Milinda takes `8 1/3` hours more when she works alone in comparison of when she works with Bill. While Bill takes `5 1/3` hours more when he work alone in comparison to when he works with Milinda. How long it will take Bill to complete the work alone?

Options

A. 10 hours
B. 15 hours
C. 18 hours
D. 12 hours

Answer

To solve the problem, we need to establish the relationship between the time taken by Milinda and Bill when they work alone compared to when they work together.

Let:
- \( x \) = time taken by Milinda and Bill to complete the work together.
- \( M \) = time taken by Milinda to complete the work alone.
- \( B \) = time taken by Bill to complete the work alone.

From the problem statement:
1. Milinda takes \( 8 \frac{1}{3} \) hours more when she works alone compared to when she works with Bill. This can be expressed as:
\[ M = x + 8 \frac{1}{3} = x + \frac{25}{3} \]

2. Bill takes \( 5 \frac{1}{3} \) hours more when he works alone compared to when he works with Milinda. This can be expressed as:
\[ B = x + 5 \frac{1}{3} = x + \frac{16}{3} \]

Now, we know that the work done by Milinda and Bill together in one hour is the sum of their individual work rates:
\[ \frac{1}{M} + \frac{1}{B} = \frac{1}{x} \]

Substituting the expressions for \( M \) and \( B \):
\[ \frac{1}{x + \frac{25}{3}} + \frac{1}{x + \frac{16}{3}} = \frac{1}{x} \]

To solve this equation, we first find a common denominator for the left-hand side:
\[ \frac{(x + \frac{16}{3}) + (x + \frac{25}{3})}{(x + \frac{25}{3})(x + \frac{16}{3})} = \frac{1}{x} \]

This simplifies to:
\[ \frac{2x + \frac{41}{3}}{(x + \frac{25}{3})(x + \frac{16}{3})} = \frac{1}{x} \]

Cross-multiplying gives:
\[ x(2x + \frac{41}{3}) = (x + \frac{25}{3})(x + \frac{16}{3}) \]

Expanding both sides:
Left side:
\[ 2x^2 + \frac{41x}{3} \]

Right side:
\[ x^2 + \frac{16x}{3} + \frac{25x}{3} + \frac{400}{9} = x^2 + \frac{41x}{3} + \frac{400}{9} \]

Setting both sides equal:
\[ 2x^2 + \frac{41x}{3} = x^2 + \frac{41x}{3} + \frac{400}{9} \]

Subtracting \( x^2 + \frac{41x}{3} \) from both sides:
\[ x^2 = \frac{400}{9} \]

Taking the square root:
\[ x = \frac{20}{3} \]

Now substituting \( x \) back to find \( M \) and \( B \):
\[ M = \frac{20}{3} + \frac{25}{3} = \frac{45}{3} = 15 \text{ hours} \]
\[ B = \frac{20}{3} + \frac{16}{3} = \frac{36}{3} = 12 \text{ hours} \]

Thus, the time taken by Bill to complete the work alone is:
\[ \boxed{12 \text{ hours}} \]