A, B and C together can complete $66\frac{2}{3}$% of work in 2 days and A & B together can complete 30% of the work in 3 days. Find the time taken by C alone to complete $4\frac{2}{3}$ of total work.

The correct answer is 20 days.

Step 1: Calculate the total time taken by (A + B + C) together to complete the entire work.
We are given that A, B, and C together can complete $66\frac{2}{3}$% of the work in 2 days.
First, convert $66\frac{2}{3}$% into a fraction:

$66\frac{2}{3}\%=\frac{200}{3}\%=\frac{200}{3\times 100}=\frac{2}{3}$

If (A + B + C) complete $\frac{2}{3}$ of the work in 2 days, then the time taken by them to complete the total (1 unit) work is:

$\text{Time taken by (A + B + C)}=2\times \frac{3}{2}=3\text{days}$

Step 2: Calculate the 1-day work rate of (A + B + C).
Since they complete 1 full work in 3 days, their combined 1-day work is:

$\text{Rate of (A + B + C)}=\frac{1}{3}\text{of the total work per day}$

Step 3: Calculate the total time taken by (A + B) together to complete the entire work.
We are given that A and B together can complete 30% of the work in 3 days.
Convert 30% into a fraction:

$30\%=\frac{30}{100}=\frac{3}{10}$

If (A + B) complete $\frac{3}{10}$ of the work in 3 days, then the time taken by them to complete the total work is:

$\text{Time taken by (A + B)}=3\times \frac{10}{3}=10\text{days}$

Step 4: Calculate the 1-day work rate of (A + B).
Since A and B complete the total work in 10 days, their combined 1-day work is:

$\text{Rate of (A + B)}=\frac{1}{10}\text{of the total work per day}$

Step 5: Find the 1-day work rate of C alone.
Subtract the 1-day work rate of (A + B) from the 1-day work rate of (A + B + C):

$\text{Rate of C}=\frac{1}{3}-\frac{1}{10}=\frac{10-3}{30}=\frac{7}{30}\text{of the work per day}$

Step 6: Calculate the time taken by C alone to complete $4\frac{2}{3}$ of the total work.
First, convert the mixed fraction $4\frac{2}{3}$ into an improper fraction:

$4\frac{2}{3}=\frac{4\times 3+2}{3}=\frac{14}{3}$

Now, find the time taken by C alone using the formula:

$\text{Time}=\frac{\text{Work to be done}}{\text{Rate of C}}$

$\text{Time}=\frac{\frac{14}{3}}{\frac{7}{30}}=\frac{14}{3}\times \frac{30}{7}=\left(\frac{14}{7}\right)\times \left(\frac{30}{3}\right)=2\times 10=20\text{days}$

Thus, C alone takes 20 days to complete $4\frac{2}{3}$ of the total work.