In the given questions, two quantities are given, one as Quantity I and another as Quantity II. You have to determine relationship between two quantities and choose the appropriate option.
A and B can together finish a piece of work in 20 days. If B starts working and after 15 days is replaced by A, A can finish the remaining work in 24 days.
Quantity I. Number of days taken by B alone to finish the same pices of work.
Quantity II. Number of days taken by A alone to finish the same piece of work.

Options

A. If Quantity I`ge` Quantity II
B. If Quantity I`gt`Quantity II
C. If Quantity I`lt`Quantity II
D. If Quantity I`=`Quantity II or the relationship cannot be established from the information that is given

Answer

To solve the problem, we need to determine the number of days taken by A and B to complete the work individually based on the information provided.

### Step-by-Step Solution:

1. **Understanding the Work Rate**:
- Let the number of days A takes to complete the work alone be \( X \).
- Let the number of days B takes to complete the work alone be \( Y \).
- The work done by A in one day is \( \frac{1}{X} \).
- The work done by B in one day is \( \frac{1}{Y} \).
- Together, A and B can complete the work in 20 days, so:
\[ \frac{1}{X} + \frac{1}{Y} = \frac{1}{20} \]

2. **Work Done by B in 15 Days**:
- B works for 15 days, so the work done by B in those 15 days is:
\[ \text{Work by B} = 15 \times \frac{1}{Y} = \frac{15}{Y} \]

3. **Work Done by A in 24 Days**:
- After B works for 15 days, A works for 24 days to finish the remaining work. The work done by A in those 24 days is:
\[ \text{Work by A} = 24 \times \frac{1}{X} = \frac{24}{X} \]

4. **Total Work Equation**:
- The total work done by both A and B must equal 1 (the whole work):
\[ \frac{15}{Y} + \frac{24}{X} = 1 \]

5. **Setting Up the Equations**:
- We now have two equations:
1. \( \frac{1}{X} + \frac{1}{Y} = \frac{1}{20} \) (Equation 1)
2. \( \frac{15}{Y} + \frac{24}{X} = 1 \) (Equation 2)

6. **Expressing Y in terms of X**:
- From Equation 1, we can express \( \frac{1}{Y} \):
\[ \frac{1}{Y} = \frac{1}{20} - \frac{1}{X} \]
- Thus,
\[ Y = \frac{20X}{X - 20} \]

7. **Substituting Y in Equation 2**:
- Substitute \( Y \) in Equation 2:
\[ \frac{15(X - 20)}{20X} + \frac{24}{X} = 1 \]
- Simplifying this gives:
\[ \frac{15(X - 20) + 480}{20X} = 1 \]
- Multiply through by \( 20X \):
\[ 15(X - 20) + 480 = 20X \]
- Expanding and rearranging gives:
\[ 15X - 300 + 480 = 20X \implies 180 = 5X \implies X = 36 \]

8. **Finding Y**:
- Substitute \( X = 36 \) back into the equation for \( Y \):
\[ \frac{1}{Y} = \frac{1}{20} - \frac{1}{36} \]
- Finding a common denominator (LCM of 20 and 36 is 180):
\[ \frac{1}{Y} = \frac{9 - 5}{180} = \frac{4}{180} \implies Y = 45 \]

### Conclusion:
- Quantity I (Days taken by B alone) = 45 days
- Quantity II (Days taken by A alone) = 36 days

### Comparison:
- Quantity I (45 days) > Quantity II (36 days)

### Final Answer:
- Quantity I is greater than Quantity II.