Correct Answer - Option 2 : 8

__Calculation:__

Efficiency of Rakesh = 1/6 = 6/36 (Using LCM of 6,9 and 12)

Efficiency of Sam = 1/9 = 4/36

Efficiency of Raj = 1/12 = 3/36

After 3 days 6/36 + 4/36 + 3/36 will be completed.i.e 13/36 part will be completed.

Let n be the number of cycles needed to complete the task.

⇒ number of cycle × parts completed in a cycle = total task

⇒ n × (13/36) = 1

⇒ n = 2.769

Part of the work completed in a day is not the same.

⇒ number of days will be between 2 × 3 to 3 × 3

∴ For completing the task more than 6 days is needed, but it will be completed in less than 9 days.

⇒ After the 7th day, 26/36 + 6/36 (Rakesh working) = 32/36 part completed.

⇒ After the 8th day, 32/36 + 4/36 (Sam working)= 36/36 part completed.

**It will take 8 days to complete the task**

In this type of questions, the answerer needs to consider working cycles and then need to find the part of the work completed in one cycle. Here three days is the number of days in one cycle and 13/36 part will be completed in one cycle.

Number of cycles × work completed in a cycle = total task ---(1)

The number of days for the work= number of cycles × number of days in a cycle. ---(2)

Also, some times the work completed in the cycle will not be uniform. Like in this case where FIrst day 6/36 part and second day 4/36 will be completed. So we need to use equation (2) to find the range of days.