{Passengers at Comfort} = \frac{x}{4} + 3 + 5 = \frac{x}{4} + 8 \] 3. Finally, the bus arrives at Down where the last 13 passengers get off. Therefore, we have: \[ \frac{x}{4} + 8 = 13 \] Now, solving for \( x \): \[ \frac{x}{4} + 8 = 13 \\ \frac{x}{4} = 13 - 8 \\ \frac{x}{4} = 5 \\ x = 20 \] So, the number of passengers aboard when the bus left Alpaca was \( \boxed{20} \).
