A, B and C together can complete 66⅔ % of work in 2 days and A & B together…

Concept

In work-and-time problems, each worker (or group) has a constant daily work rate equal to the fraction of the job they finish per day. Rates add: if a combined group's rate is known and one sub-group's rate is known, an individual's rate is the difference. Time to finish any amount of work equals that amount divided by the rate.

Application

1. Convert the given completions to rates. A + B + C finish 66⅔% = 2/3 of the work in 2 days, so their combined rate is (2/3) ÷ 2 = 1/3 of the work per day.

2. A + B finish 30% = 3/10 of the work in 3 days, so their combined rate is (3/10) ÷ 3 = 1/10 of the work per day.

3. C's own rate is the difference: 1/3 − 1/10 = 10/30 − 3/30 = 7/30 of the work per day.

4. The required amount is 4⅔ = 14/3 (in units of one whole job). Time = work ÷ rate = (14/3) ÷ (7/30) = (14/3) × (30/7) = 20 days.

Cross-check

Verify by multiplying back: in 20 days C does 20 × 7/30 = 140/30 = 14/3 of the job, which is exactly 4⅔. The rate split is also consistent because 1/10 (for A+B) is less than 1/3 (for A+B+C), so C contributes a positive rate of 7/30, as required.