Question

A railway half ticket cost $50\% $ of the full fare and the reservation charges are the same on the half ticket as on the full ticket. The cost of reserved first-class tickets from Delhi to Bangalore is Rs$2325$ and that of the one full and one half reserved first-class tickets is Rs$3532$. Find the first-class fare from Delhi to Bangalore and the reservation charges separately.

Answer 1

Hint: In the given question we have to find the full fare and reservation charges from Delhi to Bangalore. For this, we will assume that full fare is $a$, and reservation charges are$y$. Then we will go through according to the question and make the equation accordingly. Thus we will get two equations in the form of $a\& b$, after that solve these equations and find the value of $a\& b$, where a is the full fare from Delhi to Bangalore and then we will put the value of $a$ in the desired equation thus we will get the reservation charges respectively.

Complete step by step Answer:

Let the full fare $ = a$
And
Reservation charges $ = b$
Now
According to the question:
Case 1: One reservation first class full ticket cost is$ = 2325$
We get:
$a + b = 2325..........\left( 1 \right)$
Or
$b = 2325 - a$
Case2: one full and one half reserved first class ticket cost is$ = 3532$
We get:
\[
  a + a{\text{ of }}50\% + 2b = 3532 \\
   \Rightarrow a + \dfrac{a}{2} + 2b = 3532 \\
   \Rightarrow \dfrac{{2a + a}}{2} + 2b = 3532 \\
   \Rightarrow \dfrac{{3a}}{2} + 2b = 3532 \\
   \Rightarrow \dfrac{{3a + 4b}}{2} = 3532 \\
   \Rightarrow 3a + 4b = 7064........\left( 2 \right) \\
 \]
Now substitute $b = 2325 - a$in equation 2
We get:
$
  3a + 4b = 7064 \\
   \Rightarrow 3a + 4\left( {2325 - a} \right) = 7064 \\
   \Rightarrow 3a + 9300 - 4a = 7064 \\
   \Rightarrow - a + 9300 = 7064 \\
   \Rightarrow a = 2236 \\
 $
Thus we get the value of the full fare.
Now put the value of $a$ in equation 1
We get:
$
  a + b = 2325 \\
   \Rightarrow 2236 + b = 2325 \\
  b = 89 \\
 $
Thus we get the value of the full fare.
So the full fare from Delhi to Bangalore is Rs. $2236$ and reservation charges are Rs. $89$.

Note: In the question we have to keep in mind every data given in the question. First of all, read questions very carefully one to two times. Then start with the given data and make equations accordingly in the form of$a\& b$. Where $a$ is full fare and $b$ is reservation charges from Delhi to Bangalore. Substitute the values from the equation very carefully and calculate it. Thus we get the correct answer.