Ravi can do a piece of work in 15 hours while Raman can do it in 12 hours. How long will both take to do it, woking together?

Options

A. `"6 hours 40 minutes"`
B. `"7 hours 40 minutes"`
C. `"6 hours 50 minutes"`
D. `"8 hours 40 minutes"`

Answer

To solve the problem of how long Ravi and Raman will take to complete a piece of work together, we can follow these steps:

### Step 1: Determine Individual Work Rates
- **Ravi's Work Rate**: Ravi can complete the work in 15 hours. Therefore, his work rate is:
\[ \text{Ravi's work rate} = \frac{1}{15} \text{ (work per hour)} \]

- **Raman's Work Rate**: Raman can complete the work in 12 hours. Therefore, his work rate is:
\[ \text{Raman's work rate} = \frac{1}{12} \text{ (work per hour)} \]

### Step 2: Combine Work Rates
To find the combined work rate when both work together, we add their individual work rates:
\[ \text{Combined work rate} = \text{Ravi's work rate} + \text{Raman's work rate} \]
\[ = \frac{1}{15} + \frac{1}{12} \]

### Step 3: Find a Common Denominator
To add the fractions, we need a common denominator. The least common multiple (LCM) of 15 and 12 is 60. We can rewrite the fractions:
\[ \frac{1}{15} = \frac{4}{60} \quad \text{(since } 1 \times 4 = 4 \text{ and } 15 \times 4 = 60\text{)} \]
\[ \frac{1}{12} = \frac{5}{60} \quad \text{(since } 1 \times 5 = 5 \text{ and } 12 \times 5 = 60\text{)} \]

### Step 4: Add the Fractions
Now we can add the two fractions:
\[ \text{Combined work rate} = \frac{4}{60} + \frac{5}{60} = \frac{9}{60} \]

### Step 5: Simplify the Combined Work Rate
We can simplify \(\frac{9}{60}\):
\[ \frac{9}{60} = \frac{3}{20} \text{ (dividing both numerator and denominator by 3)} \]

### Step 6: Calculate Time Taken to Complete the Work
To find out how long it will take for them to complete the work together, we take the reciprocal of the combined work rate:
\[ \text{Time taken} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{3}{20}} = \frac{20}{3} \text{ hours} \]

### Step 7: Convert to Mixed Number
To convert \(\frac{20}{3}\) into a mixed number:
\[ 20 \div 3 = 6 \quad \text{(with a remainder of 2)} \]
Thus, \(\frac{20}{3} = 6 \frac{2}{3}\) hours.

### Final Answer
Therefore, Ravi and Raman will take **6 hours and 40 minutes** (or \(6 \frac{2}{3}\) hours) to complete the work together.

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