The taxi charges in a city comprise of a fixed charge, together with the charge of the distance covered. For a journey of 16 km, the charges paid are ₹ 156 and for a journey of 24 km, the charges paid are ₹ 204. What will a person have to pay for travelling a distance of 30 km?

### Concept & Formula This problem represents a system of linear equations. The total cost is a function of a constant ($y$-intercept) and a variable rate ($slope$). $$ \text{Total Cost} = \text{Fixed Charge} + (\text{Distance} \times \text{Rate per km}) $$ ### Step-by-Step Solution * **Set up the variables:** Let the fixed charge be $F$ and the rate per km be $R$. * **Create Equation 1:** For 16 km, the cost is 156. * $F + 16R = 156$ * **Create Equation 2:** For 24 km, the cost is 204. * $F + 24R = 204$ * **Solve for R (Rate):** Subtract Equation 1 from Equation 2 to eliminate $F$. * $(F + 24R) - (F + 16R) = 204 - 156$ * $8R = 48$ * $R = 6$ (The rate is ₹ 6 per km). * **Solve for F (Fixed Charge):** Substitute $R=6$ back into Equation 1. * $F + 16(6) = 156$ * $F + 96 = 156$ * $F = 60$ (The fixed charge is ₹ 60). * **Calculate cost for 30 km:** * Total Cost = $60 + 30(6) = 60 + 180 = 240$. ### Exam Strategy & Shortcut Skip solving for the fixed charge entirely to save time. You know the cost increases by ₹ 48 for an extra 8 km ($24\text{km} - 16\text{km}$). Therefore, the rate is $48/8 = \text{₹ } 6$ per km. You need the cost for 30 km. 30 km is exactly 6 km more than the 24 km journey. Just add the cost of the extra 6 km to the 24 km fare: $204 + (6 \times 6) = 204 + 36 = 240$. ### Common Pitfall A common trap is assuming direct proportionality. A student might divide $156 / 16 = 9.75$ and then multiply by 30 to get $292.5$. This completely ignores the fixed charge component of the fare. ### Final Answer **Therefore, the correct answer is ₹ 240.**