Time, Work, Speed and Distance: Formulas and Worked Examples

Time and work

Treat the whole job as 1 unit. If A finishes a job in a days, A's rate is 1/a of the job per day.

• A and B together: combined rate = 1/a + 1/b, so time together = a × b ÷ (a + b).
• If A and B together take t days and A alone takes a days, B's rate = 1/t - 1/a.
• Work done = rate × time; remaining work = 1 - work done.
• A quick alternative is to take the LCM of the days as the total work in units.

Pipes and cisterns

A filling pipe is a worker with a positive rate; an emptying pipe (outlet or leak) has a negative rate. Net rate = sum of inlet rates - sum of outlet rates, and time to fill = 1 ÷ net rate.

Work and wages

Wages for a job done together are shared in the ratio of work done. If everyone works for the same number of days, that is the ratio of daily rates, which is the inverse of the ratio of days taken alone.

Speed, distance and time

• Distance = Speed × Time, Speed = Distance ÷ Time, Time = Distance ÷ Speed.
• To convert km/h to m/s multiply by 5/18; to convert m/s to km/h multiply by 18/5.
• Average speed = Total distance ÷ Total time. For equal distances at speeds x and y it is 2xy ÷ (x + y), never (x + y) ÷ 2.

Relative speed and trains

• A train passing a pole or a standing person covers its own length.
• A train passing a platform or bridge covers train length + platform length.
• Two trains passing each other cover the sum of their lengths at the relative speed.

Boats and streams

Let b be the speed of the boat in still water and s the speed of the stream.

• Downstream speed = b + s; upstream speed = b - s.
• b = (downstream + upstream) ÷ 2 and s = (downstream - upstream) ÷ 2.

Worked examples

1. Combined work. A takes 12 days and B takes 24 days. Combined rate = 1/12 + 1/24 = 3/24 = 1/8, so together they take 8 days.
2. Pipes. A fills in 20 minutes, B fills in 30 minutes and outlet C empties in 60 minutes. Net rate = 1/20 + 1/30 - 1/60 = (3 + 2 - 1)/60 = 1/15, so the tank fills in 15 minutes.
3. Average speed. Going at 40 km/h and returning at 60 km/h: 2 × 40 × 60 ÷ 100 = 48 km/h. Check with a distance of 120 km each way: time = 3 + 2 = 5 hours and 240 ÷ 5 = 48.
4. Trains. Trains 120 m and 180 m long run in opposite directions at 50 km/h and 58 km/h. Relative speed = 108 km/h = 108 × 5/18 = 30 m/s. Time to cross = 300 ÷ 30 = 10 seconds.
5. Boat. 30 km upstream takes 5 hours (6 km/h) and 30 km downstream takes 3 hours (10 km/h). Stream = (10 - 6) ÷ 2 = 2 km/h and boat = 8 km/h.

Exam pointers

• Add rates, never times. Two workers together always take less time than the faster one alone.
• Keep units consistent: metres with seconds, kilometres with hours. Convert before dividing.
• For a platform or a second train, add the lengths before dividing by the speed.
• Opposite directions means add the speeds; same direction means subtract.
• Wages follow work done. The slower worker, who needs more days alone, gets the smaller share.
• When a worker leaves midway, compute the work already done first, then give the rest to whoever remains.