69. 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 4/3 days. What is n equal to?

1. A. 3
2. B. 4
3. C. 5
4. D. 6

Answer: B

Explanation

Let’s find the time each person takes to complete the full work:
* X completes 1/3 work in 6 days => X completes full work in 6 × 3 = 18 days.
* Y completes 1/3 work in 8 days => Y completes full work in 8 × 3 = 24 days.
* Z completes 3/4 work in 12 days => Z completes full work in 12 × (4/3) = 16 days.

Now, let’s find their individual work rates (efficiency). Assume the total work is the LCM of 18, 24, and 16.
LCM(18, 24, 16) = 144 units.
* X’s efficiency = 144 units / 18 days = 8 units/day.
* Y’s efficiency = 144 units / 24 days = 6 units/day.
* Z’s efficiency = 144 units / 16 days = 9 units/day.

Combined efficiency of X, Y, Z = 8 + 6 + 9 = 23 units/day.

They work together for ‘n’ days. Work done in ‘n’ days = 23n units.
Remaining work = 144 – 23n units.

Y alone finishes the remaining work in 8 4/3 days. Assuming 8 4/3 days is a typo and means 8 and 2/3 days (as 4/3 is improper, and the solution uses 26/3), then time = (8 × 3 + 2)/3 = 26/3 days.
Work done by Y in 26/3 days = Y’s efficiency × time = 6 units/day × (26/3) days = 2 × 26 = 52 units.

So, the remaining work (144 – 23n) must be equal to 52 units.
144 – 23n = 52
23n = 144 – 52
23n = 92
n = 92 / 23
n = 4

Therefore, n is equal to 4. This question tests the application of time and work concepts, including calculating efficiencies and combining work rates.