A and B can do a piece of work in 12 days and 15 days respectively. They began to work together but A left after 4 days. In how many more days would B alone complete the remaining work ?

Options

A. `(20)/(3)` days
B. `(25)/(3)` days
C. 6 days
D. 5 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the work rates of A and B
- A can complete the work in 12 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{12} \text{ (work per day)} \]
- B can complete the work in 15 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{15} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
- When A and B work together, their combined work rate is:
\[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{15} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 15 is 60. Thus:
\[ \frac{1}{12} = \frac{5}{60} \quad \text{and} \quad \frac{1}{15} = \frac{4}{60} \]
- Therefore, the combined work rate is:
\[ \text{Combined work rate} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \]

### Step 3: Calculate the amount of work done in 4 days
- In 4 days, the amount of work done by A and B together is:
\[ \text{Work done in 4 days} = \text{Combined work rate} \times 4 = \frac{3}{20} \times 4 = \frac{12}{20} = \frac{3}{5} \]
- To express this in terms of total work, if we consider the total work to be 60 units, then:
\[ \text{Work done in 4 days} = 60 \times \frac{3}{5} = 36 \text{ units} \]

### Step 4: Calculate the remaining work
- The total work is 60 units, and after 4 days, 36 units have been completed. Therefore, the remaining work is:
\[ \text{Remaining work} = 60 - 36 = 24 \text{ units} \]

### Step 5: Determine how long it will take B to complete the remaining work
- B's work rate is \(\frac{1}{15}\) of the work per day. Therefore, in one day, B can complete:
\[ \text{Work done by B in 1 day} = \frac{1}{15} \text{ of the total work} \]
- To find out how many days B needs to complete the remaining 24 units, we can set up the equation:
\[ \text{Days required by B} = \frac{\text{Remaining work}}{\text{Work done by B in 1 day}} = \frac{24}{\frac{1}{15}} = 24 \times 15 = 6 \text{ days} \]

### Final Answer
B will take **6 more days** to complete the remaining work.

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