The ratio of the speed of a boat in still water to the speed of the stream is $4 : 1$ respectively. The difference between the time taken by the boat to cover $D$ km downstream and upstream is $10$ hours. (Note: Speed of the stream is $5 { km/hr}$)

Quantity I: Find the time taken by the boat to cover $(D - 75)$ km downstream

Quantity II: Find the time taken by the boat to cover $(D + 45)$ km upstream

Compare Quantity I and Quantity II:

Question

The ratio of the speed of a boat in still water to the speed of the stream is $4 : 1$ respectively. The difference between the time taken by the boat to cover $D$ km downstream and upstream is $10$ hours. (Note: Speed of the stream is $5 	{ km/hr}$)

Quantity I: Find the time taken by the boat to cover $(D - 75)$ km downstream

Quantity II: Find the time taken by the boat to cover $(D + 45)$ km upstream

Compare Quantity I and Quantity II:

Accepted Answer

**Solution:**

1. **Given:**
   - Ratio of speed of boat in still water to speed of stream = $4 : 1$
   - Speed of stream, $s = 5 	ext{ km/hr}$
   - Difference in time taken to cover $D$ km downstream and upstream = 10 hours

2. **Find the speed of the boat in still water:**
   Since ratio is $4:1$ and stream speed is $5 	ext{ km/hr}$,
   
   $$	ext{Speed of boat in still water} = 4 	imes 5 = 20 	ext{ km/hr}$$

3. **Calculate downstream and upstream speeds:**
   - Downstream speed = boat speed + stream speed = $20 + 5 = 25 	ext{ km/hr}$
   - Upstream speed = boat speed - stream speed = $20 - 5 = 15 	ext{ km/hr}$

4. **Use the given difference in time to find $D$:**
   
   Time downstream = $\frac{D}{25}$
   
   Time upstream = $\frac{D}{15}$

Given:
   
   $$\frac{D}{15} - \frac{D}{25} = 10$$

5. **Solve for $D$:**
   
   Find common denominator:
   
   $$\frac{5D - 3D}{75} = 10$$

$$\frac{2D}{75} = 10$$

$$2D = 750$$

$$D = 375 	ext{ km}$$

6. **Calculate Quantity I (time to cover $D-75$ km downstream):**
   
   $$D - 75 = 375 - 75 = 300 	ext{ km}$$

Time downstream = $\frac{300}{25} = 12 	ext{ hours}$

7. **Calculate Quantity II (time to cover $D+45$ km upstream):**
   
   $$D + 45 = 375 + 45 = 420 	ext{ km}$$

Time upstream = $\frac{420}{15} = 28 	ext{ hours}$

8. **Compare Quantity I and Quantity II:**
   
   Quantity I = 12 hours

Quantity II = 28 hours

Clearly, Quantity I < Quantity II

**Answer:** (b) Quantity I < Quantity II

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