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 $8\frac{2}{3}$ days. What is n equal to?

The correct answer is Option 2: 4.

Step 1: Calculate the total time required by each person to complete the whole work individually.

For X:
X completes 1/3 of the work in 6 days.
Therefore, X takes 6 × 3 = 18 days to complete the full work.

For Y:
Y completes 1/3 of the work in 8 days.
Therefore, Y takes 8 × 3 = 24 days to complete the full work.

For Z:
Z completes 3/4 of the work in 12 days.
Therefore, Z takes 12 × (4/3) = 16 days to complete the full work.

Step 2: Determine their 1-day work rate.

X's 1-day work = $\frac{1}{18}$

Y's 1-day work = $\frac{1}{24}$

Z's 1-day work = $\frac{1}{16}$

Step 3: Calculate the combined 1-day work of X, Y, and Z.

Combined 1-day work of (X + Y + Z) = $\frac{1}{18}+\frac{1}{24}+\frac{1}{16}$

Find the LCM of denominators 18, 24, and 16, which is 144.

$\frac{1}{18}+\frac{1}{24}+\frac{1}{16}=\frac{8+6+9}{144}=\frac{23}{144}$

So, together they do $\frac{23}{144}$ unit of work per day.

Step 4: Calculate the work completed by Y alone in the remaining days.

Y works alone for $8\frac{2}{3}=\frac{26}{3}$ days.

Work done by Y alone in $\frac{26}{3}$ days = $\frac{26}{3}\times \frac{1}{24}=\frac{13}{36}$

Step 5: Calculate the remaining work done together by X, Y, and Z.

Remaining work (work done in the initial n days by X, Y, and Z together) = $1-\frac{13}{36}=\frac{23}{36}$

Step 6: Find the value of n.

Since X, Y, and Z work together for n days, the work done in n days is equal to n × (Combined 1-day work).

$n\times \frac{23}{144}=\frac{23}{36}$

$n=\frac{23}{36}\times \frac{144}{23}=\frac{144}{36}=4$

Therefore, the value of n is equal to 4.