A can do in one day three times the work done by B in one day. They together finish `(2)/(5)` of the work in 9 days. The number of days by which B can do the work alone is :

Options

A. 90 days
B. 120 days
C. 100 days
D. 30 days

Answer

To solve the problem step-by-step, we will follow the information given in the question and apply the concepts of work and efficiency.

### Step 1: Define the work done by A and B
Let the amount of work done by B in one day be \( x \). According to the problem, A can do three times the work done by B in one day. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = 3x \]

### Step 2: Calculate the combined work done by A and B
When A and B work together, their combined work done in one day is:
\[ \text{Combined work} = \text{Work done by A} + \text{Work done by B} = 3x + x = 4x \]

### Step 3: Determine the total work done in 9 days
According to the problem, A and B together finish \( \frac{2}{5} \) of the total work in 9 days. Therefore, the total work done by A and B in 9 days is:
\[ \text{Total work done in 9 days} = \text{Combined work} \times \text{Number of days} = 4x \times 9 = 36x \]
Since this represents \( \frac{2}{5} \) of the total work, we can set up the equation:
\[ 36x = \frac{2}{5} \text{Total Work} \]

### Step 4: Calculate the total work
To find the total work, we can rearrange the equation:
\[ \text{Total Work} = 36x \times \frac{5}{2} = 90x \]

### Step 5: Determine the number of days B can do the work alone
Now, we need to find out how many days B can complete the total work alone. The time taken by B to complete the total work is given by the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Work done by B in one day}} = \frac{90x}{x} = 90 \text{ days} \]

### Final Answer
Thus, the number of days by which B can do the work alone is:
\[ \boxed{90} \]