Arithmetic

Time, Speed and Distance formulas

Learn Time, Speed and Distance step by step

At a glance

D = S × T     D fixed: speed ratio = inverse time ratio
km/h × 5/18 = m/s
equal distances: average = 2ab / (a + b);  three: 3abc / (ab + bc + ca)
relative speed: a + b opposite, a − b same way
train crossing: (total length to clear) / relative speed
boats: D = b + s, U = b − s;  b = (D + U)/2, s = (D − U)/2
track: first meeting = L / relative speed

D = S × T, and thinking in ratios

D = S × T
D fixed: S₁ / S₂ = T₂ / T₁
km/h → m/s: × 5/18     m/s → km/h: × 18/5

Average speed

average speed = total distance / total time
equal distances: 2ab / (a + b)   (the harmonic mean of a and b)
three equal distances: 3abc / (ab + bc + ca)
equal times: (a + b) / 2

Relative speed

opposite directions: relative speed = a + b
same direction: relative speed = a − b
time to meet = gap / relative speed

Shortcut: In a meeting, distances covered are in the ratio of speeds. 60 : 40 = 3 : 2, so the meeting point splits 200 km as 120 : 80. No time needed.

Trains: poles, platforms and each other

Clearing a platform: the train covers its own length plus the platform

pole or person: time = train length / speed
platform or bridge: time = (train + platform) / speed
two trains: time = (L₁ + L₂) / relative speed

A train of length L at speed s passing something:

It passes	Distance to cover	Speed to use
A pole or a standing man	L	s
A platform or bridge of length P	L + P	s
A man walking at v, same direction	L	s − v
A man walking at v, towards it	L	s + v
Train of length M at u, same direction	L + M	s − u
Train of length M at u, opposite direction	L + M	s + u

Time = distance to cover ÷ speed to use. Change km/h to m/s (× 5/18) first.

Boats and streams

b = boat in still water, s = stream, D = downstream speed, U = upstream speed.

D = b + s     U = b − s
b = (D + U) / 2     s = (D − U) / 2
there and back over d:  time = d/(b + s) + d/(b − s) = 2bd / (b² − s²)

Circular tracks

first meeting = L / (a + b) opposite,  L / (a − b) same way
speeds p : q in lowest terms → p + q meeting points (opposite), |p − q| (same way)
first time both at start = LCM of the lap times

Formulas stick when you use them

The Time, Speed and Distance lesson has worked problems and quick checks for every formula here. Free to start.

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