A and B can separately do a piece of work in 18 days and 12 days respectively. How long will they together take to do the work if working together?

Options

A. 9.4 days
B. 10 days
C. 6 days
D. 7.2 days

Answer

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

### Step 1: Determine the individual work rates of A and B
- A can complete the work in 18 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ work}}{18 \text{ days}} = \frac{1}{18} \text{ work/day} \]

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

### Step 2: Calculate the combined work rate of A and B
- To find the combined work rate of A and B, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{18} + \frac{1}{12} \]

### Step 3: Find a common denominator and add the fractions
- The least common multiple (LCM) of 18 and 12 is 36. We convert the fractions:
\[ \frac{1}{18} = \frac{2}{36} \quad \text{and} \quad \frac{1}{12} = \frac{3}{36} \]
- Now, add the two fractions:
\[ \text{Combined work rate} = \frac{2}{36} + \frac{3}{36} = \frac{5}{36} \text{ work/day} \]

### Step 4: Calculate the time taken to complete the work together
- If A and B work together, the time taken to complete 1 unit of work is the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1 \text{ work}}{\text{Combined work rate}} = \frac{1}{\frac{5}{36}} = \frac{36}{5} \text{ days} \]
- Simplifying this gives:
\[ \frac{36}{5} = 7.2 \text{ days} \]

### Conclusion
- Therefore, A and B together will take **7.2 days** to complete the work.

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