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?

Let's determine the number of passengers aboard the bus when it left Alpaca by working through each stop step-by-step. 1. **Leaving Alpaca:** - Let \( x \) be the number of passengers initially on the bus. 2. **Arriving at Billow:** - **Passengers getting off:** Half of \( x \) leave, so \( \frac{x}{2} \) remain. - **Passengers getting on:** 6 new passengers board the bus. - **Total after Billow:** \( \frac{x}{2} + 6 \). 3. **Arriving at Comfort:** - **Passengers getting off:** Half of \( \frac{x}{2} + 6 \) leave, so \( \frac{1}{2} \left( \frac{x}{2} + 6 \right) = \frac{x}{4} + 3 \) remain. - **Passengers getting on:** 5 new passengers board the bus. - **Total after Comfort:** \( \frac{x}{4} + 3 + 5 = \frac{x}{4} + 8 \). 4. **Arriving at Down:** - All remaining passengers get off, and we know that there were 13 passengers at this point. - Therefore, \( \frac{x}{4} + 8 = 13 \). 5. **Solving for \( x \):** \[ \frac{x}{4} + 8 = 13 \\ \frac{x}{4} = 5 \\ x = 20 \] So, **20** passengers were aboard the bus when it left Alpaca. **Answer:** 20