To solve the problem, we need to find out how many days B can finish the work alone. Let's break it down step by step.
### Step 1: Determine the work rates of A, B, and C
1. **A and B's work rate**: A and B together can complete the work in 12 days. Therefore, their combined work rate is:
\[
\text{Work rate of A + B} = \frac{1}{12} \text{ (work per day)}
\]
2. **B and C's work rate**: B and C together can complete the work in 16 days. Therefore, their combined work rate is:
\[
\text{Work rate of B + C} = \frac{1}{16} \text{ (work per day)}
\]
### Step 2: Set up equations for individual work rates
Let:
- A's work rate = \( a \)
- B's work rate = \( b \)
- C's work rate = \( c \)
From the above information, we can write the following equations:
1. \( a + b = \frac{1}{12} \) (1)
2. \( b + c = \frac{1}{16} \) (2)
### Step 3: Work done by A, B, and C
Now, we know:
- A works for 5 days, so the work done by A is:
\[
\text{Work done by A} = 5a
\]
- B works for 7 days, so the work done by B is:
\[
\text{Work done by B} = 7b
\]
- C finishes the remaining work in 13 days, so the work done by C is:
\[
\text{Work done by C} = 13c
\]
### Step 4: Total work equation
The total work done by A, B, and C together should equal 1 (the whole work):
\[
5a + 7b + 13c = 1 \quad \text{(3)}
\]
### Step 5: Substitute for c
From equation (2), we can express \( c \) in terms of \( b \):
\[
c = \frac{1}{16} - b
\]
### Step 6: Substitute c into equation (3)
Substituting \( c \) into equation (3):
\[
5a + 7b + 13\left(\frac{1}{16} - b\right) = 1
\]
Expanding this gives:
\[
5a + 7b + \frac{13}{16} - 13b = 1
\]
Combining like terms:
\[
5a - 6b + \frac{13}{16} = 1
\]
Rearranging gives:
\[
5a - 6b = 1 - \frac{13}{16} = \frac{3}{16} \quad \text{(4)}
\]
### Step 7: Solve the system of equations
Now we have two equations:
1. \( a + b = \frac{1}{12} \) (1)
2. \( 5a - 6b = \frac{3}{16} \) (4)
From equation (1), we can express \( a \):
\[
a = \frac{1}{12} - b
\]
Substituting this into equation (4):
\[
5\left(\frac{1}{12} - b\right) - 6b = \frac{3}{16}
\]
Expanding gives:
\[
\frac{5}{12} - 5b - 6b = \frac{3}{16}
\]
Combining terms:
\[
\frac{5}{12} - 11b = \frac{3}{16}
\]
Now, convert \( \frac{5}{12} \) to a fraction with a denominator of 48:
\[
\frac{5}{12} = \frac{20}{48}, \quad \frac{3}{16} = \frac{9}{48}
\]
So, we have:
\[
\frac{20}{48} - 11b = \frac{9}{48}
\]
Rearranging gives:
\[
11b = \frac{20}{48} - \frac{9}{48} = \frac{11}{48}
\]
Thus:
\[
b = \frac{11}{48 \times 11} = \frac{1}{48}
\]
### Step 8: Calculate the days B can finish the work
Since \( b \) is the work rate of B, the number of days B can finish the work alone is:
\[
\text{Days for B} = \frac{1}{b} = 48 \text{ days}
\]
### Final Answer
B can finish the work in **48 days**.
