UPSC CSAT Quantitative Aptitude: Time-Speed-Distance, Time-Work, Averages, Profit-Loss

Must?Know (20–25 detailed bullets)

• Speed = Distance / Time; units must be consistent – km/h to m/s requires multiplication by 5/18, e.g., 72 km/h = 20 m/s.
• Relative speed of two objects moving in same direction = difference of speeds; opposite direction = sum of speeds, used in train crossing problems.
• Average speed = Total Distance / Total Time; not arithmetic mean unless time intervals are equal, e.g., going 60 km/h and returning 40 km/h-average speed = 2×60×40/(60+40) = 48 km/h.
• If a person covers half distance at x km/h and half at y km/h, average speed = 2xy/(x+y).
• In a race, if A gives B a start of 10 m in 100 m race, A runs 100 m while B runs 90 m.
• Two trains of lengths L1 and L2 cross each other in time T when moving in opposite directions at speeds S1 and S2: T = (L1 + L2)/(S1 + S2) in m/s.
• A train crosses a pole in time equal to its length divided by its speed; crosses a platform in time = (length of train + length of platform)/speed.
• If speed increases by 20%, time decreases by 1/6th for same distance, derived from inverse proportionality.
• Work = Rate × Time; if A can complete work in 10 days, daily rate = 1/10 of work per day.
• If A is twice as efficient as B, A takes half the time; ratio of work done A:B = 2:1, time ratio = 1:2.
• Pipes and cisterns follow same logic as time-work: inlet pipe fills, outlet empties; net rate = sum of inlets – sum of outlets.
• If 3 men and 4 women can do work in 10 days, and 2 men and 3 women in 12 days, solve using linear equations in work units.
• When workers leave or join mid-way, compute work done in phases using individual rates.
• Daily wage is proportional to work rate; if A earns ?600 for 12 days, daily wage ?50; if B is 1.5× efficient, B earns ?75/day.
• Average = Sum of observations / Number of observations; adding a number equal to current average does not change average.
• If average of 5 numbers is 20 and one number 25 is replaced by 35, new average = 20 + (35–25)/5 = 22.
• Weighted average accounts for frequency, e.g., average price of 3 kg rice at ?40/kg and 2 kg at ?50/kg = (3×40 + 2×50)/5 = ?44/kg.
• Profit = SP – CP; Loss = CP – SP; profit % = (Profit / CP) × 100, not SP.
• If CP of 10 articles = SP of 8 articles, profit % = (10–8)/8 × 100 = 25%.
• Successive discounts of 20% and 10% equivalent to single discount of 28%, not 30%; calculated as 1 – (0.8×0.9) = 0.28.
• Marked price = CP + Markup; discount applied on marked price; final SP = MP × (1 – d1/100) × (1 – d2/100).
• If a shopkeeper uses false weight of 900 g instead of 1 kg, profit % = (1000–900)/900 × 100 = 11.11%.
• Time taken by A and B together = (A×B)/(A+B) days if A and B alone take A and B days respectively.
• If A is 50% more efficient than B, time ratio A:B = 2:3; e.g., if B takes 15 days, A takes 10 days.
• For upstream/downstream: speed downstream = boat speed + stream speed; upstream = boat speed – stream speed.

Difficulty Level

Intermediate – problems require multi-step reasoning and unit conversions, but formulas are standard; UPSC emphasizes conceptual clarity over complexity.

Common UPSC Traps (3–5 factual traps)

Trap: Average speed is arithmetic mean of two speeds – Fact: Average speed is total distance divided by total time; only equals harmonic mean when distances are equal (e.g., same distance at two speeds).
Trap: Profit percentage is calculated on selling price – Fact: Profit % is always calculated on cost price unless stated otherwise (UPSC often tests this distinction).
Trap: If A is twice as fast as B, time ratio is 2:1 – Fact: Speed and time are inversely proportional; if speed ratio is 2:1, time ratio is 1:2.
Trap: Relative speed in same direction is sum of speeds – Fact: Relative speed in same direction is difference of speeds; sum applies only in opposite directions.

