Sam can complete a job in 20 days when working alone. Mohit is twice as fast as Sam and thrice as fast as Ayna in the same job. They undertake a job with an arrangement where Sam and Mohit work together on the first day, Sam and Ayna on the second day, Mohit and Ayna on the third day, and this three-day pattern is repeated till the work gets completed. Then, the fraction of total work done by Sam is:

The correct option is 3/10.

To find the fraction of the total work done by Sam, let us analyze the individual work rates (efficiencies) of Sam, Mohit, and Ayna by assuming a convenient value for the total work.

Let the total work to be completed be 60 units (which is a convenient common multiple to avoid fractions during calculations).

Since Sam can complete the job alone in 20 days, Sam's rate of work per day is:
$S=\frac{60}{20}=3\text{ units/day}$

Mohit is twice as fast as Sam. Therefore, Mohit's rate of work per day is:
$M=2\times 3=6\text{ units/day}$

Mohit is thrice as fast as Ayna. Thus, Ayna's rate of work per day is:
$A=\frac{6}{3}=2\text{ units/day}$

Now, let us trace the work done day-by-day based on the repeating three-day pattern:

Day 1: Sam and Mohit work together
Work done on Day 1 = $3+6=9\text{ units}$
Sam's contribution = 3 units

Day 2: Sam and Ayna work together
Work done on Day 2 = $3+2=5\text{ units}$
Sam's contribution = 3 units

Day 3: Mohit and Ayna work together
Work done on Day 3 = $6+2=8\text{ units}$
Sam's contribution = 0 units

In one complete 3-day cycle:
Total work completed = $9+5+8=22\text{ units}$
Work contributed by Sam = $3+3+0=6\text{ units}$

Now, we calculate how many cycles are needed to complete the total work of 60 units:

In 2 full cycles (6 days):
Total work completed = $22\times 2=44\text{ units}$
Remaining work = $60-44=16\text{ units}$
Work done by Sam so far = $6\times 2=12\text{ units}$

Day 7 (Day 1 of Cycle 3): Sam and Mohit work
Work completed on Day 7 = 9 units
Remaining work = $16-9=7\text{ units}$
Work done by Sam on this day = 3 units

Day 8 (Day 2 of Cycle 3): Sam and Ayna work
Work completed on Day 8 = 5 units
Remaining work = $7-5=2\text{ units}$
Work done by Sam on this day = 3 units

Day 9 (Day 3 of Cycle 3): Mohit and Ayna work
Since Mohit and Ayna can complete 8 units in a day, they will easily finish the remaining 2 units on this day. Sam does not work on this day, so Sam's contribution on Day 9 is 0 units.

Now, we sum the total work done by Sam across all days:
$\text{Total work done by Sam}=12\text{ (from first 2 cycles)}+3\text{ (from Day 7)}+3\text{ (from Day 8)}=18\text{ units}$

Finally, we calculate the fraction of the total work done by Sam:
$\text{Fraction}=\frac{18}{60}=\frac{3}{10}$