Quantity I: A and B together can do a pices of work in 4 days. If A alone can do the same work in 6 days, then B alone can do the same work in?
Quantity II: A can do a pices of work in 4 hours, B and C together can do it in 3 hours, while A and C together can do it in 2 hours. How long will B alone take to do it ?

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 will analyze both quantities step by step.

### Quantity I:
1. **Understanding the Work Rates**:
- Let the total work be represented as 1 unit of work.
- A and B together can complete the work in 4 days. Therefore, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1 \text{ unit}}{4 \text{ days}} = \frac{1}{4} \text{ units/day} \]

2. **A's Work Rate**:
- A alone can complete the work in 6 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ unit}}{6 \text{ days}} = \frac{1}{6} \text{ units/day} \]

3. **Finding B's Work Rate**:
- Let B's work rate be \( b \) units/day. From the combined work rate, we have:
\[ \frac{1}{4} = \frac{1}{6} + b \]
- Rearranging gives:
\[ b = \frac{1}{4} - \frac{1}{6} \]
- To subtract these fractions, find a common denominator (which is 12):
\[ b = \frac{3}{12} - \frac{2}{12} = \frac{1}{12} \text{ units/day} \]

4. **Calculating Time for B Alone**:
- If B's work rate is \( \frac{1}{12} \) units/day, then the time taken by B to complete the work alone is:
\[ \text{Time for B} = \frac{1 \text{ unit}}{\frac{1}{12} \text{ units/day}} = 12 \text{ days} \]

### Quantity II:
1. **Understanding the Work Rates**:
- A can do the work in 4 hours. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ unit}}{4 \text{ hours}} = \frac{1}{4} \text{ units/hour} \]

2. **B and C Together**:
- B and C together can do the work in 3 hours. Therefore, their combined work rate is:
\[ \text{Work rate of B and C} = \frac{1 \text{ unit}}{3 \text{ hours}} = \frac{1}{3} \text{ units/hour} \]

3. **A and C Together**:
- A and C together can do the work in 2 hours. Therefore, their combined work rate is:
\[ \text{Work rate of A and C} = \frac{1 \text{ unit}}{2 \text{ hours}} = \frac{1}{2} \text{ units/hour} \]

4. **Setting Up the Equations**:
- Let B's work rate be \( b \) units/hour and C's work rate be \( c \) units/hour.
- From the equations, we have:
1. \( \frac{1}{4} + c = \frac{1}{2} \) (from A and C)
2. \( b + c = \frac{1}{3} \) (from B and C)

5. **Solving for C**:
- From the first equation:
\[ c = \frac{1}{2} - \frac{1}{4} = \frac{1}{4} \text{ units/hour} \]

6. **Finding B's Work Rate**:
- Substitute \( c \) into the second equation:
\[ b + \frac{1}{4} = \frac{1}{3} \]
- Rearranging gives:
\[ b = \frac{1}{3} - \frac{1}{4} \]
- Finding a common denominator (which is 12):
\[ b = \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \text{ units/hour} \]

7. **Calculating Time for B Alone**:
- If B's work rate is \( \frac{1}{12} \) units/hour, then the time taken by B to complete the work alone is:
\[ \text{Time for B} = \frac{1 \text{ unit}}{\frac{1}{12} \text{ units/hour}} = 12 \text{ hours} \]

### Conclusion:
- **Quantity I**: B alone takes 12 days.
- **Quantity II**: B alone takes 12 hours.

### Comparison:
- 12 days (Quantity I) is greater than 12 hours (Quantity II).

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