Quantity: present ago of A and B are in the ratio of `4:5` respectively. Five years hence the ratio of their ages becomes `5:6` respectively. What is A's present age ?
Quantity II: If the average age of A, B and C are 24 years, and the average of B and C is 20, then the age of A:

Options

A. Quantity - I `gt` Quantity -Iis
B. Quantity `-I lt` Quantituy - II
C. Quantity `-Ige` Quantity - II
D. Quantity `- I le` Quantity - II

Explanation

To solve the problem, we will break it down into two parts corresponding to Quantity I and Quantity II.

### Quantity I: Present Age of A

1. **Set up the ratio of ages**:
Let the present age of A be \(4x\) and the present age of B be \(5x\), based on the ratio given (4:5).

2. **Set up the equation for future ages**:
In 5 years, A's age will be \(4x + 5\) and B's age will be \(5x + 5\). According to the problem, the ratio of their ages in 5 years will be \(5:6\). This gives us the equation:
\[ \frac{4x + 5}{5x + 5} = \frac{5}{6} \]

3. **Cross-multiply to eliminate the fraction**:
\[ 6(4x + 5) = 5(5x + 5) \]
Expanding both sides:
\[ 24x + 30 = 25x + 25 \]

4. **Rearranging the equation**:
\[ 24x + 30 - 25 = 25x \]
\[ 24x + 5 = 25x \]
\[ 5 = 25x - 24x \]
\[ 5 = x \]

5. **Finding A's present age**:
Substitute \(x\) back into the expression for A's age:
\[ A's \, present \, age = 4x = 4 \times 5 = 20 \, years \]

### Quantity II: Age of A

1. **Set up the average age equation**:
The average age of A, B, and C is 24 years. Thus, the total age of A, B, and C is:
\[ A + B + C = 24 \times 3 = 72 \, years \]

2. **Set up the average age of B and C**:
The average age of B and C is 20 years. Thus, the total age of B and C is:
\[ B + C = 20 \times 2 = 40 \, years \]

3. **Finding A's age**:
We know:
\[ A + (B + C) = 72 \]
Substitute \(B + C\) from the previous step:
\[ A + 40 = 72 \]
\[ A = 72 - 40 = 32 \, years \]

### Conclusion
- **Quantity I**: A's present age = 20 years
- **Quantity II**: A's age = 32 years

### Comparison
Since 32 years (Quantity II) is greater than 20 years (Quantity I), we conclude that:

**Quantity II is greater than Quantity I.**