A and B can do a work in 30 days, B and C do same work in 24 days. If first 16 days A work alone, next 24 days B work alone and in the last C complete the remaining work in 12 days then how many days B complete the whole work alone

Options

A. 60 days
B. 45 days
C. 90 days
D. 120 days

Answer

To solve the problem step-by-step, we will first determine the efficiencies of A, B, and C based on the information given, and then calculate how long it would take for B to complete the work alone.

### Step 1: Determine Total Work
Given:
- A and B can complete the work in 30 days.
- B and C can complete the same work in 24 days.

To find the total work, we can use the least common multiple (LCM) of the days taken by A and B, and B and C.

\[ \text{Total Work} = \text{LCM}(30, 24) = 120 \text{ units} \]

### Step 2: Calculate Efficiencies
1. **Efficiency of A and B**:
\[ \text{Efficiency of A + B} = \frac{120 \text{ units}}{30 \text{ days}} = 4 \text{ units/day} \]

2. **Efficiency of B and C**:
\[ \text{Efficiency of B + C} = \frac{120 \text{ units}}{24 \text{ days}} = 5 \text{ units/day} \]

### Step 3: Set Up Equations for Individual Efficiencies
Let:
- Efficiency of A = a
- Efficiency of B = b
- Efficiency of C = c

From the above, we have:
1. \( a + b = 4 \) (Equation 1)
2. \( b + c = 5 \) (Equation 2)

### Step 4: Calculate Work Done by A, B, and C
Now, let's analyze the work done over the specified periods:
- A works alone for 16 days.
- B works alone for 24 days.
- C completes the remaining work in 12 days.

1. **Work done by A in 16 days**:
\[ \text{Work by A} = 16a \]

2. **Work done by B in 24 days**:
\[ \text{Work by B} = 24b \]

3. **Work done by C in 12 days**:
\[ \text{Work by C} = 12c \]

### Step 5: Total Work Equation
The total work done by A, B, and C should equal the total work:
\[ 16a + 24b + 12c = 120 \text{ units} \quad \text{(Equation 3)} \]

### Step 6: Substitute and Solve
Now we will substitute \( c \) from Equation 2 into Equation 3:
From Equation 2:
\[ c = 5 - b \]
Substituting \( c \) into Equation 3:
\[ 16a + 24b + 12(5 - b) = 120 \]
Expanding this gives:
\[ 16a + 24b + 60 - 12b = 120 \]
Simplifying:
\[ 16a + 12b + 60 = 120 \]
\[ 16a + 12b = 60 \quad \text{(Equation 4)} \]

### Step 7: Solve Equations 1 and 4
Now we have two equations:
1. \( a + b = 4 \) (Equation 1)
2. \( 16a + 12b = 60 \) (Equation 4)

From Equation 1, express \( a \):
\[ a = 4 - b \]
Substituting into Equation 4:
\[ 16(4 - b) + 12b = 60 \]
Expanding:
\[ 64 - 16b + 12b = 60 \]
Combining like terms:
\[ 64 - 4b = 60 \]
\[ -4b = -4 \]
\[ b = 1 \]

### Step 8: Find Efficiencies of A and C
Substituting \( b \) back into Equation 1:
\[ a + 1 = 4 \implies a = 3 \]
Substituting \( b \) into Equation 2:
\[ 1 + c = 5 \implies c = 4 \]

### Step 9: Calculate Time for B to Complete the Work Alone
Now we know:
- Efficiency of B = 1 unit/day
- Total work = 120 units

Time taken by B to complete the work alone:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of B}} = \frac{120}{1} = 120 \text{ days} \]

### Final Answer
B can complete the whole work alone in **120 days**.