Some passengers travel in a train `1/3` rd of the passengers stepped down on the first station and 560 passengers entered the train, half of the total passengers stepped down on the second station and 24 more passengers entered in the train and on the third station all 496 passengers stepped down. How many passengers were travelling in the train in the beginning?

Options

A. 529
B. 576
C. 626
D. 625

Answer

To solve the problem step by step, let's denote the initial number of passengers in the train as \( x \).

### Step 1: Passengers stepping down at the first station
At the first station, \( \frac{1}{3} \) of the passengers stepped down. Therefore, the number of passengers who stepped down is:
\[ \text{Passengers stepped down} = \frac{1}{3}x \]
The remaining passengers after the first station will be:
\[ \text{Remaining passengers} = x - \frac{1}{3}x = \frac{2}{3}x \]

### Step 2: Passengers entering the train
After \( 560 \) passengers entered the train, the total number of passengers now becomes:
\[ \text{Total passengers after first station} = \frac{2}{3}x + 560 \]

### Step 3: Passengers stepping down at the second station
At the second station, half of the total passengers stepped down. The total number of passengers before the second station is \( \frac{2}{3}x + 560 \). Therefore, the number of passengers who stepped down at the second station is:
\[ \text{Passengers stepped down} = \frac{1}{2} \left( \frac{2}{3}x + 560 \right) \]
The remaining passengers after the second station will be:
\[ \text{Remaining passengers} = \left( \frac{2}{3}x + 560 \right) - \frac{1}{2} \left( \frac{2}{3}x + 560 \right) \]
Calculating this gives:
\[ \text{Remaining passengers} = \frac{1}{2} \left( \frac{2}{3}x + 560 \right) = \frac{1}{3}x + 280 \]

### Step 4: Passengers entering the train again
After \( 24 \) more passengers entered the train, the total number of passengers now becomes:
\[ \text{Total passengers after second station} = \left( \frac{1}{3}x + 280 \right) + 24 = \frac{1}{3}x + 304 \]

### Step 5: Passengers stepping down at the third station
At the third station, all \( 496 \) passengers stepped down. Therefore, we set up the equation:
\[ \frac{1}{3}x + 304 = 496 \]

### Step 6: Solving for \( x \)
Now we can solve for \( x \):
\[ \frac{1}{3}x + 304 = 496 \]
Subtract \( 304 \) from both sides:
\[ \frac{1}{3}x = 496 - 304 \]
\[ \frac{1}{3}x = 192 \]
Now, multiply both sides by \( 3 \):
\[ x = 192 \times 3 = 576 \]

### Conclusion
The initial number of passengers in the train was \( 576 \).

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