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:

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1. **Work Rates of Individuals:**  
   - Sam can complete the job in 20 days, so his rate of work per day is $$ \frac{1}{20} $$. Let this be $$ 3k $$.  
   - Mohit is twice as fast as Sam, so his rate of work is $$ 6k $$.  
   - Ayna is one-third as fast as Mohit, so her rate of work is $$ 2k $$.

2. **Work Done in One Cycle (3 Days):**  
   - **Day 1:** Work done by Sam and Mohit = $$ 3k + 6k = 9k $$.  
   - **Day 2:** Work done by Sam and Ayna = $$ 3k + 2k = 5k $$.  
   - **Day 3:** Work done by Mohit and Ayna = $$ 6k + 2k = 8k $$.  
   - Total work done in one cycle (3 days) = $$ 9k + 5k + 8k = 22k $$.

3. **Total Work to Be Done (LCM Approach):**  
   - The total work is assumed to be a multiple of the cycle's work (22k) and divisible by all rates of work. Let it be $$ 60k $$ (LCM of individual rates).

4. **Number of Cycles to Complete Work:**  
   - In two cycles (6 days), $$ 44k $$ work is completed. After two cycles, $$ 16k $$ work remains ($$ 60k - 44k = 16k $$).

5. **Distribution of Remaining Work:**  
   - Day 7: Sam and Mohit complete $$ 9k $$, leaving $$ 7k $$ remaining.  
   - Day 8: Sam and Ayna complete $$ 5k $$, leaving $$ 2k $$ remaining.  
   - Day 9: Mohit and Ayna complete the remaining $$ 2k $$.

6. **Work Done by Sam:**  
   - In two full cycles, Sam works for two days per cycle ($$ Day~1~\&~Day~2$$), contributing $$ (3k + 3k) = 6k $$ per cycle. Over two cycles, he completes $$ 12k $$.  
   - In the remaining days (Day 7 & Day 8), he contributes another $$ (3k + 3k) = 6k $$.  
   - Total work done by Sam = $$ 12k + 6k = 18k $$.

7. **Fraction of Total Work Done by Sam:**  
   - Fraction = $$ \frac{	ext{Work done by Sam}}{	ext{Total Work}} = \frac{18k}{60k} = \frac{3}{10} $$.

Thus, the fraction of total work done by Sam is **$$ \frac{3}{10} $$**.

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