A and B together can complete a work in 12 days and B and C can do it in 16 days. A work for 5 days and B for 7 days then C complete the remaining work in 13 days. In how many days B can finish the work?

Options

A. 48
B. 60
C. 24
D. 30

Answer

To solve the problem step by step, let's break down the information given and calculate the required values.

### Step 1: Determine the total work
A and B together can complete the work in 12 days, and B and C can do it in 16 days. We can find the total work in terms of work units.

- Work done by A and B in one day = \( \frac{1}{12} \) of the work
- Work done by B and C in one day = \( \frac{1}{16} \) of the work

To find the total work, we can take the least common multiple (LCM) of 12 and 16.

**LCM of 12 and 16 = 48 units of work.**

### Step 2: Calculate the efficiencies
Now, we can calculate the efficiencies of A, B, and C.

- Efficiency of A and B together = \( \frac{48}{12} = 4 \) units/day
- Efficiency of B and C together = \( \frac{48}{16} = 3 \) units/day

### Step 3: Work done by A and B
A works for 5 days and B works for 7 days.

- Work done by A and B in 5 days = \( 5 \times 4 = 20 \) units

### Step 4: Work done by B and C
Now, since A has worked for 5 days, we need to find out how many days B and C worked together. B worked for 7 days, so they worked together for 2 days (7 - 5).

- Work done by B and C in 2 days = \( 2 \times 3 = 6 \) units

### Step 5: Total work done
Now, we can calculate the total work done so far:

- Total work done = Work done by A and B + Work done by B and C
- Total work done = \( 20 + 6 = 26 \) units

### Step 6: Remaining work
Now, we can find the remaining work:

- Remaining work = Total work - Work done
- Remaining work = \( 48 - 26 = 22 \) units

### Step 7: Work done by C
C completes the remaining work in 13 days. Therefore, we can find C's efficiency:

- Efficiency of C = \( \frac{22}{13} \) units/day

### Step 8: Calculate B's efficiency
From the previous steps, we know that the combined efficiency of B and C is 3 units/day. Let B's efficiency be \( b \) and C's efficiency be \( c \).

- \( b + c = 3 \)
- We already found \( c = \frac{22}{13} \).

Substituting the value of \( c \):

- \( b + \frac{22}{13} = 3 \)

To solve for \( b \):

- \( b = 3 - \frac{22}{13} \)
- Convert 3 to a fraction: \( 3 = \frac{39}{13} \)
- \( b = \frac{39}{13} - \frac{22}{13} = \frac{17}{13} \) units/day

### Step 9: Calculate the number of days B can finish the work
Now, we can find out how many days B can finish the total work:

- Total work = 48 units
- Efficiency of B = \( \frac{17}{13} \) units/day

Number of days B can finish the work = \( \frac{48}{\frac{17}{13}} = 48 \times \frac{13}{17} = \frac{624}{17} \approx 36.71 \) days.

### Conclusion
B can finish the work in approximately 36.71 days, which can be rounded to 37 days.

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