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 first class ticket from the station A to B costs Rs. 2530. Also, one reserved first class ticket and one reserved first class half ticket from A to B costs Rs. 3810. Find the full first class fare from station A to B.
(a) Rs. 1280
(b) Rs. 2500
(c) Rs. 3570
(d) Rs. 2250

Answer 1

Hint: Form the relations between the fare of first class full ticket and half ticket and reservation charges according to the conditions given in question. After forming the above relations, we will find two simultaneous equations in terms of first class full ticket and reservation charges. Solve them.

Complete step-by-step answer:
Let us assume the fare of a first class half ticket as “h” and the fare of a first class full ticket as “f”. Now, we are writing the first condition from the question that half ticket costs half the full fare as:
$\text{h = }\dfrac{f}{2}$ ………..Eq. (1)
As reserved charges are the same for both the half and full ticket so let us assume reserved charges be “r”.
We are writing the given fare of reserved full first class ticket as follows:
f + r = 2530…………Eq. (2)
The fare of a first class half ticket can be written as h + r.
Now, we are writing the given combined fare of reserved first class full and half ticket as follows:
f + r + h + r = 3810
f + h + 2r = 3810
Using the relation between h and f which is given in Eq. (1) in the above equation we get,
$\begin{align}
  & f+\dfrac{f}{2}+2r=3810 \\
 & \Rightarrow \dfrac{3f}{2}+2r=3810 \\
 & \Rightarrow \dfrac{3f+4r}{2}=3810 \\
 & \Rightarrow 3f+4r=3810(2) \\
 & \Rightarrow 3f+4r=7620 \\
\end{align}$

3f + 4r = 7620……….Eq. (3)
Now, Eq. (1) and Eq. (3) are simultaneous equations.
f + r = 2530
3f + 4r = 7620
Solving the above equations by elimination method
Multiply Eq. (1) by 3 then subtract this multiplication result from Eq. (3).
$\begin{align}
  3f+4r=7620 & \\
  -\dfrac{3f+3r=7590}{r=30} & \\
\end{align}$
Substituting this value of r in Eq. (1) we get,
f + r = 2530
$\Rightarrow $ f + 30 = 2530
$\Rightarrow $f = 2500
Hence, the full fare of a first class ticket is Rs. 2500 and the correct answer is (b).

Note: Two points to be taken care of: first, Rs. 3810 is the combined cost of reserved first class full and half ticket and another one is we asked to find the basic full fare of first class reserved ticket excluding the reservation charges. And one more the thing, while solving the simultaneous equations, we can use the substitution method as well.