Finding Basic Fare and Reservation Charge

Video Explanation

Question

A half ticket costs half the full fare, but the reservation charge is same for both. One full reserved ticket costs ₹216. One full and one half reserved ticket together cost ₹327. Find the basic full fare and the reservation charge.

Solution

Step 1: Let Variables

Let basic full fare = \(x\)

Let reservation charge = \(y\)

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Step 2: Form Equations

Full ticket:

\[ x + y = 216 \quad (1) \]

Half ticket = \( \frac{x}{2} + y \) Second condition:

\[ x + y + \frac{x}{2} + y = 327 \]

\[ \frac{3x}{2} + 2y = 327 \quad (2) \]

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Step 3: Simplify

Multiply (2) by 2:

\[ 3x + 4y = 654 \quad (3) \]

Multiply (1) by 3:

\[ 3x + 3y = 648 \quad (4) \]

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Step 4: Solve Linear Equations

Subtract (4) from (3):

\[ y = 6 \]

Substitute into (1):

\[ x + 6 = 216 \]

\[ x = 210 \]

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Conclusion

\[ \text{Basic full fare} = ₹210,\quad \text{Reservation charge} = ₹6 \]

Verification

Full ticket: \(210 + 6 = 216\) ✔

Full + half: \(216 + (105 + 6) = 327\) ✔