Quantity I: The age of teacher, if the average age of 36 students is 14. When teacher's age is included the average increases by 1.
Quantity II: The age of teacher, if the average age of 19 students is 35. When teacher's age is included the average increases by `0.5.`

Options

A. Quantity `Igt` Quantity II
B. Quantity `I ge` Quantity II
C. Quantity `II gt` Quantity II
D. Quantity `II ge` Quantity I

Explanation

To solve the problem, we need to calculate the age of the teacher in both quantities and then compare them.

### Quantity I:
1. **Given Data**:
- Average age of 36 students = 14
- When the teacher's age is included, the average increases by 1 (to 15).

2. **Calculate Total Age of Students**:
\[ \text{Total age of 36 students} = \text{Average age} \times \text{Number of students} = 14 \times 36 = 504 \]

3. **Set Up the Equation**:
- Let \( T \) be the age of the teacher.
- When the teacher's age is included, the total number of individuals becomes 37.
- The new average age becomes 15.
\[ \frac{504 + T}{37} = 15 \]

4. **Solve for Teacher's Age**:
- Multiply both sides by 37:
\[ 504 + T = 15 \times 37 \]
\[ 504 + T = 555 \]
- Subtract 504 from both sides:
\[ T = 555 - 504 = 51 \]

### Quantity II:
1. **Given Data**:
- Average age of 19 students = 35
- When the teacher's age is included, the average increases by 0.5 (to 35.5).

2. **Calculate Total Age of Students**:
\[ \text{Total age of 19 students} = 35 \times 19 = 665 \]

3. **Set Up the Equation**:
- Let \( T \) be the age of the teacher.
- When the teacher's age is included, the total number of individuals becomes 20.
- The new average age becomes 35.5.
\[ \frac{665 + T}{20} = 35.5 \]

4. **Solve for Teacher's Age**:
- Multiply both sides by 20:
\[ 665 + T = 35.5 \times 20 \]
\[ 665 + T = 710 \]
- Subtract 665 from both sides:
\[ T = 710 - 665 = 45 \]

### Conclusion:
- **Quantity I**: Teacher's age = 51
- **Quantity II**: Teacher's age = 45

### Comparison:
- Quantity I (51) is greater than Quantity II (45).

### Answer:
**Quantity I is greater than Quantity II.**

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