Time, Distance and Work

Time, Distance and Work

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

• The master equation is Distance = Speed × Time, where speed is expressed as distance per unit time (e.g. km/h, m/s). Convert km/h → m/s by multiplying by 5/18.
• Work-rate identity: if a person completes a job in T days, the one-day work fraction is 1/T. When two agents work together, their rates add: 1/T = 1/T₁ + 1/T₂, so the combined time T is the harmonic sum.
• Relative speed governs meeting/overtaking problems: objects moving towards each other close at S₁ + S₂; objects moving in the same direction close at S₁ − S₂.
• Average speed for a round trip at two different speeds (equal time or equal distance) is the harmonic mean, never the arithmetic mean.
• HAT-UG pointer: 2–4 questions per paper, mostly numerical MCQs, target the above four traps.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Relationships

Three variables — distance D, speed S, and time T — are linked by the linear identity D = S × T. Holding any one constant makes the other two directly or inversely proportional: doubling speed halves travel time for a fixed route; doubling distance doubles the time at constant speed.

Work as a Rate

Treat work like a fluid. If agent A finishes a task in Tₐ days, A completes 1/Tₐ of the task per day. Two agents working simultaneously contribute 1/Tₐ + 1/Tᵦ per day, so the time to finish together satisfies:

1/T = 1/Tₐ + 1/Tᵦ, hence T = (Tₐ × Tᵦ) / (Tₐ + Tᵦ).

Extension to n agents: 1/T = Σ (1/Tᵢ). The classic “man-days” form restates this as M₁D₁W₂ = M₂D₂W₁ for equal work, useful when crew sizes change.

Relative Speed and Trains

Two trains of lengths L₁, L₂ moving in opposite directions fully cross each other in time t = (L₁ + L₂) / (S₁ + S₂); in the same direction, t = (L₁ + L₂) / (S₁ − S₂). Boats-and-streams adds a current speed c: effective speed upstream = b − c, downstream = b + c, where b is the boat’s still-water speed.

Average Speed Trap

For a round trip where equal distances are covered at speeds S₁ and S₂, the average is 2S₁S₂ / (S₁ + S₂) — the harmonic mean. The arithmetic mean (S₁ + S₂)/2 is wrong and is the most-tested trap in HAT-UG.

Typical Question Patterns

1. “A and B can paint a wall in 12 and 18 days. With C they finish in 4 days. Find C’s solo time.”
2. “A train 120 m long crosses a pole in 8 s; find speed in km/h.”
3. “A man rows 24 km upstream and 24 km back in 10 hours; find stream speed.”

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Efficiency

Efficiency vs. time: if A is k times as efficient as B, A’s solo time is Tᵦ/k, not k × Tᵦ. A common reverse trap is “A is twice as fast as B, so A takes twice as long” — false.

Partial work and remaining work: track the fraction completed, not absolute output. If A does 1/3 of the work in 5 days at rate r, the remaining 2/3 needs 2/(3r) days. Pipe-and-cistern problems follow the same reciprocal-addition rule, with filling rates positive and leakage rates negative.

Unit Conversion Discipline

Speed may be given in km/h while distances are in metres, or times in seconds. Convert before substituting: km/h → m/s via ×5/18; m/s → km/h via ×18/5. Mixing units is the single most frequent arithmetic error in HAT-UG numericals.

Adjacent Topic Links

Time-Distance-Work overlaps with Ratio & Proportion (work split among agents), Percentages (efficiency comparisons), and Basic Algebra (setting up linear equations for combined rates). Mastering reciprocal addition unlocks all three.

Common Mistakes

• Using arithmetic mean for round-trip average speed.
• Adding times directly instead of reciprocals.
• Forgetting train length when two trains “cross” vs. when one “passes a pole”.
• Ignoring the stream current in boat problems during upstream/downstream swaps.

Worked Example

A can finish a task in 20 days, B in 30 days. They start together, but A leaves after 5 days. How many more days does B need?

A’s 5-day work = 5/20 = 1/4. Remaining = 3/4. B’s rate = 1/30 per day, so extra days = (3/4) ÷ (1/30) = 22.5 days.

Practice Prompts

1. Two pipes fill a tank in 15 min and 20 min; a third empties it in 30 min. Find the time to fill when all three are open.
2. A car travels 60 km at 40 km/h and returns at 60 km/h. Compute the correct average speed for the round trip.

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.