Practice MCQs (5–7 questions)

Question: A train 120 m long crosses a pole in 12 seconds. What is its speed in km/h?
A) 36 km/h
B) 48 km/h
C) 60 km/h
D) 72 km/h
Answer: A
Explanation: Speed = 120 m / 12 s = 10 m/s = 10 × (18/5) = 36 km/h.
Why others fail: Option D (72 km/h) results from incorrectly using 120/12 = 10 and multiplying by 2 instead of 18/5.

Question: A can complete a work in 15 days, B in 20 days. With C’s help, they finish in 5 days. How many days would C alone take?
A) 10 days
B) 12 days
C) 15 days
D) 18 days
Answer: B
Explanation: Combined rate = 1/5; A+B rate = 1/15 + 1/20 = 7/60; C’s rate = 1/5 – 7/60 = 5/60 = 1/12-12 days.
Why others fail: Option C (15 days) arises from miscalculating A+B rate as 1/15 + 1/20 = 1/35.

Question: The average weight of 8 persons increases by 2.5 kg when a new person replaces one weighing 65 kg. What is the weight of the new person?
A) 75 kg
B) 80 kg
C) 85 kg
D) 90 kg
Answer: C
Explanation: Total increase = 8 × 2.5 = 20 kg; new person = 65 + 20 = 85 kg.
Why others fail: Option B (80 kg) results from assuming only 6 kg increase per person.

Question: A shopkeeper marks goods 40% above cost and gives 20% discount. His net profit % is:
A) 12%
B) 16%
C) 18%
D) 20%
Answer: A
Explanation: Let CP = ?100; MP = ?140; SP = 140 × 0.8 = ?112; profit = 12%.
Why others fail: Option B (16%) comes from adding 40% and subtracting 20% directly.

Question: A boat goes 30 km downstream in 2 hours and same distance upstream in 6 hours. Speed of boat in still water is:
A) 5 km/h
B) 8 km/h
C) 10 km/h
D) 12 km/h
Answer: C
Explanation: Downstream speed = 15 km/h, upstream = 5 km/h; boat speed = (15 + 5)/2 = 10 km/h.
Why others fail: Option B (8 km/h) results from averaging times instead of speeds.

Last?Minute Revision (20–25 one?liners)

• Average speed for equal distances = 2xy/(x+y); not (x+y)/2.
• Relative speed same direction: |S1 – S2|; opposite: S1 + S2.
• Train crossing platform: time = (train length + platform length)/speed.
• Profit % = (Profit / CP) × 100; never on SP unless specified.
• If CP of n items = SP of m items, profit % = ((n–m)/m) × 100.
• Successive discounts: net discount = 1 – (1–d1)(1–d2); not d1 + d2.
• False weight: if uses 900g for 1kg, profit % = (100/900) × 100 = 11.11%.
• Work rate: more efficiency-less time; inverse proportion.
• A 50% more efficient than B-time ratio A:B = 2:3.
• Pipes: inlet positive rate, outlet negative; net rate = sum.
• If A and B together take T days, T = (A×B)/(A+B); not (A+B)/2.
• Upstream speed = boat speed – stream speed.
• Downstream speed = boat speed + stream speed.
• When a person is replaced, change in average × count = difference in individual values.
• Weighted average: (w1x1 + w2x2)/(w1 + w2); used in mixture problems.
• If speed ratio A:B = 3:4, time ratio = 4:3.
• Time and speed inversely proportional; distance constant.
• 1 m/s = 18/5 km/h.
• 1 km/h = 5/18 m/s.
• If two people work alternately, compute work per cycle.
• Daily wage-work efficiency.
• For same distance, average speed = harmonic mean of speeds.
• If a tap fills tank in 6h, rate = 1/6 per hour.
• In race, "gives start of 10m" means runner starts 10m ahead.
• Verify from standard source: profit on SP is rare; always confirm basis.