**Direction:** Given below are two quantities named I and II. Based on the given information, you have to determine the relation between the two quantities. You should use the given data and your knowledge of Mathematics to choose among the possible answers.

**Quantity I:** One day Sandeep on tour travels first $160 ext{ km}$ at $64 ext{ km/hr}$ and the next $160 ext{ km}$ at $80 ext{ km/hr}$. The average speed of the tour is:

**Quantity II:** Sandeep went from Patna to Giridih with the speed of $60 ext{ km/hr}$ and returned back with the speed of $90 ext{ km/hr}$. Find the average speed.

Question

**Direction:** Given below are two quantities named I and II. Based on the given information, you have to determine the relation between the two quantities. You should use the given data and your knowledge of Mathematics to choose among the possible answers.

**Quantity I:** One day Sandeep on tour travels first $160 	ext{ km}$ at $64 	ext{ km/hr}$ and the next $160 	ext{ km}$ at $80 	ext{ km/hr}$. The average speed of the tour is:

**Quantity II:** Sandeep went from Patna to Giridih with the speed of $60 	ext{ km/hr}$ and returned back with the speed of $90 	ext{ km/hr}$. Find the average speed.

Accepted Answer

**Solution:**

1. **Quantity I:**
   - Distance 1 = $160 	ext{ km}$, Speed 1 = $64 	ext{ km/hr}$
   - Distance 2 = $160 	ext{ km}$, Speed 2 = $80 	ext{ km/hr}$

Calculate time taken for each part:
   $$t_1 = \frac{160}{64} = 2.5 	ext{ hours}$$
   $$t_2 = \frac{160}{80} = 2 	ext{ hours}$$

Total time = $2.5 + 2 = 4.5$ hours

Total distance = $160 + 160 = 320 	ext{ km}$

Average speed = $$\frac{	ext{Total distance}}{	ext{Total time}} = \frac{320}{4.5} = 71.11 	ext{ km/hr}$$

2. **Quantity II:**
   - Speed going = $60 	ext{ km/hr}$
   - Speed returning = $90 	ext{ km/hr}$

Average speed for round trip when distances are equal is given by harmonic mean:
   $$	ext{Average speed} = \frac{2 	imes 60 	imes 90}{60 + 90} = \frac{10800}{150} = 72 	ext{ km/hr}$$

3. **Comparison:**
   - Quantity I average speed = $71.11 	ext{ km/hr}$
   - Quantity II average speed = $72 	ext{ km/hr}$

Therefore, Quantity II > Quantity I.

**Answer:** Quantity II > Quantity I

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