The wording of this problem is terrible and it can't really be solved as written, but making a few assumptions, we can say:

12 children * 16 days = 192 days, so each kid does 1/192 of the job per day

8 adults * 12 days = 96 days, so each adult does 1/96 of the job per day, or DOUBLE the rate of each kid

From here:

first 3 days:
W = 3 * (1 / 96) * 16 => 48/96 => 1/2

so we have 1/2 of the job left. Now

W = 1/2
R = 6*(1/96) + 4*(1/192)
T = t

1/2 = (6/96 + 4/192)*t

6 = t

Joy Shaha wrote:Q. 12 children take 16 days to complete the work which can be completed by 8 adults in 12 days.16 adults started work and after 3 days 10 adults left & 4 children join them.How many days will they take to complete the work.
A.3;
B.4;
C.6;
D.8;
E.None

Assuming the children and adults all work at identical rates (the actual GMAT would probably not require you to make such an assumption; it would be made explicit in the question prompt)...

Since 8 adults take 12 days to complete the task, 16 adults would take 6 days to complete the same task. Thus, after 3 days, the 16 adults would accomplish exactly 1/2 of the task. So when the 10 adults and the 4 children join, 1/2 of the task has already been completed. Eliminate choice (E).

Now we have 6 adults and 4 children working on the remaining 1/2 of the task. Note that the 6 adults working alone would complete the remaining 1/2 of the task in 6x(4/3)=8 days. So (D) can also be eliminated, as we have 4 children also assisting that time frame will be less than that. The 4 children working alone could complete the remaining 1/2 of the task in 8x3=24 days.

To make things a little easier algebraically, I set up the rates for the adults & children as a collective unit and defined the task as the remaining 1/2 of the work.

x/8 + x/24 = 1
3x/24 + x/24 = 1
4x/24 = 1
4x = 24
x = 6 days

Answer choice:

C

Joy Shaha wrote:Q. 12 children take 16 days to complete the work which can be completed by 8 adults in 12 days.16 adults started work and after 3 days 10 adults left & 4 children join them.How many days will they take to complete the work.
A.3;
B.4;
C.6;
D.8;
E.None

Let the job = 96 units.

12 children take 16 days:
Work produced each day = (total job)/(number of days) = 96/16 = 6 units per day.
Thus, the daily rate for each child = (total work per day)/(number of children) = 6/12 = 1/2 unit per day.

8 adults take 12 days:
Work produced each day = (total job)/(number of days) = 96/12 = 8 units per day.
Thus, the daily rate for each adult = (total work per day)/(number of adults) = 8/8 = 1 unit per day.

For the first 3 days, 16 adults work:
Since each adult produces 1 unit per day, the daily rate for 16 adults = 16 units per day.
At a rate of 16 units per day, the amount of work produced over 3 days = rt = 16*3 = 48 units.
Remaining units = 96-48 = 48 units.

After 3 days 10 adults left & 4 children join them:
After 10 adults leave, the daily rate for the remaining 6 adults = 6 units per day.
Since each child produces 1/2 unit per day, the daily rate for 4 children = 2 units per day.
Combined rate for 6 adults and 4 children = 6+2 = 8 units per day.
At a combined rate of 8 units per day, the time to produce the remaining 48 units = w/r = 48/8 = 6 days.

The correct answer is C.

Joy Shaha wrote:Q. 12 children take 16 days to complete the work which can be completed by 8 adults in 12 days.16 adults started work and after 3 days 10 adults left & 4 children join them.How many days will they take to complete the work.
A.3;
B.4;
C.6;
D.8;
E.None

Assume total units of work = 100x
Rate of work y children = C
Rate of work by adults = A

12C*16 = 8A*12
A = 2C

Total work = 96A or 192C

Work done by 1 adults in 3 days = 16A*3 = 48A
Amount of work left = 48A

People doing this work = 6 adults and 4 children.
Hence total units of work done each day = 6A + 4C or 6A = 2A = 8A

Days required to do the remaining job = 48A/8A = 6

Correct Option: C