John started from his home to his office, After 2 hours, his friend Ali started from office towards john's home. By the time John travelled one-fifth of the total distance, Ali had also travelled the same. Ali's speed is thrice of that of john's speed.
Quantity I: Difference in time (in hours) taken by john and Ali to reach their respective destinations.
Quantity II: 12 hours

Options

A. Quantity `I=` Quantity II or No relation
B. Quantity `I le` Quantity II
C. Quantity `I ge` Quantity II
D. Quantity `I gt` Quantity II

Explanation

To solve the problem, we need to analyze the information provided step-by-step.

### Step 1: Define Variables
Let John's speed be \( x \) km/h. Since Ali's speed is thrice John's speed, Ali's speed will be \( 3x \) km/h.

### Step 2: Determine Distances and Times
- Let the total distance from John's home to his office be \( D \) km.
- John starts traveling towards his office, and after 2 hours, Ali starts traveling towards John's home.
- In 2 hours, John will have traveled \( 2x \) km.

### Step 3: Distance Covered by John and Ali
By the time John has traveled one-fifth of the total distance \( D \), we can express this as:
\[ \frac{D}{5} = x \cdot t \quad \text{(where \( t \) is the time taken by John to travel \( \frac{D}{5} \))} \]
Thus,
\[ t = \frac{D}{5x} \]

### Step 4: Time Taken by Ali
Since Ali starts 2 hours after John, the time Ali has traveled when John has covered \( \frac{D}{5} \) is:
\[ t - 2 = \frac{D}{5x} - 2 \]

### Step 5: Distance Covered by Ali
In the same time, Ali also covers \( \frac{D}{5} \):
\[ \frac{D}{5} = 3x \cdot (t - 2) \]
Substituting \( t \):
\[ \frac{D}{5} = 3x \left(\frac{D}{5x} - 2\right) \]

### Step 6: Simplifying the Equation
Expanding the equation:
\[ \frac{D}{5} = \frac{3D}{5} - 6x \]
Rearranging gives:
\[ 6x = \frac{3D}{5} - \frac{D}{5} = \frac{2D}{5} \]
Thus,
\[ D = 15x \]

### Step 7: Calculate Time Taken by John and Ali
- Time taken by John to cover the total distance \( D \):
\[ \text{Time}_{\text{John}} = \frac{D}{x} = \frac{15x}{x} = 15 \text{ hours} \]
- Time taken by Ali to cover the same distance \( D \):
\[ \text{Time}_{\text{Ali}} = \frac{D}{3x} = \frac{15x}{3x} = 5 \text{ hours} \]

### Step 8: Calculate the Difference in Time
The difference in time taken by John and Ali is:
\[ \text{Difference} = \text{Time}_{\text{John}} - \text{Time}_{\text{Ali}} = 15 - 5 = 10 \text{ hours} \]

### Step 9: Compare with Quantity II
- Quantity I: 10 hours
- Quantity II: 12 hours

Since \( 10 < 12 \), we conclude that Quantity II is greater than Quantity I.

### Final Answer
**Quantity I is less than Quantity II.**

---