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
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.