Question & Answer
QUESTION

A half-ticket issued by railways costs half the full fare but the reservation charge is the same on half-ticket as on full-ticket. One full reserved first-class ticket for a journey between two stations costs Rs.362 and one full and one half reserved first class ticket costs Rs.554. Find the reservation charge.

ANSWER Verified Verified
Hint: First of all, consider the ticket fare and reservation charge for a full-ticket as variables. Then find the ticket fare for Half-ticket. Then equal the sum of total fares for one full-ticket and one- half-ticket to Rs.554 to get the charge of reservation.

Complete step-by-step answer:
Let \[x\] be the fare of a full ticket and \[y\] be the charge of reservation.
As one full reserved first-class ticket for a journey between two stations costs Rs.362, we have
\[x + y = 362....................................\left( 1 \right)\]
Given half-ticket costs half the cost of full fare, but the reservation charge is the same as a full ticket.
So, ticket fare for half-ticket \[ = \dfrac{x}{2}\]
And reservation charge for half ticket \[ = y\]
Thus, the total fare for reserved first class \[ = \dfrac{x}{2} + y..........................................\left( 2 \right)\]
Given the fare for one full and one half reserved first class ticket costs Rs.554
So, this fare is equal to the sum of fares of a hall-ticket and full-ticket.
From equations (1), (2) and the above data, we have
\[
   \Rightarrow x + y + \dfrac{x}{2} + y = 554 \\
   \Rightarrow 2x + 2y + x + 2y = 2\left( {554} \right) \\
   \Rightarrow 3x + 4y = 1108 \\
  \therefore x = \dfrac{{1108 - 4y}}{3}..............................................\left( 3 \right) \\
\]
Substituting equation (3) in (1), we have
\[
   \Rightarrow \dfrac{{1108 - 4y}}{3} + y = 362 \\
   \Rightarrow \dfrac{{1108 - 4y + 3y}}{3} = 362 \\
   \Rightarrow 1108 - y = 3\left( {362} \right) = 1086 \\
   \Rightarrow 1108 - 1086 = y \\
  \therefore y = 22 \\
\]
Thus, the reservation charge is Rs.22.

Note: The charge of reservation and ticket fare cannot be negative. The total fare for one half-ticket must be less than the total fare for one full-ticket. The full fare for a full ticket is 362 – 22 = Rs.340.