The charges for a five-day trip by a tourist bus for one full ticket and a half-ticket are ₹ 1440 inclusive of boarding charges which are same for a full ticket and a half-ticket. The charges for the same trip for 2 full tickets and one half-ticket inclusive of boarding charges are ₹ 2220. The fare for a half-ticket is $75\%$ of the full ticket. Find the fare and the boarding charges separately for one full ticket.

### Concept & Strategy This problem translates directly into a system of linear equations. The crucial element is recognizing that "boarding charges" apply per person (per ticket, regardless of full or half), not per transaction. However, solving simultaneous decimal equations is slow; **Option Elimination** is vastly superior here. ### Step-by-Step Solution * **Given:** Let Full Fare = $F$ and Boarding Charge = $B$. Half Fare = $0.75F$. Total cost = (Fares) + (Boarding Charges for each ticket). Scenario 1: 1 Full + 1 Half ticket = ₹ $1440$. Equation 1: $(F + B) + (0.75F + B) = 1440 \Rightarrow 1.75F + 2B = 1440$ Scenario 2: 2 Full + 1 Half ticket = ₹ $2220$. Equation 2: $2(F + B) + (0.75F + B) = 2220 \Rightarrow 2.75F + 3B = 2220$ * **Calculation / Deduction (Standard Method):** Subtract Equation 1 from Equation 2 to eliminate some variables: $(2.75F + 3B) - (1.75F + 2B) = 2220 - 1440$ $1.0F + B = 780$ This means the cost of one full ticket *including* boarding is ₹ $780$. Substitute $(F + B) = 780$ back into the first scenario's logic: Total Trip 1 Cost = (1 Full with Boarding) + (1 Half with Boarding) $1440 = 780 + (0.75F + B)$ $660 = 0.75F + B$ Now we have: $F + B = 780$ $0.75F + B = 660$ Subtracting these: $0.25F = 120 \Rightarrow F = 480$. If $F = 480$, then $480 + B = 780 \Rightarrow B = 300$. ### Exam Strategy & Shortcut **Option Verification!** It takes seconds. We know the total for 1 Full ($F$) + 1 Boarding ($B$) + 1 Half ($0.75F$) + 1 Boarding ($B$) must equal $1440$. Let's test Option (c): $F = 480, B = 300$. Half Fare = $0.75 \times 480 = 360$. Cost of Full Passenger = $480 + 300 = 780$. Cost of Half Passenger = $360 + 300 = 660$. Total = $780 + 660 = 1440$. (Matches Scenario 1 perfectly!) You don't even need to test Scenario 2 if it matches this uniquely. ### Common Pitfall The most common trap is misinterpreting the boarding charge. Students often write Equation 1 as $F + 0.75F + B = 1440$, assuming the boarding charge is a flat fee for the whole booking rather than a per-person fee. The text "inclusive of boarding charges which are same for a full ticket and a half-ticket" dictates it applies to each ticket individually. ### Final Answer **Therefore, the correct answer is ₹ 480, ₹ 300.**