When a train started from station A, there were certain number of passengers in it. On the next station, `(1)/(11)` of these passengers got down and 20% of these got down passengers got into the train. Now if the number of passengers in the train is 510, how many passengers were there at station A?

Options

A. 600
B. 580
C. 560
D. 550

Answer

To solve the problem step by step, we will denote the number of passengers at station A as \( X \).

### Step 1: Determine the number of passengers who got down
When the train reaches the next station, \( \frac{1}{11} \) of the passengers got down. Therefore, the number of passengers who got down is:
\[ \text{Passengers got down} = \frac{1}{11} \times X = \frac{X}{11} \]

### Step 2: Calculate the remaining passengers after some got down
After \( \frac{X}{11} \) passengers got down, the number of passengers remaining in the train is:
\[ \text{Remaining passengers} = X - \frac{X}{11} = \frac{11X}{11} - \frac{X}{11} = \frac{10X}{11} \]

### Step 3: Determine the number of new passengers who got in
Next, 20% of the passengers who got down (which is \( \frac{X}{11} \)) got into the train. The number of new passengers who got in is:
\[ \text{New passengers} = 20\% \times \frac{X}{11} = \frac{20}{100} \times \frac{X}{11} = \frac{X}{55} \]

### Step 4: Calculate the total number of passengers after new passengers got in
Now, the total number of passengers in the train after some got down and new ones got in is:
\[ \text{Total passengers} = \text{Remaining passengers} + \text{New passengers} = \frac{10X}{11} + \frac{X}{55} \]

### Step 5: Find a common denominator to combine the fractions
To combine these fractions, we need a common denominator. The least common multiple of 11 and 55 is 55. We can rewrite \( \frac{10X}{11} \) as:
\[ \frac{10X}{11} = \frac{10X \times 5}{11 \times 5} = \frac{50X}{55} \]
Now we can add the two fractions:
\[ \text{Total passengers} = \frac{50X}{55} + \frac{X}{55} = \frac{50X + X}{55} = \frac{51X}{55} \]

### Step 6: Set up the equation based on the information given
According to the problem, the total number of passengers in the train is 510. Therefore, we can set up the equation:
\[ \frac{51X}{55} = 510 \]

### Step 7: Solve for \( X \)
To solve for \( X \), we can multiply both sides by 55:
\[ 51X = 510 \times 55 \]
Calculating the right side:
\[ 510 \times 55 = 28050 \]
So we have:
\[ 51X = 28050 \]
Now, divide both sides by 51:
\[ X = \frac{28050}{51} = 550 \]

### Conclusion
The number of passengers at station A is \( \boxed{550} \).

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