A can do a piece of work in 4 days and B can do it in 12 days. In how many days will they finish the work, both working together ?

Options

A. 4 days
B. 6 days
C. 2 days
D. 3 days

Answer

To solve the problem of how many days A and B will take to finish the work together, we can follow these steps:

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 4 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{4} \]
- B can complete the work in 12 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{12} \]

### Step 2: Calculate the total work done by A and B together in one day.
- When A and B work together, the total work done in one day is the sum of their individual work rates:
\[ \text{Total work done in one day} = \frac{1}{4} + \frac{1}{12} \]

### Step 3: Find a common denominator to add the fractions.
- The least common multiple of 4 and 12 is 12. We can convert \(\frac{1}{4}\) to have a denominator of 12:
\[ \frac{1}{4} = \frac{3}{12} \]
- Now we can add the fractions:
\[ \frac{3}{12} + \frac{1}{12} = \frac{4}{12} \]

### Step 4: Simplify the total work done in one day.
- Simplifying \(\frac{4}{12}\):
\[ \frac{4}{12} = \frac{1}{3} \]
- This means A and B together can complete \(\frac{1}{3}\) of the work in one day.

### Step 5: Calculate the total number of days to complete the work.
- If A and B together can complete \(\frac{1}{3}\) of the work in one day, the total number of days required to complete the entire work is the reciprocal of \(\frac{1}{3}\):
\[ \text{Total days} = \frac{1}{\frac{1}{3}} = 3 \text{ days} \]

### Final Answer:
A and B will finish the work together in **3 days**.

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