QUESTION

# A railway half-ticket costs half the full fare but the reservation charges are the same on a half-ticket as on a full ticket. One reserved full first class ticket from station A to station B costs Rs. 2125. Also, one reserved a full first class ticket and a half first class ticket from A to B costs Rs. 3200. Find the full fare from station A to station B and also the reservation charges for a ticket.

Hint: Assume that the full fare of the ticket is Rs. $x$ and reservation charges on a ticket is Rs. $y$. Form two sets of linear equations in two variables with the help of provided conditions and solve them to get the answer.

Let us assume that the full fare of the ticket is Rs. $x$ and reservation charges on a ticket is Rs. $y$. It is given that, half-ticket costs half the full fare and the reservation charges are the same on a half-ticket as on a full ticket. Therefore,

Cost of a full ticket $=x+y$,

And, Cost of a half ticket $=\dfrac{x}{2}+y$.

Now, one reserved full first class ticket from station A to station B costs Rs. 2125. Therefore, mathematically,

$x+y=2125........................(i)$

Also, one reserved full first class ticket and a half first class ticket from A to B costs Rs. 3200. Therefore, mathematically,

\begin{align} & x+y+\dfrac{x}{2}+y=3200 \\ & \dfrac{3x}{2}+2y=3200...................(ii) \\ \end{align}

Now, multiplying equation (i) by 2 and subtracting it from equation (ii) we get,

\begin{align} & \dfrac{3x}{2}+2y-(2x+2y)=3200-2\times 2125 \\ & \dfrac{3x}{2}-2x=3200-4250 \\ & \dfrac{-x}{2}=-1050 \\ \end{align}

Cancelling the minus sign and cross multiplying, we get,

\begin{align} & x=2\times 1050 \\ & x=2100 \\ \end{align}

Substituting the value of $x$ in equation (i) we get,

\begin{align} & y=2125-2100 \\ & y=25 \\ \end{align}

Hence, full fare and reservation charge for a ticket from station A to B are Rs. 2100 and Rs. 25 respectively.

Note: One may get confused in understanding such a long question. The main problem here is to form linear equations with the help of provided information. So, we have to go thoroughly through the question.