X, Y , and Z can complete a project in 20, 40, and 60 days respectively. X works every day, while Y and Z join X every third day. How long will it take to finish the project?

The correct option is $15\frac{5}{6}$ days.

Let's solve the problem step-by-step to understand how long it takes X, Y, and Z to complete the project.

Step 1: Calculate the total work and individual daily work rates

Let the total work be the Least Common Multiple (LCM) of the days taken by X, Y, and Z, which are 20, 40, and 60 days respectively.

LCM(20, 40, 60) = 120 units.

Thus, total work = 120 units.

Now, calculate the work done per day by each person:

Work done by X per day = 120 / 20 = 6 units/day

Work done by Y per day = 120 / 40 = 3 units/day

Work done by Z per day = 120 / 60 = 2 units/day

Step 2: Understand the work pattern

The work happens in a 3-day cycle pattern:

Day 1: Only X works = 6 units

Day 2: Only X works = 6 units

Day 3: X, Y, and Z work together = 6 + 3 + 2 = 11 units

Total work completed in one 3-day cycle:

$6+6+11=23$ units

Step 3: Calculate the number of complete 3-day cycles

To find how many full 3-day cycles are required to complete most of the 120 units of work:

$120÷23=5$ cycles with a remainder.

Work done in 5 full cycles (which takes $5\times 3=15$ days):

$5\times 23=115$ units

Remaining work after 15 days:

$120-115=5$ units

Step 4: Calculate time taken for the remaining work

On the 16th day (the 1st day of the 6th cycle), only X works.

X's rate of work is 6 units per day.

Time required by X to complete the remaining 5 units of work:

$\frac{5}{6}$ days

Step 5: Total time to complete the project

Total time = 15 days + $\frac{5}{6}$ days = $15\frac{5}{6}$ days.