A and B working together can do 30% of the work in 6 days. B alone can do the same work in 25 days. How long will A alone take to complete the same work?

Options

A. 75 days
B. 60 days
C. 100 days
D. 80 days

Answer

To solve the problem step by step, let's break it down clearly.

### Step 1: Determine the work done by A and B together
A and B together can do 30% of the work in 6 days. To find out how long they take to complete the entire work, we can set up the following calculation:

\[ \text{Total work (100%)} = \frac{30\%}{6 \text{ days}} \times 100 = \frac{30}{6} \times 100 = 5 \times 100 = 500 \text{ days} \]

Thus, A and B together can complete the entire work in 20 days.

### Step 2: Calculate the work done by B alone
B can complete the work alone in 25 days. Therefore, B's work rate is:

\[ \text{B's work rate} = \frac{1}{25} \text{ (work per day)} \]

### Step 3: Calculate the work done by A and B together
From Step 1, we found that A and B together can do the work in 20 days, so their combined work rate is:

\[ \text{A + B's work rate} = \frac{1}{20} \text{ (work per day)} \]

### Step 4: Set up the equation for A's work rate
Let A's work rate be \( \frac{1}{x} \) where \( x \) is the number of days A takes to complete the work alone. Therefore, we can write:

\[ \frac{1}{x} + \frac{1}{25} = \frac{1}{20} \]

### Step 5: Solve for \( x \)
To solve for \( x \), we first find a common denominator for the fractions on the left side. The common denominator for 20 and 25 is 100. Thus, we rewrite the equation:

\[ \frac{5}{100} + \frac{4}{100} = \frac{5}{100} \]

This simplifies to:

\[ \frac{1}{x} = \frac{1}{20} - \frac{1}{25} \]

Finding a common denominator (100):

\[ \frac{5}{100} - \frac{4}{100} = \frac{1}{100} \]

Thus, we have:

\[ \frac{1}{x} = \frac{1}{100} \]

### Step 6: Find \( x \)
Taking the reciprocal gives us:

\[ x = 100 \]

### Conclusion
A alone will take 100 days to complete the work.