Question

A takes 6 days less than the time taken by B to finish a piece of work. If both A and B together can finish it in 4 days, then the time taken by B to finish the work is

• A. 12 days
• B. 14 days
• C. 16 days
• D. 18 days

Answer 1

The Correct Option is A

Step 1: Define variables.

Let the time taken by B to finish the work be \( x \) days. Then the time taken by A is \( x - 6 \) days.

Step 2: Express their work rates.

The work rate of B is \( \frac{1}{x} \) (work per day), and the work rate of A is \( \frac{1}{x - 6} \).

Together, their combined work rate is \( \frac{1}{4} \) (since they finish the work in 4 days).

Step 3: Set up the equation.

\[ \frac{1}{x} + \frac{1}{x - 6} = \frac{1}{4}. \]

Step 4: Solve the equation.

Multiply through by \( 4x(x - 6) \) to eliminate the denominators:

\[ 4(x - 6) + 4x = x(x - 6). \]

Simplify:

\[ 4x - 24 + 4x = x^2 - 6x \implies x^2 - 14x + 24 = 0. \]

Factorize:

\[ (x - 12)(x - 2) = 0. \]

Thus, \( x = 12 \) or \( x = 2 \). Since \( x = 2 \) would make \( x - 6 \) negative (impossible), we have \( x = 12 \).

Final Answer: The time taken by B to finish the work is \( \mathbf{12 \text{ days}} \), which corresponds to option \( \mathbf{(1)} \).