Combined work and remaining work

Learning outcome

Find joint completion times, track work remaining after a departure, and recover an unknown worker’s rate from a staged schedule.

Concepts and assumptions

Count the whole job as 1. Each worker has a constant rate and works the same fixed hours per day. Assume their contributions add without interference or duplicated effort, and the job can be shared. Work stops at completion.

If A needs a days alone, A’s rate is 1/a job per day; similarly, B’s rate is 1/b. In one joint day, the completed fractions add. Thus their joint rate is 1/a + 1/b, and joint time is 1 divided by that rate. Averaging their individual completion times does not represent their combined output.

For a changing arrangement, calculate each stage separately:

Stage work = stage rate × stage duration.

Subtract completed work from 1, then divide the remainder by the next stage’s rate. A departure changes future output, not work already completed.

Alternatively, choose a convenient whole-job size, such as a common multiple of individual times. Dividing this size by each time gives integer daily outputs. Changing the work unit simplifies arithmetic but cannot change the answer. Keep rates and remaining work in compatible units. Fractional days represent part of a standard working day.

Worked examples

Example 1 — Joint work. A needs 12 days alone and B 18 days. Find their time together.

Take the job as 36 units. A produces 36/12 = 3 units daily; B produces 36/18 = 2. Joint output = 3 + 2 = 5 units daily. Required time = 36/5 = 7 1/5 days, exactly.

Example 2 — One worker leaves. A needs 15 days alone and B 20 days. They work together for 4 days; then B leaves. Find the total completion time.

Joint rate = 1/15 + 1/20 = 7/60. Work completed = 4 × 7/60 = 7/15. Remaining work = 1 − 7/15 = 8/15. A’s additional time = (8/15)/(1/15) = 8 days. Total time = 4 + 8 = 12 days.

Example 3 — Finding an unknown rate. A needs 24 days alone. A and B work together for 6 days; B then leaves, and A finishes in another 9 days. Find B’s individual completion time.

The final 9 days complete 9/24 = 3/8 of the job. Thus the first stage completes 1 − 3/8 = 5/8. Joint rate = (5/8)/6 = 5/48. B’s rate = 5/48 − 1/24 = 3/48 = 1/16. B alone needs 16 days. Check: A contributes 15/24 = 5/8 overall; B contributes 6/16 = 3/8.

Common mistakes

Adding completion times; giving a departed worker credit for later days; dividing the whole job instead of the remainder; reporting additional time as total time.

Practice questions

1. A needs 16 days alone and B 24 days. How long do they need together?
2. A needs 10 days and B 15 days alone. After 3 joint days, A leaves. Find B’s additional time and the total time.
3. A and B together need 12 days; A alone needs 30 days. Find B’s time alone.
4. A, B and C alone need 18, 27 and 54 days respectively. A and B work for 3 days; then C replaces B. Find total time.

Worked solutions

1. Joint rate = 1/16 + 1/24 = 5/48. Time = 1/(5/48) = 48/5 = 9 3/5 days.
2. Joint rate = 1/10 + 1/15 = 1/6. Three days complete 1/2. B needs (1/2)/(1/15) = 7 1/2 more days. Total = 10 1/2 days.
3. B’s rate = 1/12 − 1/30 = 3/60 = 1/20. Therefore B needs 20 days.
4. First-stage work = 3(1/18 + 1/27) = 5/18. Remainder = 13/18. A and C’s rate = 1/18 + 1/54 = 2/27. Additional time = (13/18)/(2/27) = 39/4 days. Total = 3 + 39/4 = 51/4 = 12 3/4 days.