A' completes a work in 12 days 'B' completes the same work in 15 days .A started working alone and after 3 days B joined him .How many days will they now take together to complete the remaining work ?

Options

A. 6
B. 8
C. 5
D. 4

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the work done by A and B in one day.
- A completes the work in 12 days, so A's work in one day is:
\[ \text{Work done by A in 1 day} = \frac{1}{12} \]
- B completes the work in 15 days, so B's work in one day is:
\[ \text{Work done by B in 1 day} = \frac{1}{15} \]

**Hint:** To find the work done in one day, take the reciprocal of the number of days taken to complete the work.

### Step 2: Calculate the work done by A in the first 3 days.
- In 3 days, A will complete:
\[ \text{Work done by A in 3 days} = 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \]

**Hint:** Multiply the work done in one day by the number of days worked to find the total work done.

### Step 3: Determine the remaining work after A has worked for 3 days.
- The total work is considered as 1 (whole work). Therefore, the remaining work after A has worked for 3 days is:
\[ \text{Remaining work} = 1 - \frac{1}{4} = \frac{3}{4} \]

**Hint:** Subtract the work done from the total work to find the remaining work.

### Step 4: Calculate the combined work done by A and B in one day.
- When A and B work together, their combined work in one day is:
\[ \text{Combined work} = \frac{1}{12} + \frac{1}{15} \]
- To add these fractions, find the least common multiple (LCM) of 12 and 15, which is 60:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
- Therefore,
\[ \text{Combined work} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \]

**Hint:** To add fractions, convert them to have a common denominator.

### Step 5: Calculate the number of days required to complete the remaining work.
- Let \( X \) be the number of days required to finish the remaining work of \( \frac{3}{4} \):
\[ X \times \frac{3}{20} = \frac{3}{4} \]
- To find \( X \), solve the equation:
\[ X = \frac{3/4}{3/20} = \frac{3}{4} \times \frac{20}{3} = \frac{20}{4} = 5 \]

**Hint:** To isolate \( X \), divide the remaining work by the combined work done in one day.

### Conclusion
- A and B together will take **5 days** to complete the remaining work.

**Final Answer:** 5 days