A does half as much work as B in three fourth of the times. If together they take 18 days to complete the work, how much time shall B alone take to do it?

Options

A. 50 days
B. 45 days
C. 40 days
D. 30 days

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the relationship between A and B
A does half as much work as B in three-fourths of the time. This means:
- If B does a certain amount of work in time \( T_B \), then A does half of that work in time \( T_A = \frac{3}{4} T_B \).

### Step 2: Set up the equations for work done
Let the work done by B in one day be \( R_B \) (rate of work of B). Then the work done by A in one day, \( R_A \), can be expressed as:
\[ R_A = \frac{1}{2} R_B \]

### Step 3: Express the time taken by A and B
Using the relationship established:
- The time taken by B to complete the work is \( T_B \).
- The time taken by A to complete the work is \( T_A = \frac{3}{4} T_B \).

### Step 4: Calculate the combined work rate
When A and B work together, their combined rate of work is:
\[ R_A + R_B = \frac{1}{2} R_B + R_B = \frac{3}{2} R_B \]

### Step 5: Use the total time to find the combined work rate
Given that A and B together take 18 days to complete the work:
Let the total work be \( W \). The work done together in 18 days is:
\[ W = (R_A + R_B) \times 18 = \frac{3}{2} R_B \times 18 = 27 R_B \]

### Step 6: Express the work done by B alone
The work done by B alone in \( T_B \) days is:
\[ W = R_B \times T_B \]

### Step 7: Set the equations equal to each other
Since both expressions represent the same total work \( W \):
\[ 27 R_B = R_B \times T_B \]

### Step 8: Solve for \( T_B \)
Dividing both sides by \( R_B \) (assuming \( R_B \neq 0 \)):
\[ 27 = T_B \]

### Conclusion
Thus, B alone will take **27 days** to complete the work.