Quantity I: A man on tour travels first 160 km at 64km/hr and the next 160 km at 80 km/hr. The average speed of the tour is:
Quantity II: A went from P to Q with the speed of 60km/hr. and retrun back with the speed of 90km/hr. Find the average speed.

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 calculate the average speed for both quantities step by step.

### Quantity I:
A man travels the first 160 km at a speed of 64 km/hr and the next 160 km at a speed of 80 km/hr. We need to find the average speed for the entire journey.

1. **Calculate the total distance:**
\[ \text{Total Distance} = 160 \text{ km} + 160 \text{ km} = 320 \text{ km} \]

2. **Calculate the time taken for the first part of the journey:**
\[ \text{Time}_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{160 \text{ km}}{64 \text{ km/hr}} = 2.5 \text{ hours} \]

3. **Calculate the time taken for the second part of the journey:**
\[ \text{Time}_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{160 \text{ km}}{80 \text{ km/hr}} = 2 \text{ hours} \]

4. **Calculate the total time taken:**
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = 2.5 \text{ hours} + 2 \text{ hours} = 4.5 \text{ hours} \]

5. **Calculate the average speed:**
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{320 \text{ km}}{4.5 \text{ hours}} \approx 71.11 \text{ km/hr} \]

### Quantity II:
A goes from P to Q at a speed of 60 km/hr and returns back at a speed of 90 km/hr. We need to find the average speed for the round trip.

1. **Let the distance from P to Q be \(d\) km.**

2. **Calculate the time taken to go from P to Q:**
\[ \text{Time}_{PQ} = \frac{d}{60} \text{ hours} \]

3. **Calculate the time taken to return from Q to P:**
\[ \text{Time}_{QP} = \frac{d}{90} \text{ hours} \]

4. **Calculate the total time taken for the round trip:**
\[ \text{Total Time} = \text{Time}_{PQ} + \text{Time}_{QP} = \frac{d}{60} + \frac{d}{90} \]
To add these fractions, we find a common denominator (which is 180):
\[ \text{Total Time} = \frac{3d}{180} + \frac{2d}{180} = \frac{5d}{180} = \frac{d}{36} \text{ hours} \]

5. **Calculate the total distance for the round trip:**
\[ \text{Total Distance} = d + d = 2d \text{ km} \]

6. **Calculate the average speed:**
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{36}} = 2d \times \frac{36}{d} = 72 \text{ km/hr} \]

### Conclusion:
- **Quantity I Average Speed:** 71.11 km/hr
- **Quantity II Average Speed:** 72 km/hr

### Comparison:
Since 72 km/hr (Quantity II) is greater than 71.11 km/hr (Quantity I), we conclude that:

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

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