Directions: Read the table carefully and answer the following question. The table shows the data about distance (km) covered from one point to another point by the vehicle used between each point and speed of each vehicle (km/h). Note: (i) If one person wants to travel from P to S, then he/she can only use a vehicle available at the point P. (ii) ** means there is no direct connection between given points. | | P | Q | R | S | Transport Vehicle | Speed (km/h) | |---|---|---|---|---|---|---| | P | 0 | 360 | ** | 780 | Train | 60 | | Q | 540 | 0 | ** | 600 | Taxi | 50 | | R | 420 | 720 | 0 | ** | Bus | 40 | | S | 630 | ** | 900 | 0 | Bike | 45 | If Hemant travels from P to S via Q and the fair of the vehicle per km from point Q to S is 25% more than that of from P to Q, then the total fair collected is ₹19980. Find the train fare per km.

### Concept & Linear Equations This problem combines distance interpretation with proportional algebra. We can set up a linear equation representing the total cost of the journey by assigning a variable to the base fare. $$ \text{Total Fare} = (\text{Distance}_1 \times \text{Fare}_1) + (\text{Distance}_2 \times \text{Fare}_2) $$ ### Step-by-Step Solution 1. **Define the variables:** Let the fare per km from P to Q (which uses the Train) be $x$. 2. The fare per km from Q to S is 25% more than from P to Q, which makes it $1.25x$. 3. **Extract the distances from the table:** - Distance $P \to Q = 360$ km. - Distance $Q \to S = 600$ km. 4. **Formulate the equation for total fare:** - Cost for $P \to Q = 360 \times x = 360x$ - Cost for $Q \to S = 600 \times 1.25x = 750x$ - Total Cost $= 360x + 750x = 1110x$ 5. **Solve for $x$:** - $1110x = 19980$ - $x = \frac{19980}{1110} = \frac{1998}{111} = 18$ 6. The train fare per km (from P to Q) is ₹18. ### Exam Strategy & Shortcut To calculate $600 \times 1.25$ quickly mentally, break it down: $1.25 = 1 + \frac{1}{4}$. So, $600 + (600 / 4) = 600 + 150 = 750$. This avoids messy decimal multiplication. ### Common Pitfall A common error is applying the 25% increase to the total distance or total cost rather than strictly to the per-km rate for the specific second leg of the trip. ### Final Answer Therefore, the correct answer is **₹18**.