Time, Speed and Distance Question 3 - CAT 2024
Two places A and B are 45 km apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is:
Step-by-Step Explanation
Analyze the Crossing Event:
The total distance between A and B is 45 km. Anil and Sunil start at the same time and cross each other after 1 hour 30 minutes (90 minutes).
Apply the Crossing Time Theorem:
For two objects moving toward each other from opposite ends of a straight line, the time taken from the start to the meeting point (t) is the geometric mean of the individual times they take to reach their destinations after crossing each other (t₁ and t₂):
t = √(t₁ × t₂)
We are given that t = 90 minutes.
Set up the Time Variables:
Let the time taken by Sunil to reach A after crossing be x minutes.
Anil reaches B exactly 1 hour 15 minutes (75 minutes) after Sunil reaches A. Since they cross at the same moment, Anil's remaining time to reach B after the crossing point must be (x + 75) minutes.
90 = √[x(x + 75)]
8100 = x² + 75x
x² + 75x − 8100 = 0
Solve the Quadratic Equation:
(x + 135)(x − 60) = 0
Since a time duration must be positive, x = 60 minutes.
Calculate Anil's Speed:
Anil’s total travel time is the time before crossing plus his time after crossing:
Total Time = 90 minutes + (60 + 75) minutes = 225 minutes
Total Time in Hours = 225/60 = 3.75 hours
Anil's Speed = Total Distance / Total Time
= 45/3.75 = 12 km/h
Correct Option: D
💡 FundaMakers Tip:
The formula t = √(t₁ × t₂) is an essential shortcut for crossing problems. It eliminates the need to carry individual relative speed variables. Once you find x = 60 minutes, you can quickly write Anil's total time as 225 minutes, or 15/4 hours. Speed is then simply 45 × 4/15 = 12 km/h.