A can do a plece 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, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{4} \text{ of the work per day} \]
- B can complete the work in 12 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{12} \text{ of the work per day} \]

### Step 2: Calculate the combined work rate of A and B.
- To find the total work rate when A and B work together, we add their individual work rates:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} = \frac{1}{4} + \frac{1}{12} \]

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

### Step 4: Calculate the time taken to complete the work together.
- If A and B together can complete \(\frac{1}{3}\) of the work in one day, then the time taken to complete the entire work is the reciprocal of their combined work rate:
\[ \text{Time taken} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{1}{3}} = 3 \text{ days} \]

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

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