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.

What is the difference between the number of tickets booked to Station C and the number of tickets booked to Station D?

The correct answer is 40.

Let's define the number of tickets booked from station $X$ to station $Y$ as $T_{XY}$.

The train capacity is 200 seats. We are given that segment C-D had an occupancy factor of 95%, which means $0.95\times 200=190$ seats were reserved in this segment.

We also know that only segment B-C had a higher occupancy factor than C-D. Since all ticket amounts are multiples of 10, the occupancy of segment B-C must be exactly 100%, which corresponds to 200 seats.

From the provided conditions, we know certain ticket values directly:

$T_{BC}=40$
$T_{BE}=30$
$T_{AB}=0$
$T_{BD}=0$
$T_{DE}=0$

The passengers present on segment B-C are all those individuals who started their journey at or before station B (which includes stations A and B) and are traveling to station C or beyond (which includes stations C, D, and E). Thus, the sum of tickets comprising segment B-C is:

$T_{AC}+T_{AD}+T_{AE}+T_{BC}+T_{BD}+T_{BE}=200$

Substitute the known values into the equation:

$T_{AC}+T_{AD}+T_{AE}+40+0+30=200$

$T_{AC}+T_{AD}+T_{AE}=130$

Similarly, the passengers taking up seats on segment D-E are those traveling from any station prior to D (stations A, B, and C) and disembarking at station E:

$\text{Total on D-E}=T_{AE}+T_{BE}+T_{CE}+T_{DE}$

Substitute the known values for the D-E segment:

$\text{Total on D-E}=T_{AE}+30+T_{CE}+0$

We are told that exactly $\frac{4}{7}$ of these D-E reservations were from stations before C (which means tickets originating from A and B, ending in E). The number of these tickets is $T_{AE}+30$. We can set up the proportion:

$T_{AE}+30=\frac{4}{7}(T_{AE}+30+T_{CE})$

For $T_{AE}+30$ to be a fraction of $\frac{4}{7}$ of a whole integer number of passengers, $T_{AE}+30$ must be a multiple of 4.

We also know that $T_{AC}=T_{AE}$ and this value is strictly greater than $T_{BE}$ (which is 30). Since all tickets are multiples of 10, the possible values for $T_{AE}$ are 40, 50, 60, and so on.

Let's test these possible values to determine which one makes $T_{AE}+30$ perfectly divisible by 4:

If $T_{AE}=40$, then $40+30=70$ (Not divisible by 4).

If $T_{AE}=50$, then $50+30=80$ (Divisible by 4).

If $T_{AE}=60$, then $60+30=90$ (Not divisible by 4).

Thus, it must be that $T_{AE}=50$. This also consequently means $T_{AC}=50$.

We can now find $T_{CE}$ using the 4/7 proportion established earlier:

$80=\frac{4}{7}(80+T_{CE})$

$80+T_{CE}=140$

$T_{CE}=60$

Next, we find $T_{AD}$ using our earlier segment B-C equation:

$T_{AC}+T_{AD}+T_{AE}=130$

$50+T_{AD}+50=130$

$T_{AD}=30$

Finally, we analyze segment C-D to find $T_{CD}$. The total tickets reserving seats on segment C-D was 190. Passengers on this segment started their journey at or before C, and are ending their journey at D or E:

$T_{AD}+T_{AE}+T_{BD}+T_{BE}+T_{CD}+T_{CE}=190$

Substitute the known values:

$30+50+0+30+T_{CD}+60=190$

$170+T_{CD}=190$

$T_{CD}=20$

Now, we have all the required ticket combinations to calculate the final question:

The total number of tickets booked to Station C is:

$T_{AC}+T_{BC}=50+40=90$

The total number of tickets booked to Station D is:

$T_{AD}+T_{BD}+T_{CD}=30+0+20=50$

The absolute difference between the two destinations is:

$90-50=40$