A do `(4)/(9)`th as much work as B in `(2)/(5)`th of time. If they together take 27 days to complete a work, how much time shall B take alone to do it?

Options

A. 57
B. 38
C. 46
D. 54

Answer

To solve the problem, we need to find out how much time B will take to complete the work alone. Let's break it down step by step.

### Step 1: Understand the relationship between A and B
From the problem, we know that A does \( \frac{4}{9} \) of the work that B does in \( \frac{2}{5} \) of the time.

Let’s denote:
- Work done by B in 1 day = \( b \)
- Work done by A in 1 day = \( a \)

From the information given:
\[ a = \frac{4}{9}b \]
And since A takes \( \frac{2}{5} \) of the time that B takes to do the same amount of work, we can express this as:
\[ \text{Time taken by A} = \frac{2}{5} \times \text{Time taken by B} \]

### Step 2: Set up the equations
Let \( T_B \) be the time taken by B to complete the work alone. Then, the time taken by A is:
\[ T_A = \frac{2}{5} T_B \]

### Step 3: Calculate the total work done
If A and B together take 27 days to complete the work, we can express the total work done in terms of their efficiencies:
\[ \text{Total Work} = (a + b) \times 27 \]

### Step 4: Substitute A's work in terms of B's work
Substituting \( a = \frac{4}{9}b \) into the total work equation:
\[ \text{Total Work} = \left(\frac{4}{9}b + b\right) \times 27 \]
\[ = \left(\frac{4}{9}b + \frac{9}{9}b\right) \times 27 \]
\[ = \left(\frac{13}{9}b\right) \times 27 \]
\[ = \frac{13 \times 27}{9} b \]
\[ = 39b \]

### Step 5: Calculate the individual work rates
Now, we know that:
- A's work rate is \( a = \frac{4}{9}b \)
- B's work rate is \( b \)

The total work can also be expressed in terms of B's time:
\[ \text{Total Work} = b \times T_B \]

### Step 6: Equate the two expressions for total work
Setting the two expressions for total work equal to each other:
\[ 39b = b \times T_B \]

### Step 7: Solve for \( T_B \)
Dividing both sides by \( b \) (assuming \( b \neq 0 \)):
\[ 39 = T_B \]

Thus, B will take **39 days** to complete the work alone.

### Final Answer
B will take **39 days** to complete the work alone.