A, B and C together can complete a work in 16*4/11  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.

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 work rates
Let the total work be represented by $W$.
A, B, and C together can complete the work in $16\frac{4}{11}$ days.
First, let's convert this mixed fraction into an improper fraction:
$16\frac{4}{11}=\frac{16\times 11+4}{11}=\frac{176+4}{11}=\frac{180}{11}$ days.

Thus, the combined work rate of A, B, and C (work done per day) is:
$R_{A+B+C}=\frac{1}{180/11}=\frac{11}{180}$ of the work per day.

Similarly, A and B together can complete the same work in 20 days.
So, the combined work rate of A and B is:
$R_{A+B}=\frac{1}{20}$ of the work per day.

Step 2: Find the work rate of C alone
The work rate of C can be found by subtracting the combined rate of A and B from the combined rate of all three:
$R_C=R_{A+B+C}-R_{A+B}$

Substitute the values we calculated:
$R_C=\frac{11}{180}-\frac{1}{20}$

To subtract these fractions, find a common denominator, which is 180:
$R_C=\frac{11}{180}-\frac{9}{180}=\frac{2}{180}=\frac{1}{90}$ of the work per day.

This means C alone can complete the entire work in 90 days.

Step 3: Calculate the time taken by C to complete 60% of the work
Since C takes 90 days to do 100% of the work, the time required to complete 60% of the work is:
$\text{Required Time}=60\%\times 90\text{days}$
$\text{Required Time}=\frac{60}{100}\times 90=0.6\times 90=54\text{days}$.

Therefore, C alone can complete 60% of the work in 54 days.