Solving the Bus Stop Enigma: A Logical Mathematical Riddle

When we encounter mathematical problems that involve a bus stop and bus passengers, they often require a careful analysis of given data points and logical deduction. In this case, we are presented with a riddle that asks us to determine the initial number of people at a bus stop. Let's break down the scenario and solve it step by step.

The Scenario and Key Data Points

The riddle presents a situation where people get on and off a bus at a bus stop. Specifically, 19 people board a bus, and 17 people come to the bus stop. Following this, there are 63 people at the bus stop in total. We need to determine how many people were at the bus stop initially.

Analysis and Solution

Let's denote the initial number of people at the bus stop as P.

Step 1: Additionally, we know that 19 people board the bus, and 17 people come to the bus stop. After these events, there are 63 people at the bus stop. Step 2: The equation can be set up as follows: Step 3:

P - passengers who got off   passengers who boarded  total number of people at the bus stop
P - 17   19  63

Step 4: Further simplifying the equation: Step 5:

P   2  63
P  63 - 2
P  61

This means that the initial number of people at the bus stop was 61. Let's explore some alternative methods that might also lead us to the same conclusion.

Alternative Solutions

Here are a few other ways to solve the problem:

Alternative Method 1

In another approach, we are told that 11 new passengers boarded the bus, and this, combined with the remaining passengers, made 20 in total. We can use this information as well:

Step 1: Let the initial number of passengers be P. When 11 people boarded, the total became 20. Initially, there were P - 11 passengers on the bus. Step 2: The equation would be: Step 3:

P - 11   11  20

Step 4: This simplifies to: Step 5:

P  20

Step 6: The problem states that 19 people got on at the bus stop, and 17 people left. Therefore, we can set up the equation: Step 7:

P - 17   19  63

Step 8: Simplifying further: Step 9:

P   2  63

Step 10:

P  61

Alternative Method 2

Another approach involves using a logical deduction:

Step 1: We assume the original number of people on the bus as 2x. Step 2: When x people got off, the remaining people on the bus were x. Step 3: Then, 11 more people got on, making the total number of people on the bus 20. Step 4: Therefore: Step 5:

x   11  20

Step 6:

x  20 - 11

Step 7:

x  9

Step 8: Since x represents half of the original number of passengers, the initial number of people on the bus would be: Step 9:

2x  2 * 9  18

Step 10: We then need to calculate the initial number of people at the bus stop, adding the 17 who came to the bus stop: Step 11:

P  18   17  35

However, this approach does not match the given solution, indicating potential confusion with the number of people getting off or on the bus.

Consideration of Additional Information

A critical point to note in the problem is the inclusion of the driver. Often, in such scenarios, the driver is not considered a passenger but part of the total number of people counted at the bus stop. This distinction should not be overlooked, as it slightly affects the equation.

Conclusion

After analyzing various methods, the most accurate solution, considering all data points, is that the initial number of people at the bus stop was 61. This solution is derived by accounting for the passengers who boarded and those who left the bus, as well as those who arrived at the bus stop.

Key Takeaways

The solution involves logical deduction and careful accounting of people's movements at the bus stop. Include the driver in the total number of people when counting the initial population at the bus stop. Always verify the assumptions to ensure the correct answer.