SOF IMO · Bite-size lessons

Time, Speed and Distance

About 7 min · Everyday Mathematics · English

Time worked out from the lesson's 501 words at 150 words a minute, plus a minute for each question.

Time, Speed and Distance (TSD) forms the backbone of motion-based word problems in competitive mathematics. In SOF IMO, this topic tests your ability to translate real-world scenarios—trains crossing platforms, boats rowing upstream, cyclists meeting on roads—into clean mathematical equations. Mastery here requires fluency with the fundamental relationship Distance = Speed × Time and its rearrangements, plus pattern recognition across three classic sub-types: relative motion (trains), upstream/downstream (boats and streams), and average speed calculations.

Key ideas

• Core Formula: Distance (D) = Speed (S) × Time (T). Any one can be found if the other two are known. Speed = D/T, Time = D/S.
• Unit Conversion: Always match units. 1 km/h = 5/18 m/s; to convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.
• Relative Speed (Same Direction): When two objects move in the same direction, their relative speed is the difference: S₁ − S₂.
• Relative Speed (Opposite Direction): When two objects move towards each other, their relative speed is the sum: S₁ + S₂.
• Average Speed: Average speed ≠ arithmetic mean of speeds. Use Average Speed = Total Distance / Total Time. Never average two speeds directly unless the times are equal.

Watch out for

• Averaging speeds arithmetically — Students compute (S₁ + S₂)/2 instead of Total Distance / Total Time. Always use the definition of average speed. Fix: Calculate total distance and total time separately, then divide.
• Forgetting unit conversion — Mixing km/h and m/s in the same calculation, or applying the wrong factor. Fix: Convert all speeds to the same unit before computing. Remember 1 km/h = 5/18 m/s (= 1/3.6 m/s).
• Adding train lengths incorrectly — For a train crossing a platform, students use only train length or only platform length. Fix: Train + platform when crossing a platform; train length alone when crossing a pole; sum of both train lengths when two trains cross.

Quick reference

• D = S × T — All TSD problems reduce to this. Isolate the unknown.
• 1 km/h = 5/18 m/s — Memorize this conversion factor cold.
• Opposite directions → add speeds; same direction → subtract speeds.
• Average speed = Total Distance ÷ Total Time — never the arithmetic mean of two speeds unless times are identical.
• Trains crossing: pole → train length; platform → train + platform; another train → sum of both lengths.

Check yourself: 3 questions

This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.

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