A bus left with some definite number of passengers. At the first stop, half the passengers left the bus and 35 boarded the bus. At the second stop `(1)/(5)` th of the passengers left and 40 boarded the bus. Then, the bus moved with 80 passengers towards its destination without stopping anywhere. How many passengers were there originally?

Options

A. 25
B. 30
C. 40
D. 50

Explanation

To solve the problem step by step, we will define the original number of passengers as \( x \).

### Step 1: First Stop
At the first stop:
- Half of the passengers left the bus: \( \frac{x}{2} \)
- Therefore, the remaining passengers after half left: \( x - \frac{x}{2} = \frac{x}{2} \)
- Then, 35 passengers boarded the bus.

So, the total number of passengers after the first stop becomes:
\[ \text{Passengers after first stop} = \frac{x}{2} + 35 \]

### Step 2: Second Stop
At the second stop:
- One-fifth of the passengers left the bus: \( \frac{1}{5} \left(\frac{x}{2} + 35\right) \)
- The remaining passengers after one-fifth left:
\[ \text{Remaining passengers} = \left(\frac{x}{2} + 35\right) - \frac{1}{5} \left(\frac{x}{2} + 35\right) \]
This can be simplified as:
\[ = \left(\frac{4}{5}\right) \left(\frac{x}{2} + 35\right) \]
- Then, 40 passengers boarded the bus.

So, the total number of passengers after the second stop becomes:
\[ \text{Passengers after second stop} = \left(\frac{4}{5}\right) \left(\frac{x}{2} + 35\right) + 40 \]

### Step 3: Final Passengers
According to the problem, after the second stop, the bus moved with 80 passengers:
\[ \left(\frac{4}{5}\right) \left(\frac{x}{2} + 35\right) + 40 = 80 \]

### Step 4: Solve the Equation
Now we will solve the equation:
1. Start with:
\[ \left(\frac{4}{5}\right) \left(\frac{x}{2} + 35\right) + 40 = 80 \]
2. Subtract 40 from both sides:
\[ \left(\frac{4}{5}\right) \left(\frac{x}{2} + 35\right) = 40 \]
3. Multiply both sides by \( \frac{5}{4} \):
\[ \frac{x}{2} + 35 = 50 \]
4. Subtract 35 from both sides:
\[ \frac{x}{2} = 15 \]
5. Multiply both sides by 2:
\[ x = 30 \]

### Conclusion
The original number of passengers was \( \boxed{30} \).