A and B can do a piece of work in 12 days, B and C in 8 days and C and A in 6 days. How long would B take to do the same work alone?

Options

A. 24 days
B. 32 days
C. 40 days
D. 48 days

Answer

To solve the problem, we need to determine how long B would take to complete the work alone based on the information given about the work done by A, B, and C together. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the total work
We know that:
- A and B can complete the work in 12 days.
- B and C can complete the work in 8 days.
- C and A can complete the work in 6 days.

To find a common measure of work, we can take the Least Common Multiple (LCM) of the days taken by each pair to complete the work.

The LCM of 12, 8, and 6 is 24. Therefore, we can consider the total work to be 24 units.

### Step 2: Calculate the work done by each pair in one day
Now we can find the work done by each pair in one day:
- Work done by A and B in one day = Total work / Days = 24 / 12 = 2 units.
- Work done by B and C in one day = Total work / Days = 24 / 8 = 3 units.
- Work done by C and A in one day = Total work / Days = 24 / 6 = 4 units.

### Step 3: Set up equations based on the work done
Let:
- A's work in one day = a
- B's work in one day = b
- C's work in one day = c

From the above calculations, we can set up the following equations:
1. \( a + b = 2 \) (Equation 1)
2. \( b + c = 3 \) (Equation 2)
3. \( c + a = 4 \) (Equation 3)

### Step 4: Solve the equations
We can add all three equations:
\[ (a + b) + (b + c) + (c + a) = 2 + 3 + 4 \]
This simplifies to:
\[ 2a + 2b + 2c = 9 \]
Dividing by 2 gives:
\[ a + b + c = 4.5 \] (Equation 4)

Now we can find the individual work rates:
From Equation 1:
\[ c = 4.5 - (a + b) = 4.5 - 2 = 2.5 \]

Substituting \( c \) back into Equation 2:
\[ b + 2.5 = 3 \]
Thus,
\[ b = 3 - 2.5 = 0.5 \]

Now substituting \( b \) into Equation 1:
\[ a + 0.5 = 2 \]
Thus,
\[ a = 2 - 0.5 = 1.5 \]

### Step 5: Calculate B's time to complete the work alone
Now we have:
- A's work rate (a) = 1.5 units/day
- B's work rate (b) = 0.5 units/day
- C's work rate (c) = 2.5 units/day

To find out how long B would take to complete the entire work alone (24 units), we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{B's Work Rate}} \]
\[ \text{Time} = \frac{24}{0.5} = 48 \text{ days} \]

### Final Answer
B would take **48 days** to complete the work alone.

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