Rajan can do a piece of work in 24 days while Amit can do it in 30 days. In how many days can they complete it they work together?

Answer

To solve the problem of how many days Rajan and Amit can complete the work together, we can follow these steps:

### Step-by-Step Solution:

1. **Determine Individual Work Rates**:
- Rajan can complete the work in 24 days. Therefore, his work rate is:
\[ \text{Rajan's work rate} = \frac{1}{24} \text{ (work per day)} \]
- Amit can complete the work in 30 days. Therefore, his work rate is:
\[ \text{Amit's work rate} = \frac{1}{30} \text{ (work per day)} \]

2. **Combine Work Rates**:
- When working together, their combined work rate is the sum of their individual work rates:
\[ \text{Combined work rate} = \frac{1}{24} + \frac{1}{30} \]

3. **Find a Common Denominator**:
- The least common multiple (LCM) of 24 and 30 is 120. We can convert the fractions:
\[ \frac{1}{24} = \frac{5}{120} \quad \text{(since } 120 \div 24 = 5\text{)} \]
\[ \frac{1}{30} = \frac{4}{120} \quad \text{(since } 120 \div 30 = 4\text{)} \]

4. **Add the Work Rates**:
- Now, we can add the two fractions:
\[ \text{Combined work rate} = \frac{5}{120} + \frac{4}{120} = \frac{9}{120} \]
- Simplifying \(\frac{9}{120}\):
\[ \frac{9}{120} = \frac{3}{40} \text{ (dividing numerator and denominator by 3)} \]

5. **Calculate Time to Complete Work**:
- The time taken to complete the work when they work together is the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{3}{40}} = \frac{40}{3} \text{ days} \]

6. **Convert to Mixed Number**:
- To express \(\frac{40}{3}\) in mixed number form:
\[ 40 \div 3 = 13 \quad \text{with a remainder of } 1 \]
- Thus, \(\frac{40}{3} = 13 \frac{1}{3}\) days.

### Final Answer:
Rajan and Amit can complete the work together in **13 \(\frac{1}{3}\) days**.

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