A and B together can complete a work in 12 days. A alone works for 8 days and B alone can complete the remaining work in 20 days. Then in how many days B alone can complete the whole work?

Options

A. 60 days
B. 36 days
C. 45 days
D. 24 days

Answer

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

### Step 1: Determine the combined work rate of A and B
A and B together can complete the work in 12 days. Therefore, their combined work rate (work done per day) can be calculated as:
\[ \text{Combined work rate} = \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \text{ work per day} \]

### Step 2: Express the work rates of A and B
Let the work rate of A be \( A \) (units of work per day) and the work rate of B be \( B \) (units of work per day). From the combined work rate, we have:
\[ A + B = \frac{1}{12} \]

### Step 3: Calculate the work done by A in 8 days
If A works alone for 8 days, the amount of work done by A is:
\[ \text{Work done by A in 8 days} = 8A \]

### Step 4: Determine the remaining work
The remaining work after A has worked for 8 days can be expressed as:
\[ \text{Remaining work} = 1 - 8A \]

### Step 5: Calculate the work done by B in 20 days
B alone can complete the remaining work in 20 days, so we can express this as:
\[ \text{Work done by B in 20 days} = 20B \]
Since this work is equal to the remaining work, we have:
\[ 20B = 1 - 8A \]

### Step 6: Set up the equations
Now we have two equations:
1. \( A + B = \frac{1}{12} \)
2. \( 20B = 1 - 8A \)

### Step 7: Substitute \( A \) from the first equation into the second
From the first equation, we can express \( A \) in terms of \( B \):
\[ A = \frac{1}{12} - B \]
Substituting this into the second equation gives:
\[ 20B = 1 - 8\left(\frac{1}{12} - B\right) \]
Simplifying this:
\[ 20B = 1 - \frac{8}{12} + 8B \]
\[ 20B = 1 - \frac{2}{3} + 8B \]
\[ 20B = \frac{3}{3} - \frac{2}{3} + 8B \]
\[ 20B = \frac{1}{3} + 8B \]

### Step 8: Solve for \( B \)
Now, isolate \( B \):
\[ 20B - 8B = \frac{1}{3} \]
\[ 12B = \frac{1}{3} \]
\[ B = \frac{1}{36} \]

### Step 9: Find the work rate of A
Now substituting \( B \) back to find \( A \):
\[ A + \frac{1}{36} = \frac{1}{12} \]
\[ A = \frac{1}{12} - \frac{1}{36} \]
Finding a common denominator (36):
\[ A = \frac{3}{36} - \frac{1}{36} = \frac{2}{36} = \frac{1}{18} \]

### Step 10: Calculate the total work
The total work can be calculated as:
\[ \text{Total work} = \text{(A + B) work rate} \times \text{time} = \left(\frac{1}{12}\right) \times 12 = 1 \text{ work unit} \]

### Step 11: Determine how many days B alone can complete the whole work
Since B's work rate is \( \frac{1}{36} \) work units per day, the total time for B to complete the whole work is:
\[ \text{Time} = \frac{1 \text{ work}}{B} = \frac{1}{\frac{1}{36}} = 36 \text{ days} \]

### Final Answer
B alone can complete the whole work in **36 days**.

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