Most doubts in my live sessions are about Maths, and these two chapters come up again and again. Both look formula-heavy, but in the exam you need only a few ideas. Learn the method once, then practise each type until you stop thinking about the method.
What this topic covers
Time and Work: the LCM method
Take the total work = LCM of the days given. Then each person's work per day is a whole number.
- A does a job in 10 days, B in 15 days. Together?
Total work = LCM(10, 15) = 30 units. A = 3 units/day, B = 2 units/day. Together 5 units/day → 30 ÷ 5 = 6 days.
- A and B together take 12 days; A alone takes 20. B alone?
Total = LCM(12, 20) = 60. (A + B) = 5/day, A = 3/day, so B = 2/day → 60 ÷ 2 = 30 days.
Pipes and cisterns
The same method. A pipe that empties does negative work.
- Pipe A fills a tank in 20 hours, pipe B empties it in 30 hours. Both open?
Total = 60. A = +3/hour, B = −2/hour. Net +1/hour → 60 hours.
Wages
Wages are shared in the ratio of work done, not in the ratio of days. In the first example, A and B share wages 3 : 2.
Speed, Time and Distance
- Distance = Speed × Time.
- km/h → m/s: × 5/18. m/s → km/h: × 18/5. (72 km/h = 20 m/s.)
- Average speed for equal distances at x and y: 2xy/(x + y).
Trains
- Crossing a pole or a person: distance = length of the train.
- Crossing a platform or bridge: distance = train + platform.
- Two trains: relative speed = sum if moving in opposite directions, difference if in the same direction; distance = sum of both lengths.
Boats and streams
- Downstream speed = u + v; upstream = u − v (u = boat in still water, v = stream).
- Still water speed = (down + up) / 2; stream speed = (down − up) / 2.
How it is asked in the exam
Example 1: train and pole, then platform
A 200 m train runs at 72 km/h. How long does it take to cross a pole, and a 300 m platform?
72 km/h = 20 m/s. Pole: 200 ÷ 20 = 10 s. Platform: (200 + 300) ÷ 20 = 25 s.
Example 2: two trains, opposite directions
Trains of 150 m and 100 m run towards each other at 50 km/h and 40 km/h. How long do they take to cross each other?
Relative speed = 90 km/h = 25 m/s. Distance = 250 m. Time = 10 s.
Example 3: average speed
A man drives to a town at 40 km/h and returns at 60 km/h. What is his average speed?
2 × 40 × 60 ÷ 100 = 48 km/h.
Example 4: boats
A boat goes 15 km/h downstream and 9 km/h upstream. Find the speed in still water and of the stream.
Still water = (15 + 9) ÷ 2 = 12 km/h. Stream = (15 − 9) ÷ 2 = 3 km/h.
Points to remember
- Average speed is never the simple average when distances are equal.
- Always convert units first. Train lengths are in metres, speeds usually in km/h.
- In time and work, efficiency and time are inversely proportional: if A is twice as fast as B, A takes half the time.
Common mistakes
- Forgetting to add the platform length.
- Adding speeds for trains moving in the same direction.
- Splitting wages by days worked instead of work done.
Practise straight after reading
List the types above (work together, one alone, pipes, wages, pole, platform, two trains, average speed, boats). Solve 15 questions of each type from R.S. Aggarwal or your own book, then 100 to 200 mixed questions from the Maths quizzes on this site. Revise your wrong answers after 1, 7 and 30 days.