Quantity I : Time taken by Manoj and shubham to complete the work together. When Manoj works alone, he takes 4 . 8 hrs more than the time taken by Manoj and Shubham to complete the work together . When Shubham works along he takes 10 . 8 hrs more than the time taken by both of them to complete the work together
Quantity II : 7 . 4 hrs.

Options

A. Quantity I ` gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge `Quantity II
D. Quantity I `le` QuantityII

Answer

To solve the problem step by step, we need to find the time taken by Manoj and Shubham to complete the work together (let's denote this time as \( t \)).

### Step 1: Define the Variables
Let:
- \( t \) = time taken by Manoj and Shubham to complete the work together.
- \( M \) = time taken by Manoj to complete the work alone.
- \( S \) = time taken by Shubham to complete the work alone.

### Step 2: Set Up the Equations
According to the problem:
1. Manoj takes 4.8 hours more than the time taken by both together:
\[ M = t + 4.8 \]
2. Shubham takes 10.8 hours more than the time taken by both together:
\[ S = t + 10.8 \]

### Step 3: Write the Work Rates
The work rates for Manoj and Shubham can be expressed as:
- Rate of Manoj = \( \frac{1}{M} = \frac{1}{t + 4.8} \)
- Rate of Shubham = \( \frac{1}{S} = \frac{1}{t + 10.8} \)
- Rate of both together = \( \frac{1}{t} \)

### Step 4: Set Up the Combined Work Rate Equation
When they work together, their combined work rate is the sum of their individual work rates:
\[ \frac{1}{t} = \frac{1}{t + 4.8} + \frac{1}{t + 10.8} \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can multiply through by \( t(t + 4.8)(t + 10.8) \):
\[ (t + 4.8)(t + 10.8) = t(t + 10.8) + t(t + 4.8) \]

### Step 6: Expand and Simplify
Expanding both sides:
\[ t^2 + 15.6t + 51.84 = t^2 + 10.8t + t^2 + 4.8t \]
Combine like terms:
\[ t^2 + 15.6t + 51.84 = 2t^2 + 15.6t \]

### Step 7: Rearranging the Equation
Subtract \( 15.6t \) from both sides:
\[ t^2 + 51.84 = 2t^2 \]
Rearranging gives:
\[ 0 = t^2 - 51.84 \]
This simplifies to:
\[ t^2 = 51.84 \]

### Step 8: Solve for \( t \)
Taking the square root:
\[ t = \sqrt{51.84} = 7.2 \text{ hours} \]

### Step 9: Compare with Quantity II
Now we compare Quantity I (7.2 hours) with Quantity II (7.4 hours):
- Quantity I: 7.2 hours
- Quantity II: 7.4 hours

### Conclusion
Since \( 7.2 < 7.4 \), we conclude that:
- Quantity I is less than Quantity II.

### Final Answer
The correct option is that Quantity I is less than Quantity II.

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