A and B can complete a work in 12 days and 20 days, respectively They work together for 4 days. C alone completes the remaining work in 14 days. B and C together can complete the same work in:

Options

A. 16 days
B. 12 days
C. 15 days
D. 10 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 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 20 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{20} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
- The combined work rate of A and B when they work together is:
\[ \text{Combined work rate of A and B} = \frac{1}{12} + \frac{1}{20} \]
- To add these fractions, we find a common denominator, which is 60:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
- Therefore,
\[ \text{Combined work rate of A and B} = \frac{5}{60} + \frac{3}{60} = \frac{8}{60} = \frac{2}{15} \text{ (work per day)} \]

### Step 3: Calculate the work done by A and B in 4 days
- In 4 days, the amount of work done by A and B together is:
\[ \text{Work done in 4 days} = 4 \times \frac{2}{15} = \frac{8}{15} \]

### Step 4: Determine the remaining work
- The total work can be considered as 1 (or 60 units for easier calculation). The remaining work after A and B have worked for 4 days is:
\[ \text{Remaining work} = 1 - \frac{8}{15} = \frac{15}{15} - \frac{8}{15} = \frac{7}{15} \]

### Step 5: Calculate C's work rate
- C completes the remaining work in 14 days. Therefore, C's work rate is:
\[ \text{Work rate of C} = \frac{\frac{7}{15}}{14} = \frac{7}{15 \times 14} = \frac{7}{210} = \frac{1}{30} \text{ (work per day)} \]

### Step 6: Calculate the combined work rate of B and C
- Now we find the combined work rate of B and C:
\[ \text{Combined work rate of B and C} = \text{Work rate of B} + \text{Work rate of C} = \frac{1}{20} + \frac{1}{30} \]
- To add these fractions, we find a common denominator, which is 60:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
- Therefore,
\[ \text{Combined work rate of B and C} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \text{ (work per day)} \]

### Step 7: Calculate the time taken by B and C to complete the work
- The time taken by B and C to complete the entire work (1 unit) is:
\[ \text{Time} = \frac{1}{\text{Combined work rate of B and C}} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

### Final Answer
B and C together can complete the work in **12 days**.

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