A, B and C together can complete a work in $16\frac{4}{11}\text{days}$, while A and B together can complete the same work in 20 days. Find in how many days C alone can do 60% of the same work.

The correct answer is 54 days.

Let's break down the problem step-by-step to find the time taken by C to complete 60% of the work.

Step 1: Understand the given rates of work
Let the total work be represented by $W$.
The time taken by A, B, and C together to complete the work is:
$16\frac{4}{11}=\frac{16\times 11+4}{11}=\frac{176+4}{11}=\frac{180}{11}\text{days}$
Therefore, the combined rate of work of A, B, and C (work done per day) is:
$R_{A+B+C}=\frac{11}{180}\text{of the work per day}$

Step 2: Find the rate of work of A and B together
The time taken by A and B together to complete the work is 20 days.
Therefore, the combined rate of work of A and B is:
$R_{A+B}=\frac{1}{20}\text{of the work per day}$

Step 3: Determine the rate of work of C alone
The rate of work of C alone is the difference between the combined rate of (A + B + C) and the rate of (A + B):
$R_C=R_{A+B+C}-R_{A+B}$
Substituting the values we have:
$R_C=\frac{11}{180}-\frac{1}{20}$
To subtract these fractions, we find a common denominator, which is 180:
$R_C=\frac{11}{180}-\frac{9}{180}=\frac{2}{180}=\frac{1}{90}\text{of the work per day}$
This means C alone can complete the entire work in 90 days.

Step 4: Calculate the time taken by C to complete 60% of the work
We need to find the time required for C to complete 60% (or 0.6) of the total work:
$\text{Time taken}=\frac{60\%\text{of Work}}{R_C}=0.6\times 90=54\text{days}$
Thus, C alone can complete 60% of the work in 54 days.