A and B can complete a job in 24 days working together. A alone can complete it in 32 days. Both of them worked together for 8 days and then A left. The number of days B will take to complete the remaining job is

Options

A. 16
B. 32
C. 64
D. 128

Answer

To solve the problem step by step, we will follow the details provided in the video transcript.

### Step 1: Determine the efficiencies of A and B
- A and B together can complete the job in 24 days. Therefore, their combined efficiency (E_AB) is:
\[ E_{AB} = \frac{1}{24} \text{ (work per day)} \]
- A alone can complete the job in 32 days, so A's efficiency (E_A) is:
\[ E_A = \frac{1}{32} \text{ (work per day)} \]

### Step 2: Calculate B's efficiency
- We know that:
\[ E_{AB} = E_A + E_B \]
- Rearranging gives us B's efficiency (E_B):
\[ E_B = E_{AB} - E_A = \frac{1}{24} - \frac{1}{32} \]
- To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 24 and 32 is 96. Thus:
\[ E_B = \frac{4}{96} - \frac{3}{96} = \frac{1}{96} \]

### Step 3: Calculate the work done in 8 days by A and B together
- The work done by A and B in 8 days is:
\[ \text{Work done} = E_{AB} \times \text{time} = \frac{1}{24} \times 8 = \frac{8}{24} = \frac{1}{3} \]

### Step 4: Calculate the remaining work
- The total work is considered as 1 unit. Therefore, the remaining work after 8 days is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 5: Calculate the time taken by B to complete the remaining work
- Since B alone will finish the remaining work, we can find the time taken by B:
\[ \text{Time} = \frac{\text{Remaining work}}{E_B} = \frac{\frac{2}{3}}{\frac{1}{96}} = \frac{2}{3} \times 96 = 64 \text{ days} \]

### Final Answer
- B will take **64 days** to complete the remaining job.

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