Multiple choice

Two men are running in the same direction with a speed of $6$ km/hr and $\displaystyle 7\frac{1}{2}$ km/hr. A train running in the same direction crosses them in $5$ sec and $\displaystyle 5\frac{1}{2}$ sec respectively. The length and the speed of the train are

  1. $22.92$ m (approx ) and $22$ km/hr
  2. $22$ m (approx) and $22.5$ km/hr
  3. $22.90$ m(approx) and $20.5$ km/hr
  4. $22.92$ m (approx) and $22.5$ km/hr
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D Correct answer
Explanation

Let L be length and V be speed. Convert speeds to m/s: 6 km/h = 5/3 m/s, 7.5 km/h = 25/12 m/s. Relative speeds are (V - 5/3) and (V - 25/12). Using L = (V - v) * t, we get two equations for L and V. Solving yields V = 22.5 km/h and L approx 22.92 m.

AI explanation

Let the train's speed be y m/s and its length be x meters. The relative speed formula gives x equals 5 times y minus 6 times 5/18 and x equals 5.5 times y minus 7.5 times 5/18. Equating the two expressions gives 5 times y minus 1.67 equals 5.5 times y minus 2.08, which solves to a train speed of 6.25 m/s. Multiplying 6.25 m/s by 18/5 converts it to 22.5 km/hr, and the train's length is 5 times 6.25 minus 1.67, which is approximately 22.92 meters.

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