DIRECTIONS for questions: Read the information given below and answer the question that follows.

A train travels from Station A to Station E, passing through stations B, C, and D, in that order. The train has a seating capacity of 200. A ticket may be booked from any station to any other station ahead on the route, but not to any earlier station.

A ticket from one station to another reserves one seat on every intermediate segment of the route. For example, a ticket from B to E reserves a seat in the intermediate segments B – C, C – D, and D – E.

The occupancy factor for a segment is the total number of seats reserved in the segment as a percentage of the seating capacity. The total number of seats reserved for any segment cannot exceed 200.

The following information is known.

1. Segment C – D had an occupancy factor of 95%. Only segment B – C had a higher occupancy factor.

2. Exactly 40 tickets were booked from B to C and 30 tickets were booked from B to E.

3. Among the seats reserved on segment D – E, exactly four-sevenths were from stations before C.

4. The number of tickets booked from A to C was equal to that booked from A to E, and it was higher than that from B to E.

5. No tickets were booked from A to B, from B to D and from D to E.

6. The number of tickets booked for any segment was a multiple of 10.

How many tickets were booked from Station A to Station E?

The correct answer is 50.

Let's break down the given information to find the number of tickets booked from Station A to Station E step-by-step. We know there are four segments along the route: A-B, B-C, C-D, and D-E. The maximum seating capacity for any segment is 200, and all ticket bookings are multiples of 10.

From condition 1, the segment C-D had an occupancy factor of 95%.

$0.95\times 200=190$

The condition also states that only segment B-C had a higher occupancy factor. Since total seat bookings must be multiples of 10 and cannot exceed the 200 maximum capacity, the number of reserved seats for segment B-C must be exactly 200.

Let's define the unknown ticket bookings. Based on condition 4, the number of tickets booked from A to C is equal to the number of tickets booked from A to E. Let this quantity be $x$. Condition 4 also states that this quantity is strictly greater than the tickets booked from B to E. From condition 2, exactly 30 tickets were booked from B to E. Therefore, we know that $x$ must be greater than 30.

$x>30$

Let the number of tickets booked from A to D be $y$.

Now, let's analyze the total number of reserved seats on segment B-C. This segment includes all tickets that start at or before B and end at or after C. According to condition 5, no tickets were booked from A to B. Thus, the valid tickets passing through segment B-C are: A to C, A to D, A to E, B to C, and B to E. From condition 2, there were 40 tickets from B to C. Adding these up to match the segment B-C capacity of 200 gives us our first equation:

$x+y+x+40+30=200$

$2x+y+70=200$

$2x+y=130$

Because $x$ and $y$ must be multiples of 10, and $x$ is strictly greater than 30, the possible values for $x$ are 40, 50, and 60. To find the exact value, we use condition 3, which states that among the seats reserved on segment D-E, exactly four-sevenths were from stations before C.

The tickets passing through segment D-E that come from stations before C are A to E ($x$) and B to E (30). This gives us a total of $x+30$ tickets. Since this sum represents exactly four-sevenths of the total tickets on segment D-E, the quantity $x+30$ must be evenly divisible by 4 (a multiple of 4).

Let's test our possible values for $x$:

If $x=40$, then $40+30=70$ (not a multiple of 4).
If $x=50$, then $50+30=80$ (is a multiple of 4).
If $x=60$, then $60+30=90$ (not a multiple of 4).

Thus, $x$ must be exactly 50. Since $x$ represents the number of tickets booked from Station A to Station E, the correct answer is 50.