Step 1: Understand the given conditions

Let the efficiencies of A, B, and C be $E_A$, $E_B$, and $E_C$ respectively. Since A can do the work in the same time as B and C together, their work rates satisfy the relation: $E_A = E_B + E_C$

$E_A = E_B + E_C$

Step 2: Express combined rate of A and B

Given that A and B together take 10 days to finish the work, their combined efficiency is: $E_A + E_B = \frac{1}{10}$

$E_A + E_B = \frac{1}{10}$

Step 3: Substitute $E_A$ in the equation

Substitute $E_A = E_B + E_C$ into the combined efficiency equation: $(E_B + E_C) + E_B = \frac{1}{10} \implies 2E_B + E_C = \frac{1}{10}$

$2E_B + E_C = \frac{1}{10}$

Step 4: Use C's individual rate

Since C alone takes 50 days, $E_C = \frac{1}{50}$. Substitute this value into the equation: $2E_B + \frac{1}{50} = \frac{1}{10}$

$2E_B + \frac{1}{50} = \frac{1}{10}$

Step 5: Solve for B's efficiency and time

Simplify the equation to find $E_B$: $2E_B = \frac{1}{10} - \frac{1}{50} = \frac{5 - 1}{50} = \frac{4}{50} = \frac{2}{25}E_B = \frac{1}{25}$ Thus, B alone can complete the work in $25$ days.

$E_B = \frac{1}{25} \implies \text{Time} = 25 \text{ days}$