Time-Speed-Distance and Time-Work Questions for UPSC CSE

Core Speed-Distance-Time Relationship

The entire topic rests on one triangle relationship: Distance equals Speed multiplied by Time. All problems are variations of this. Always check units first — if speed is in km/h and time is in minutes, convert before applying the formula. A common trick: when distance is constant, speed and time are inversely proportional. So if speed doubles, time halves.

• Distance = Speed x Time (remember: DST triangle — cover what you want)
• Speed in m/s = km/h value multiplied by 5/18
• Speed in km/h = m/s value multiplied by 18/5
• When distance is fixed: Speed1 x Time1 = Speed2 x Time2
• Average speed for equal distances = 2ab/(a+b), NOT simple average of a and b
• Relative speed: same direction = subtract speeds; opposite direction = add speeds

Trains, Boats and Races

These are applied TSD problems. For trains, the key insight is that a train must travel its own length PLUS the length of whatever it is crossing. For boats, the current either adds to or subtracts from the boat speed. In races, if A gives B a head start of X metres, A effectively runs the full distance while B runs (distance minus X).

• Train crossing a pole or person: distance = length of train only
• Train crossing a platform or bridge: distance = length of train + length of platform
• Two trains crossing each other: distance = sum of both lengths, speed = relative speed
• Boat downstream speed = boat speed + stream speed (U + V)
• Boat upstream speed = boat speed - stream speed (U - V)
• Boat speed in still water = (downstream + upstream) / 2

Time and Work — Core Logic

Think of work as a bucket to be filled. Each worker fills a fraction of the bucket per day. The key shift is to always convert time to rate (work done per day = 1/number of days). Add rates when workers work together. Subtract rates when one fills and another empties (like pipes and cisterns). The LCM method is often faster than fractions for UPSC-level questions.

• If A finishes work in N days, A does 1/N work per day
• Combined rate of A and B = 1/A + 1/B; total time together = AB/(A+B)
• LCM method: assume total work = LCM of given days; find per-day work as whole numbers
• If a pipe fills in F hours and a leak empties in L hours, net rate = 1/F - 1/L
• Efficiency is inversely proportional to time — more efficient means fewer days
• If A is twice as efficient as B, A takes half the time B takes

Work and Wages, Chain Rule and Man-Days

A common UPSC variant combines work with wages or scales up work with more workers. The key principle: wages are distributed in proportion to work done (i.e., proportional to daily rate x days worked). For scaling problems, use the Man-Day concept: total work = men x days x hours. If any variable changes, adjust the others proportionally.

• Wages ratio = ratio of work done = ratio of (rate x time worked)
• Man-Day formula: M1 x D1 x H1 = M2 x D2 x H2 (when same work is done)
• If more men are added midway, split the problem into two phases
• If A is paid more per day than B, it does NOT mean A does more work — read carefully
• Always check if workers work for the same number of days before splitting wages equally

• Using simple average instead of harmonic mean (2ab/a+b) for average speed when distances are equal — this is the single most recycled UPSC trap.
• Forgetting to add the train length when it crosses a bridge or platform — only the pole/person crossing uses train length alone.
• Not converting units before calculation — mixing km/h with seconds or metres leads to wrong answers even with correct logic.
• In Time-Work problems, adding times instead of adding rates — you can never add days directly; always convert to per-day work first.
• In wages problems, dividing total wages equally when workers worked for different durations or have different efficiencies — always split by actual work contribution.

• DST Triangle: Draw a triangle with D on top, S and T at bottom. Cover the letter you want to find — the remaining two show the operation (S x T, D/T, D/S).
• Boats: U and V — think Upstream is Unhelpful (subtract V), Downstream is Delightful (add V). Still water = (D+U)/2, Stream = (D-U)/2.
• Work rates: think FILL THE BUCKET — each worker pours 1/N buckets per day. Together they pour faster. A leak pours it out. Net rate = fill rate minus drain rate.
• SAME-OPPOSITE for trains: trains going SAME direction, subtract speeds; OPPOSITE direction, add them up — just like cars on a road.

• CSAT 2023, 2022, 2019 all had 1-2 TSD questions involving trains or average speed, and 1-2 Time-Work questions involving pipes or combined workers. Expect this pattern to continue.
• Most UPSC TSD questions are solvable in under 90 seconds using LCM method or harmonic mean — avoid long algebraic setups.
• Boats and streams questions often appear as single-step calculations (find still water speed or stream speed) — just memorise the two formulas and you get the mark.
• For Time-Work in Mains context (Data Interpretation sets), the Man-Day formula appears in larger calculation sets — practice scaling up quickly.
• If a question looks complex, assign a smart number — pick total work = LCM of given days. This converts fractions into whole numbers and saves 30-40 seconds per question.