A journey of 192 km between two cities takes 2 hours less by a fast train than by a slow train. If the average speed of the slow train is 16 km/hr less than that of the fast train, then the average speed of the fast train is

### Concept & Quadratic Equation for Speed Difference When comparing the time taken by two objects travelling the same distance with a known difference in their speeds and times, we construct an algebraic time difference equation resulting in a quadratic equation. $$\frac{Distance}{S_{slow}} - \frac{Distance}{S_{fast}} = \Delta Time$$ ### Step-by-Step Solution 1. Let the average speed of the fast train be $x$ km/hr. 2. The average speed of the slow train is $(x - 16)$ km/hr. 3. The time taken by the fast train to cover $192$ km is $\frac{192}{x}$ hours. 4. The time taken by the slow train to cover $192$ km is $\frac{192}{x - 16}$ hours. 5. The problem states the time difference is $2$ hours: $$\frac{192}{x - 16} - \frac{192}{x} = 2$$ 6. Factor out $192$ and find a common denominator: $$192 \left[ \frac{x - (x - 16)}{x(x - 16)} \right] = 2$$ $$192 \left[ \frac{16}{x^2 - 16x} \right] = 2$$ 7. Divide both sides by $2$ to simplify: $$96 \times 16 = x^2 - 16x$$ $$1536 = x^2 - 16x$$ $$x^2 - 16x - 1536 = 0$$ 8. Factor the quadratic equation. Look for two numbers that multiply to $-1536$ and add to $-16$. These numbers are $-48$ and $+32$. $$(x - 48)(x + 32) = 0$$ 9. Since speed cannot be negative, $x = 48$ km/hr. ### Exam Strategy & Shortcut Use Option Elimination based on the time difference. The calculation $\frac{192}{x - 16} - \frac{192}{x}$ must equal $2$. Test Option (c) 48 km/hr: Fast speed $= 48$. Slow speed $= 48 - 16 = 32$. Fast time $= \frac{192}{48} = 4$ hours. Slow time $= \frac{192}{32} = 6$ hours. Difference $= 6 - 4 = 2$ hours. This matches perfectly and skips solving the quadratic equation! ### Common Pitfall Setting up the subtraction backwards (e.g., $\frac{192}{x} - \frac{192}{x - 16} = 2$) is a major pitfall. It produces a negative time difference and invalidates the setup. Always subtract the smaller time (faster speed) from the larger time (slower speed). ### Final Answer Therefore, the correct answer is **48 km/hr**.