Time and work
Replace confusing day counts with rates, combine compatible rates, and convert the result back to time.
Represent one whole job#
Treat the complete task as 1. If A completes it in 6 days, A's daily rate is 1/6. If B completes it in 3 days, B's rate is 1/3.
Working together at constant rates:
1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 job per day.
They need 1 ÷ 1/2 = 2 days.
Partial work#
If A works alone for two days, A completes 2 × 1/6 = 1/3 of the job. The remaining work is 2/3. Use the appropriate later rate only for that remainder.
Efficiency#
For the same task, higher efficiency means less time. If A is twice as efficient as B, A's rate is double and A's time is half.
Inlets and outlets#
An inlet adds work while an outlet removes it, so their rates have opposite signs. State which direction is positive before combining them.
Common mistakes#
Do not add completion times. Add rates. Also avoid assuming every worker contributes equally when the problem gives different individual times.
Assumptions#
Basic problems assume constant independent rates and no interference. Real teams may not scale this way, but use the stated mathematical model.
Final check#
The combined time for two positive workers should be less than either individual time. If it is larger, inspect the rate conversion or fraction addition.
Key points
- If a job takes n days, the daily work rate is one divided by n.
- Compatible work rates add when people work together.
- Time equals total work divided by combined rate.
- Efficiency and time are inversely related for the same amount of work.