In a flight of 6000 km, an aircraft was slowed down due to bad weather. The average speed for the trip was reduced by 400 kmph and the time of flight increased by 30 minutes. The original planned duration of the flight was (M.A.T., 2006)

### Concept & Quadratic Equation for Speed and Time When speed and time change but the distance remains constant, we can construct an equation using the time difference. $$\frac{Distance}{Original \ Speed} - \frac{Distance}{New \ Speed} = \Delta Time$$ Alternatively, working directly with time: $$\frac{Distance}{Original \ Time} - \frac{Distance}{New \ Time} = \Delta Speed$$ ### Step-by-Step Solution 1. Let the original planned duration of the flight be $t$ hours. 2. The distance of the flight is $6000$ km. 3. The original speed of the aircraft is $\frac{6000}{t}$ kmph. 4. Due to bad weather, the time of flight increased by $30$ minutes, which is $0.5$ hours. The new time is $(t + 0.5)$ hours. 5. The new speed is $\frac{6000}{t + 0.5}$ kmph. 6. The problem states that the average speed was reduced by $400$ kmph: $$\frac{6000}{t} - \frac{6000}{t + 0.5} = 400$$ 7. Divide the entire equation by $400$ to simplify: $$\frac{15}{t} - \frac{15}{t + 0.5} = 1$$ 8. Multiply by $t(t + 0.5)$ to eliminate the denominators: $$15(t + 0.5) - 15t = t(t + 0.5)$$ $$15t + 7.5 - 15t = t^2 + 0.5t$$ $$t^2 + 0.5t - 7.5 = 0$$ 9. Multiply by 2 to remove decimals: $$2t^2 + t - 15 = 0$$ 10. Factor the quadratic equation: $$2t^2 + 6t - 5t - 15 = 0$$ $$2t(t + 3) - 5(t + 3) = 0$$ $$(2t - 5)(t + 3) = 0$$ 11. Time cannot be negative, so $2t - 5 = 0 \Rightarrow t = 2.5$ hours. 12. $2.5$ hours is equivalent to $2 \frac{1}{2}$ hours. ### Exam Strategy & Shortcut Option elimination is by far the fastest method for quadratic speed-time problems. Test option (a): $2.5$ hours. Original speed $= \frac{6000}{2.5} = 2400$ kmph. New time $= 2.5 + 0.5 = 3$ hours. New speed $= \frac{6000}{3} = 2000$ kmph. Difference in speed $= 2400 - 2000 = 400$ kmph. This matches the condition perfectly! ### Common Pitfall A common error is converting $30$ minutes to $0.3$ hours instead of $0.5$ hours, which leads to completely wrong quadratic equations. ### Final Answer Therefore, the correct answer is **$2 \frac{1}{2}$ hours**.