Math · Aptitude · Chapter 05

🚗 Speed, Time & Distance

Trains, buses, relative speed — and same-direction tricks.

💡 The Golden Triangle

One formula rules everything in this topic:

Speed = Distance ÷ Time

Rearrange as needed: Distance = Speed × Time, Time = Distance ÷ Speed.

km/hr → m/s
× 5/18
m/s → km/hr
× 18/5
Average speed
2xy / (x + y) for equal distances at speeds x, y
Same direction
Relative speed = a − b (faster − slower)
Opposite directions
Relative speed = a + b
Train crossing a pole
Time = length of train / speed
Train crossing platform
Time = (train + platform) / speed
Two trains crossing
Time = (L₁ + L₂) / relative speed
⚡ Average speed trap

If a man goes from A to B at 60 km/hr and returns at 40 km/hr, average speed is NOT 50 km/hr.

For equal distances, use: Average = 2xy/(x + y). Here: 2(60)(40)/100 = 48 km/hr.

⚡ Train problems — the rule

• Pole/man/tree → just the train's length matters
• Platform/bridge/tunnel → train length + platform length
• Train chasing/meeting another train → both lengths + relative speed

Always convert speeds to m/s before computing time in seconds!

⚡ Common SSC question pattern

"If speed is increased by x%, time decreases by ?"
If speed increases by 20% (becomes 6/5), time becomes 5/6 of original. Time decrease = 1/6 = 16.67%.

Rule: speed and time are inversely proportional for fixed distance.

🎬

Two Trains, Two Cases

Animation
RELATIVE SPEED — DIFFERENT WHEN MOVING TOWARDS OR SAME WAY SAME DIRECTION → relative speed = a − b Faster train catches slower — slow relative motion OPPOSITE DIRECTIONS → relative speed = a + b Trains meet quickly — fast relative motion QUICK UNIT CONVERSION 72 km/hr × 5/18 = 20 m/s Whenever you see km/hr & seconds in the same problem, convert.

Top: two trains in same direction (gap closes slowly). Bottom: opposite directions (they fly past).

🧮

STD Calculators

Calculator
📏 Basic STD

Speed km/hr · Time hr

Distance: 120 km

🔁 Unit converter

km/hr = 20 m/s

📊 Average speed (equal distances)

Speed A→B: · Return B→A: km/hr

Average: 48 km/hr (NOT 50! Use 2xy/(x+y))

🚂 Train crossing platform

Train m at km/hr crosses platform m in: 25 s

Practice (SSC): Two trains 140m and 160m long run on parallel tracks. If they run in opposite directions at 60 km/hr and 40 km/hr, in how many seconds will they completely pass each other?
Total length = 140 + 160 = 300m. Relative speed (opposite) = 60 + 40 = 100 km/hr = 100 × 5/18 = 250/9 m/s. Time = 300 ÷ (250/9) = 300 × 9/250 = 10.8 seconds.
Practice (Banking): A car covers a distance in 4 hours at 60 km/hr. By what % should speed be increased to cover the same distance in 3 hours?
Distance = 60 × 4 = 240 km. New speed = 240/3 = 80 km/hr. Increase = (80−60)/60 × 100 = 33.33%. Shortcut: time goes from 4 to 3 (ratio 4:3) → speed goes 3:4 → ↑ by 1/3 = 33.33%.
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