A, B, and C together can complete a piece of work in $12\frac{3}{5}$ days, while A and C together can finish it in 14 days. Find in how many days B alone can complete 66.67% of the same work.

To find the number of days B alone takes to complete 66.67% of the work, let us first analyze the work rates of the individuals involved.

First, we are given that A, B, and C together can complete the work in $12\frac{3}{5}$ days.
Let us convert this mixed fraction into an improper fraction:
$12\frac{3}{5}=\frac{12\times 5+3}{5}=\frac{63}{5}$ days.

Therefore, the combined daily work rate of A, B, and C is the reciprocal of the time taken:
$\text{Rate of }(A+B+C)=\frac{5}{63}$ of the total work per day.

Next, we are given that A and C together can finish the work in 14 days.
So, the combined daily work rate of A and C is:
$\text{Rate of }(A+C)=\frac{1}{14}$ of the total work per day.

We can find the individual daily work rate of B by subtracting the combined rate of A and C from the combined rate of A, B, and C:
$\text{Rate of }B=\text{Rate of }(A+B+C)-\text{Rate of }(A+C)$
$\text{Rate of }B=\frac{5}{63}-\frac{1}{14}$

To subtract these fractions, we find a common denominator for 63 and 14.
Since 63 = 9 × 7 and 14 = 2 × 7, the Least Common Multiple (LCM) is 9 × 2 × 7 = 126.
Now, rewrite the fractions with the common denominator:
$\frac{5}{63}=\frac{5\times 2}{126}=\frac{10}{126}$
$\frac{1}{14}=\frac{1\times 9}{126}=\frac{9}{126}$

Subtract the two values to get B's rate:
$\text{Rate of }B=\frac{10}{126}-\frac{9}{126}=\frac{1}{126}$

This means B alone can complete the entire (100%) work in 126 days.

The question asks for the time B alone takes to complete 66.67% of the same work.
Note that 66.67% is equivalent to the fraction:
$\frac{2}{3}$

Thus, the time required by B to complete 66.67% of the work is:
$\text{Time}=126\times \frac{2}{3}$
$\text{Time}=42\times 2=84\text{ days}$

The correct answer is 84 days.