Q. X can complete one-third of a certain work in 6 days, Y can complete one-third of the same work in 8 days and Z can complete three-fourth of the same work in 12 days. All of them work together for n days and then X and Z quit and Y alone finishes the remaining work in 8 and 2/3 days. What is n equal to?

Explanation : 

1. Compute each person’s daily work rate (fraction of the whole work per day):

• X does 1/3 of the work in 6 days ⇒ rate of X = (1/3) ÷ 6 = 1/18
• Y does 1/3 of the work in 8 days ⇒ rate of Y = (1/3) ÷ 8 = 1/24
• Z does 3/4 of the work in 12 days ⇒ rate of Z = (3/4) ÷ 12 = 1/16

2. All three work together for n days. Together they do per day:
1/18 + 1/24 + 1/16
= (8 + 6 + 9) / 144
= 23/144 of the work per day.
In n days they complete 23n/144 of the work.

3. Remaining work after n days = 1 – 23n/144 = (144 – 23n)/144.

4. Then X and Z quit, and Y alone finishes the rest in 8 2/3 days = 26/3 days.
Work done by Y in that time = rate × time = (1/24) × (26/3) = 26/72 = 13/36.

5. Set remaining work = 13/36:

(144 – 23n) / 144 = 13/36
⇒ 144·(13/36) = 144 – 23n
⇒ 4·13 = 144 – 23n
⇒ 52 = 144 – 23n
⇒ 23n = 144 – 52 = 92
⇒ n = 92 / 23 = 4.

Answer: (b) 4