Directions: In the following question, two quantities I and II are given. Compare both the quantities and choose the correct option and give answer accordingly.
Quantity I: A and B can do a work in `8/3` days, B and C can do the same work in 3 days, and A and C can do the same work in `(24)/5` days. In what time can C alone complete the work?
Quantity II: 15

Options

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

Answer

To solve the problem, we need to determine how long C alone can complete the work based on the information given about A, B, and C working together.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- Let the total work be represented in units. We can assume the total work is 24 units for easier calculations (as it is a common multiple of the denominators).

2. **Calculate Work Rates**:
- **A + B** can complete the work in \( \frac{8}{3} \) days:
\[ \text{Work rate of A + B} = \frac{24 \text{ units}}{\frac{8}{3} \text{ days}} = 24 \times \frac{3}{8} = 9 \text{ units/day} \]
- **B + C** can complete the work in 3 days:
\[ \text{Work rate of B + C} = \frac{24 \text{ units}}{3 \text{ days}} = 8 \text{ units/day} \]
- **A + C** can complete the work in \( \frac{24}{5} \) days:
\[ \text{Work rate of A + C} = \frac{24 \text{ units}}{\frac{24}{5} \text{ days}} = 24 \times \frac{5}{24} = 5 \text{ units/day} \]

3. **Setting Up Equations**:
- Let the work rates of A, B, and C be \( a \), \( b \), and \( c \) respectively.
- From the work rates calculated:
- \( a + b = 9 \) (1)
- \( b + c = 8 \) (2)
- \( a + c = 5 \) (3)

4. **Solving the Equations**:
- From equation (1), we have \( a = 9 - b \).
- Substitute \( a \) in equation (3):
\[ (9 - b) + c = 5 \implies c = 5 - 9 + b \implies c = b - 4 \quad (4) \]
- Substitute \( c \) from equation (4) into equation (2):
\[ b + (b - 4) = 8 \implies 2b - 4 = 8 \implies 2b = 12 \implies b = 6 \]
- Substitute \( b = 6 \) back into equations (1) and (4):
\[ a + 6 = 9 \implies a = 3 \]
\[ c = 6 - 4 = 2 \]

5. **Finding C's Work Rate**:
- C can complete 2 units of work in one day. To find out how many days C will take to complete the total work of 24 units:
\[ \text{Time taken by C} = \frac{24 \text{ units}}{2 \text{ units/day}} = 12 \text{ days} \]

### Conclusion:
- **Quantity I**: C can complete the work in 12 days.
- **Quantity II**: 15 days.

### Comparison:
- Since 12 days (Quantity I) is less than 15 days (Quantity II), we conclude that Quantity I is smaller than Quantity II.

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

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