Time, Speed & Distance

Overview

Time, Speed & Distance (TSD) is one of the most frequently tested topics on both CAT and GMAT. Problems involve moving objects — people jogging, trains crossing, boats sailing — and require the core relationship between distance, speed, and time. Difficulty ranges from direct formula use to multi-step scenarios involving relative motion, head starts, and circular tracks.

On CAT, TSD appears in standalone Quant problems and occasionally as part of DILR sets. On GMAT, it appears in Problem Solving. Expect 2–4 questions per exam.

Core Formula

Distance = Speed × Time (D = S × T)

Derived forms:

• S = D / T
• T = D / S

Unit conversions:

• km/h → m/s: multiply by 5/18
• m/s → km/h: multiply by 18/5

Proportionality (when one variable is constant):

• Distance ∝ Time (constant speed)
• Distance ∝ Speed (constant time)
• Speed ∝ 1/Time (constant distance) — faster speed means less time

Average Speed

Average Speed = Total Distance / Total Time

Never average speeds arithmetically unless time spent at each speed is equal.

For equal distances at two different speeds a and b:

Average Speed = 2ab / (a + b) — the harmonic mean

Example: 60 km at 30 km/h, then 60 km at 60 km/h.

• Time leg 1 = 2 hrs, time leg 2 = 1 hr. Total distance = 120 km, total time = 3 hrs.
• Average speed = 120/3 = 40 km/h, not (30+60)/2 = 45 km/h.

Relative Speed

Scenario	Relative Speed
Opposite directions (towards each other)	S₁ + S₂
Same direction	|S₁ − S₂|

• Time to meet (approaching): T = Gap / (S₁ + S₂)
• Time to catch up (same direction): T = Gap / (S₁ − S₂), where S₁ > S₂

Common Problem Types

1. Pursuit & Head Start

If B has a time head start of t at speed v_B:

• Head start distance = v_B × t
• Time for A to catch B = (v_B × t) / (v_A − v_B)

2. Trains

• Crossing a pole/person (length-less): Distance = Length of train
• Crossing a platform/bridge: Distance = Length of train + Length of platform
• Two trains crossing (opposite dirs): T = (L₁ + L₂) / (S₁ + S₂)
• Two trains crossing (same dir): T = (L₁ + L₂) / |S₁ − S₂|

3. Boats & Streams

Let b = boat speed in still water, s = stream speed.

• Downstream speed = b + s
• Upstream speed = b − s
• Still water speed = (Downstream + Upstream) / 2
• Stream speed = (Downstream − Upstream) / 2

4. Meetings on a Straight Path

Two people A and B start from opposite ends of a path of length D at the same time:

• 1st meeting: combined distance = D → Time = D / (S_A + S_B)
• 2nd meeting: combined distance = 3D
• nth meeting: combined distance = (2n−1)D

Ratio of distances covered by A and B = S_A : S_B.

Example: A and B first meet 0.6D from P (A's starting end).

• Ratio of speeds S_A : S_B = 0.6 : 0.4 = 3 : 2
• 4th meeting: combined = 7D. A covers 7D × (3/5) = 4.2D → position = 0.2D from P.

5. Circular Tracks (same start point and time)

• Opposite directions: First meet when combined distance = track length L. T = L / (S_A + S_B)
• Same direction: First meet when faster has lapped slower once. T = L / |S_A − S_B|

Common Mistakes

• Using arithmetic mean instead of harmonic mean for equal-distance journeys
• Mixing units (km/h vs m/s, or hours vs minutes)
• In pursuit problems, confusing head start in time vs head start in distance
• Forgetting to include the train's own length when it crosses a platform or another train
• Assuming both upstream and downstream speeds without checking which is which

Exam Tips

• For equal-distance legs, always use 2ab/(a+b) for average speed — never (a+b)/2
• Set speeds as ratios (e.g., 3k and 5k) to avoid messy fractions throughout
• Draw a timeline or number line for multi-leg journeys — it prevents leg-swapping errors
• For CAT problems with two objects on a circular track going opposite ways, add speeds; same direction, subtract
• In GMAT word problems, re-read the question carefully: "how fast is car Y?" means the slower car's speed, not the gap-closing speed