A boat can move at 5 km/h in still water (i.e., when water is not flowing). The speed of stream of the river is 1 km/h. A boat takes 80 minutes to go from a point A to another point B and return to the same point.
(i) What is the distance between the two points?
(ii) What is the ratio of downstream speed and upstream speed?
(iii) What is the ratio of time taken in downstream speed to the upstream speed?

Answer

To solve the problem step by step, we will address each part of the question sequentially.

### Given Data:
- Speed of the boat in still water (x) = 5 km/h
- Speed of the stream (y) = 1 km/h
- Total time taken for the round trip = 80 minutes = \( \frac{80}{60} \) hours = \( \frac{4}{3} \) hours

### (i) Finding the Distance Between Two Points (A and B)

1. **Calculate Downstream Speed:**
\[ \text{Downstream Speed} = x + y = 5 + 1 = 6 \text{ km/h} \]

2. **Calculate Upstream Speed:**
\[ \text{Upstream Speed} = x - y = 5 - 1 = 4 \text{ km/h} \]

3. **Let the distance between A and B be \( d \) km.**
- Time taken to go downstream from A to B:
\[ \text{Time}_{\text{downstream}} = \frac{d}{\text{Downstream Speed}} = \frac{d}{6} \text{ hours} \]
- Time taken to return upstream from B to A:
\[ \text{Time}_{\text{upstream}} = \frac{d}{\text{Upstream Speed}} = \frac{d}{4} \text{ hours} \]

4. **Total time for the round trip:**
\[ \text{Total Time} = \text{Time}_{\text{downstream}} + \text{Time}_{\text{upstream}} = \frac{d}{6} + \frac{d}{4} \]
- To add these fractions, find a common denominator (which is 12):
\[ \frac{d}{6} = \frac{2d}{12}, \quad \frac{d}{4} = \frac{3d}{12} \]
- Thus,
\[ \text{Total Time} = \frac{2d + 3d}{12} = \frac{5d}{12} \]

5. **Set the total time equal to \( \frac{4}{3} \) hours:**
\[ \frac{5d}{12} = \frac{4}{3} \]

6. **Cross-multiply to solve for \( d \):**
\[ 5d \cdot 3 = 4 \cdot 12 \implies 15d = 48 \implies d = \frac{48}{15} = \frac{16}{5} \text{ km} \]

### (ii) Ratio of Downstream Speed to Upstream Speed

1. **Downstream Speed = 6 km/h**
2. **Upstream Speed = 4 km/h**

3. **Ratio:**
\[ \text{Ratio} = \frac{\text{Downstream Speed}}{\text{Upstream Speed}} = \frac{6}{4} = \frac{3}{2} \]

### (iii) Ratio of Time Taken in Downstream Speed to Upstream Speed

1. **Time taken downstream:**
\[ \text{Time}_{\text{downstream}} = \frac{d}{6} = \frac{\frac{16}{5}}{6} = \frac{16}{30} = \frac{8}{15} \text{ hours} \]

2. **Time taken upstream:**
\[ \text{Time}_{\text{upstream}} = \frac{d}{4} = \frac{\frac{16}{5}}{4} = \frac{16}{20} = \frac{4}{5} \text{ hours} \]

3. **Ratio of time taken:**
\[ \text{Ratio} = \frac{\text{Time}_{\text{downstream}}}{\text{Time}_{\text{upstream}}} = \frac{\frac{8}{15}}{\frac{4}{5}} = \frac{8}{15} \cdot \frac{5}{4} = \frac{40}{60} = \frac{2}{3} \]

### Summary of Answers:
(i) Distance between A and B = \( \frac{16}{5} \) km
(ii) Ratio of downstream speed to upstream speed = \( 3:2 \)
(iii) Ratio of time taken in downstream speed to upstream speed = \( 2:3 \)