A can complete a work in 10 days and B can complete it in 15 days. If A starts the work and 5 days later B also joins him, then in how many days the work will be finished?

Options

A. 6 days
B. 7 days
C. 10 days
D. 8 days

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the work rates of A and B
- A can complete the work in 10 days. Therefore, A's work rate is:
\[ \text{Rate of A} = \frac{1}{10} \text{ work/day} \]
- B can complete the work in 15 days. Therefore, B's work rate is:
\[ \text{Rate of B} = \frac{1}{15} \text{ work/day} \]

### Step 2: Calculate the amount of work done by A in the first 5 days
- In 5 days, A will complete:
\[ \text{Work done by A in 5 days} = 5 \times \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \text{ of the work} \]

### Step 3: Determine the remaining work after 5 days
- Since A completed half of the work, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{2} = \frac{1}{2} \text{ of the work} \]

### Step 4: Calculate the combined work rate of A and B
- When B joins A, their combined work rate will be:
\[ \text{Combined Rate} = \frac{1}{10} + \frac{1}{15} \]
- To add these fractions, we find a common denominator, which is 30:
\[ \text{Combined Rate} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ work/day} \]

### Step 5: Determine how long it takes for A and B to finish the remaining work
- The remaining work is \(\frac{1}{2}\). To find the time taken to complete this work at their combined rate:
\[ \text{Time} = \frac{\text{Remaining Work}}{\text{Combined Rate}} = \frac{\frac{1}{2}}{\frac{1}{6}} = \frac{1}{2} \times 6 = 3 \text{ days} \]

### Step 6: Calculate the total time taken to complete the work
- The total time taken to complete the work is the time A worked alone plus the time A and B worked together:
\[ \text{Total Time} = 5 \text{ days (A alone)} + 3 \text{ days (A and B together)} = 8 \text{ days} \]

### Final Answer
The work will be finished in **8 days**.

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