Work is a rate, not a pile of days
Time and work questions become ordinary once you stop adding days. If a person finishes a job in n days, that person completes 1/n of the job each day. Two people working together add their daily fractions. The day count comes out only after the rates are combined.
Write the whole job as 1. Everything else is a fraction of that 1.
Two people on the same job
A finishes a job in 10 days, so A’s rate is 1/10 per day. B finishes it in 15 days, so B’s rate is 1/15 per day. Together:
1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6
They complete one-sixth of the job each day, so the job takes 6 days. Adding 10 and 15, or averaging them, does not answer the question.
The general line is: if A takes a days and B takes b days, together they take ab / (a + b) days. For 10 and 15 that is 150 / 25 = 6. Use the fraction sum when a third person joins; the product formula is only the two-person case.
Someone leaves midway
A and B work together, but A stops after 2 days. How long does B need to finish the rest alone?
In 2 days, A has done 2/10 = 1/5 of the job. If they were together for those 2 days, B has also done 2/15. Combined work so far:
2/10 + 2/15 = 3/15 + 2/15 = 5/15 = 1/3
Remaining work is 2/3. B’s rate is 1/15 per day, so B needs (2/3) ÷ (1/15) = (2/3) × 15 = 10 more days.
Read the sentence carefully. “A works alone for 2 days, then B finishes” is a different sum from “both work for 2 days, then A leaves.” The first version leaves 4/5, and B needs (4/5) × 15 = 12 days.
One of them is more efficient
If A is 50% more efficient than B, A’s rate is 1.5 times B’s rate. Suppose B finishes in 15 days. Then A finishes in 15 / 1.5 = 10 days. Efficiency and time move in opposite directions: higher efficiency means fewer days for the same job.
A clean order of work
- Turn each worker into a daily fraction, 1/n.
- Add the fractions for the days they actually work.
- Subtract from 1 to get what is left.
- Divide the remainder by the rate of whoever finishes it.
If the fractions share an awkward denominator, you can set the total work to the LCM of the day counts instead of 1. With 10 and 15, total work can be 30 units. A does 3 units a day, B does 2. The arithmetic is the same idea with whole numbers. Either way, set the rates before you touch the calculator.