A' can do a piece of work in 4 days and B in 12 days. In how many days can A and B together complete the same piece of work ?

Options

A. 7 days
B. 3 days
C. 2 days
D. 4 days

Answer

To solve the problem of how many days A and B can complete a piece of work together, we can follow these steps:

### Step 1: Determine the work rates of A and B
- A can complete the work in 4 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ work}}{4 \text{ days}} = \frac{1}{4} \text{ work per day} \]

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

### Step 2: Calculate the combined work rate of A and B
- To find the combined 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 and add the fractions
- The least common multiple (LCM) 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 two fractions:
\[ \text{Combined work rate} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \text{ work per day} \]

### 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 1 whole work is the reciprocal of their combined work rate:
\[ \text{Time taken} = \frac{1 \text{ work}}{\frac{1}{3} \text{ work per day}} = 3 \text{ days} \]

### Final Answer
A and B together can complete the work in **3 days**.

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