Time & Work, Speed & Distance — IBPS PO Quant

Time-work and speed-distance look like two chapters but run on one equation: rate × time = amount done. In time & work the "amount" is a job; in speed-distance it's distance. The professional shortcut for both is to avoid fractions: use the LCM method for work (assume the job is a whole number of units) and proportionality / relative speed for motion. IBPS asks 1–3 directly and buries more inside caselets. Learn the templates — efficiency, relative speed, trains, boats — and these are quick, reliable marks.

1. Time & work — the LCM method

Instead of "A does 1/12 of the work per day", make the total work an LCM so everyone's daily output is a whole number:

• A finishes in 12 days, B in 18 ⇒ total work = LCM(12,18) = 36 units. A does 36/12 = 3 units/day, B does 36/18 = 2 units/day.
• Together: 3 + 2 = 5 units/day ⇒ time = 36/5 = 7.2 days.
• Efficiency = units/day. "A is twice as efficient as B" means A's rate is double B's.

$\text{Total work}=\text{LCM of times};\text{rate}=\frac{\text{total}}{\text{time}};\text{time together}=\frac{\text{total}}{\sum \text{rates}}$

This kills all the fractions that make time-work slow.

2. Work variations

• Leaving/joining: track units done each phase and subtract from the total.
• Wages split in the ratio of work done (= ratio of rates × days worked).
• Pipes & cisterns: filling pipes are +rate, emptying pipes (leaks) are −rate; net rate fills/empties the tank. Same LCM method (tank = LCM litres).
• M₁D₁H₁/W₁ = M₂D₂H₂/W₂: the men–days–hours–work chain rule for scaled problems.

3. Speed & distance — proportionality

$\text{Speed}=\frac{\text{Distance}}{\text{Time}},\text{Distance}=\text{Speed}\times \text{Time}$

• Unit conversion: km/h → m/s multiply by 5/18; m/s → km/h multiply by 18/5.
• Same distance: speed and time are inversely proportional — if speed rises 25% (×5/4), time falls to 4/5. This is a percentage-in-disguise question.
• Average speed for equal distances at speeds u and v is the harmonic mean 2uv/(u+v), not the simple average.

4. Relative speed (trains, and chasing/meeting)

• Opposite directions: speeds add (closing fast).
• Same direction: speeds subtract (closing slow).
• A train crossing a pole: distance = train length. Crossing a platform: distance = train length + platform length. Two trains crossing: distance = sum of lengths, at the relative speed.

Example: two trains 120 m and 180 m, speeds 40 and 50 km/h, opposite directions. Relative speed = 90 km/h = 25 m/s; total length = 300 m ⇒ time = 300/25 = 12 s.

5. Boats & streams

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

• Downstream speed = b + s; upstream speed = b − s.
• From downstream D and upstream U: b = (D+U)/2, s = (D−U)/2.
• Same distance up and down: time is longer upstream (slower speed).

7. Common traps

• Averaging two speeds arithmetically for equal distances — use the harmonic mean 2uv/(u+v).
• Forgetting the platform/train length in crossing problems — pole = length only; platform = both.
• Wrong 5/18 direction in unit conversion.
• Same-direction vs opposite relative speed — subtract vs add; re-read which.

8. The protocol

1. Time & work: set total = LCM of times; rates = whole units/day; combine and subtract as needed.
2. Pipes: filling +, emptying −; net rate on the LCM tank.
3. Speed–distance: use inverse proportion for same-distance; harmonic mean for equal-distance average.
4. Trains: distance = sum of relevant lengths; add/subtract speeds by direction; convert with 5/18.
5. Boats: down = b+s, up = b−s; recover b and s by half-sum and half-difference.