A car starts running with the initial speed of 40 kmph, with its speed increasing every hour by 5 kmph. How many hours will it take to cover a distance of 385 km?

### Concept & Arithmetic Progression When an object travels with an initial distance in the first hour and the distance increases by a constant amount each subsequent hour, the total distance covered over a number of hours forms an Arithmetic Progression (A.P.). $$S_n = \frac{n}{2}[2a + (n-1)d]$$ ### Step-by-Step Solution * **Identify A.P. Parameters:** Initial distance (first term $a$) = $40$ km. Hourly increase (common difference $d$) = $5$ km. Total distance ($S_n$) = $385$ km. * **Apply Formula:** $385 = \frac{n}{2}[2(40) + (n-1)5]$ * **Simplify:** $770 = n[80 + 5n - 5] \Rightarrow 770 = n(75 + 5n)$ * **Form Quadratic Equation:** $5n^2 + 75n - 770 = 0 \Rightarrow n^2 + 15n - 154 = 0$ * **Factorize:** $n^2 + 22n - 7n - 154 = 0 \Rightarrow n(n + 22) - 7(n + 22) = 0$ * **Solve for n:** $(n - 7)(n + 22) = 0$. Since time cannot be negative, $n = 7$. ### Exam Strategy & Shortcut Instead of solving the quadratic equation, plug the options into the sum formula or just add manually if the numbers are small: $40 + 45 + 50 + 55 + 60 + 65 + 70$. The sum of the unit digits is $0 + 5 + 0 + 5 + 0 + 5 + 0 = 15$, ending in 5, matching $385$. Seven terms is exactly 7 hours. ### Common Pitfall Misidentifying the hourly increase as acceleration and trying to use physics kinematic equations ($s = ut + \frac{1}{2}at^2$), which applies to continuous smooth acceleration, not discrete hourly jumps. ### Final Answer Therefore, the correct answer is **7 hours**.