Time, Distance and Work

Time, Distance and Work

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Time, Distance and Work quantifies how three linked variables — speed, time, and distance — interact, and extends the same additive-rate logic to work-rate problems (men, machines, pipes). The single must-know triad is:

• Distance = Speed × Time
• Average Speed = Total Distance ÷ Total Time (use harmonic mean 2xy / (x + y) when two equal time intervals are travelled at speeds x and y)
• Relative Speed = sum if objects move towards each other, difference if they move in the same direction.

For work: if A finishes a job in a days and B in b days, together they finish in ab / (a + b) days because their per-day rates are 1/a + 1/b. For NAT-I, expect 1–2 short MCQs testing average-speed or pipe-and-cistern logic. Always convert every rate to the same time unit before adding.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Relationships

The Speed–Time–Distance (STD) triangle rests on one equation rewritten three ways:

Quantity	Formula	Units
Distance	Speed × Time	km, m, miles
Speed	Distance / Time	km/h, m/s
Time	Distance / Speed	h, s

Relative speed is the closing-rate between two moving objects. For two runners on the same straight track, their gap closes at S₁ + S₂ when running towards each other and at |S₁ − S₂| when running in the same direction. This single rule solves all chase, meeting, and train-crossing questions.

Average Speed

Average speed = total distance / total time — never the arithmetic mean of speeds unless equal distances are covered. The two important special cases:

• Equal time intervals at speeds x and y → harmonic mean 2xy / (x + y).
• Equal distance intervals at speeds x and y → arithmetic mean (x + y) / 2.

A 60 km outward trip at 30 km/h and return at 60 km/h gives average speed of 40 km/h (harmonic), not 45 km/h.

Work Rate (Pipes, Cisterns, Man-days)

If a worker (or inlet pipe) completes the whole job in t hours, the rate = 1/t job per hour. Several workers operating simultaneously contribute rates that add:

Combined rate = 1/a + 1/b + 1/c
Time together = 1 / (1/a + 1/b + 1/c)

For two workers, this simplifies to ab / (a + b) days. An outlet pipe or leak has a negative rate and is subtracted. The man-days identity M₁ × D₁ × H₁ = M₂ × D₂ × H₂ lets you swap workforce size for duration while keeping total work constant.

Worked Mini-example

A cistern has two inlet pipes (filling in 6 h and 8 h) and one leak (emptying in 12 h). Net rate per hour = 1/6 + 1/8 − 1/12 = (4 + 3 − 2)/24 = 5/24. Time to fill = 24/5 = 4.8 hours.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Subtleties

Partial work carried forward. If A works for d days then leaves, the unfinished fraction is 1 − d/a. The remaining worker B then needs (1 − d/a) × b more days, not b days from scratch.

Negative combined rate. When leak rate exceeds inlet rate, the cistern actually empties; NAT-I questions sometimes ask how long until a half-full tank drains, in which case multiply the time by the fractional fullness.

Time-gain in races. In a 100 m race, if A beats B by 10 m, it means when A finishes, B has run 90 m. Equivalently, when B runs 100 m, A has run 100 × (100/90) ≈ 111.11 m. Translate the win margin into a speed ratio, then reuse STD.

Unit conversion trap. NAT-I mixes km/h, m/s, and minutes. Memorise 1 m/s = 18/5 km/h = 3.6 km/h. A 250 m train crossing a 150 m platform at 54 km/h covers 400 m at 15 m/s → 26.67 s.

Common Mistakes

1. Averaging speeds with arithmetic mean for unequal-distance trips.
2. Treating two pipes filling the same tank as independent instead of additive.
3. Forgetting that time to fill and rate of filling are reciprocals.
4. Applying relative-speed sum when both objects run in the same direction.

Practice Prompts

1. A car travels the first half of a journey at 40 km/h and the second half at 60 km/h. Find the average speed for the equal-distance case, then recompute assuming the two speeds apply to equal times. Comment on which answer is larger.
2. Pipe P fills a tank in 9 h, Q in 12 h, and a leak empties it in 18 h. Starting with an empty tank, all three are opened for 3 h and then the leak is closed. How much additional time does Q alone need to finish?

Exam Strategy for NAT-I

Quantitative Reasoning carries about 4 % weight, yielding roughly 1 question from this cluster. Scan the answer choices first — NAT-I often offers ab / (a+b) and (x+y)/2 as deliberate distractors. Convert every rate to hours, set up the additive-rate fraction, and pick the choice matching the harmonic mean when time intervals are equal.

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