A reserved first-class full ticket from A to B costs Rs 2530. A reserved full + a reserved half together cost Rs 3810. The reservation charge is the same for both, but a half ticket has half fare. Find the full fare and the reservation charge.
Full fare = Rs 2500; Reservation charge = Rs 30 per ticket.
Step 1: Let the full fare (without reservation charge) be \(F\). Let the reservation charge be \(R\).
Step 2: For a full reserved ticket:
Full fare + reservation charge = \(F + R = 2530\).
Step 3: For a half reserved ticket:
The fare will be half of full fare = \(\tfrac{F}{2}\).
Reservation charge is the same = \(R\).
So, one half reserved ticket = \(\tfrac{F}{2} + R\).
Step 4: For one full reserved + one half reserved ticket together:
\((F + R) + (\tfrac{F}{2} + R) = 3810\).
Simplify: \(F + R + \tfrac{F}{2} + R = 3810\).
So, \(\tfrac{3F}{2} + 2R = 3810\).
Step 5: From Step 2 we know: \(F + R = 2530\). So, \(F = 2530 - R\).
Step 6: Substitute \(F = 2530 - R\) in the equation \(\tfrac{3F}{2} + 2R = 3810\):
\(\tfrac{3}{2}(2530 - R) + 2R = 3810\).
\(3795 - 1.5R + 2R = 3810\).
\(3795 + 0.5R = 3810\).
Step 7: Subtract 3795 from both sides:
\(0.5R = 15\).
So, \(R = 30\).
Step 8: Put \(R = 30\) in \(F + R = 2530\):
\(F + 30 = 2530\).
So, \(F = 2500\).
Final Answer: Full fare = Rs 2500, Reservation charge = Rs 30.