Time and work, speed and distance

Time and work

• If A finishes a job in a days, A does 1/a of it per day.
• Together: 1/a + 1/b per day, so the time taken = ab ÷ (a + b).
• LCM method (faster): take total work = LCM of the days.

Example: A takes 12 days and B takes 18 days. Total work = 36 units; A does 3/day, B 2/day; together 5/day → 36 ÷ 5 = 7.2 days.

Example: A and B together take 10 days; A alone takes 15. Work = 30 units; together 3/day, A 2/day, so B does 1/day → B alone takes 30 days.

Pipes and cisterns work the same way; a leak or outlet pipe contributes negative work.

Speed, distance and time

• Distance = speed × time. km/h to m/s: multiply by 5/18; m/s to km/h: multiply by 18/5.
• Average speed for equal distances at speeds x and y = 2xy ÷ (x + y) (not their simple average).
• Relative speed: same direction → subtract; opposite directions → add.
• Trains: crossing a pole → distance = the train's length; crossing a platform or another train → the sum of the lengths.
• Boats and streams: downstream = b + s, upstream = b − s; boat speed = (down + up) ÷ 2; stream speed = (down − up) ÷ 2.

Example: a 200 m train at 72 km/h (20 m/s) crosses a 300 m platform in (200 + 300) ÷ 20 = 25 seconds.

Example: you go to college at 30 km/h and return at 20 km/h. Average speed = 2 × 30 × 20 ÷ 50 = 24 km/h.

Check yourself

• A can do a job in 20 days, B in 30. How long together?
• Convert 54 km/h to m/s.
• Two trains of 150 m and 100 m travel in opposite directions at 40 and 50 km/h. How long to cross each other?