Shortcut Trick
Let total distance = LCM of 4 hours and 3.5 hours = 28 units.
Speed of 1st train = 28 ÷ 4 = 7 units/hr | Speed of 2nd train = 28 ÷ 3.5 = 8 units/hr.
Distance covered by 1st train in 2 hrs (6:00 a.m. to 8:00 a.m.) = 7 × 2 = 14 units.
Remaining distance = 28 − 14 = 14 units.
Time to cross after 8:00 a.m. = Distance ÷ Relative speed = 14 ÷ (7 + 8) = 14/15 hours = 56 minutes.
∴ The correct answer is 8:56 a.m.
Alternate Method
Delhi
Jainagar
Train 1 (6 a.m.)
Speed: D/4
Train 2 (8 a.m.)
Speed: 2D/7
8 a.m. Position
Meet at 8:56 a.m.
D/2 covered (6-8 a.m.)
D/2 Remaining
Given:
Train 1 leaves Delhi at 6:00 a.m. and reaches Jainagar at 10:00 a.m.
Train 2 leaves Jainagar at 8:00 a.m. and reaches Delhi at 11:30 a.m.
Formula Used:
Speed = Distance ÷ Time
Time to meet = Remaining Distance ÷ Relative Speed
Calculations:
Let the total distance between Delhi and Jainagar be D.
⇒ Time taken by Train 1 = 10:00 a.m. − 6:00 a.m. = 4 hours.
⇒ Time taken by Train 2 = 11:30 a.m. − 8:00 a.m. = 3.5 hours = 7/2 hours.
⇒ Speed of Train 1 = D/4
⇒ Speed of Train 2 = D ÷ (7/2) = 2D/7
⇒ By 8:00 a.m., Train 1 travels for 2 hours (6:00 a.m. to 8:00 a.m.).
⇒ Distance covered by Train 1 = 2 × (D/4) = D/2
⇒ Remaining distance at 8:00 a.m. = D − D/2 = D/2
Now, both trains are moving towards each other.
⇒ Relative speed = D/4 + 2D/7 = (7D + 8D) / 28 = 15D/28
⇒ Time taken to meet = (D/2) ÷ (15D/28)
⇒ Time taken = (D/2) × (28/15D) = 14/15 hours.
⇒ Convert 14/15 hours into minutes: 14/15 × 60 = 56 minutes.
⇒ The trains meet 56 minutes after 8:00 a.m., which is 8:56 a.m.
∴ The correct answer is 8:56 a.m.
Additional Information
Relative Speed Concept
When two bodies travel in opposite directions at speeds S1 and S2, their relative speed is S1 + S2. Time to meet = Distance between them ÷ Relative Speed.
Direct Formula for Meeting Time
If Train 1 takes time T1, Train 2 takes time T2, and Train 1 starts Δt hours early, the time to meet after Train 2 starts is: Time = (T1 − Δt) × T2 ÷ (T1 + T2).
LCM Method for Constant Distance
When only travel times are given for a fixed distance, assuming the total distance as the Least Common Multiple (LCM) of the given times entirely eliminates fractional calculations and speeds up the process.