A person can swim in still water at `5 m//s`.He moves in a river of velocity `3 m//s`, first down the steam and next same distance up the stream. The ratio of times taken are

Options

A. `1:1`
B. `1:2`
C. `1:4`
D. `4:1`

Explanation

To solve the problem, we need to determine the ratio of the times taken by a person swimming downstream and upstream in a river. Let's break this down step by step.

### Step 1: Define the Variables
- Speed of the swimmer in still water (v) = 5 m/s
- Speed of the river (u) = 3 m/s
- Distance to be covered (d) = d (we'll keep it as a variable for now)

### Step 2: Calculate the Effective Speeds
1. **Downstream Speed**: When swimming downstream, the effective speed of the swimmer is the sum of his swimming speed and the river's speed.
\[ v_{\text{downstream}} = v + u = 5 \, \text{m/s} + 3 \, \text{m/s} = 8 \, \text{m/s} \]

2. **Upstream Speed**: When swimming upstream, the effective speed is the swimmer's speed minus the river's speed.
\[ v_{\text{upstream}} = v - u = 5 \, \text{m/s} - 3 \, \text{m/s} = 2 \, \text{m/s} \]

### Step 3: Calculate the Time Taken for Each Direction
1. **Time Taken Downstream (t1)**:
\[ t_1 = \frac{d}{v_{\text{downstream}}} = \frac{d}{8} \]

2. **Time Taken Upstream (t2)**:
\[ t_2 = \frac{d}{v_{\text{upstream}}} = \frac{d}{2} \]

### Step 4: Calculate the Ratio of Times
Now, we can find the ratio of the times taken:
\[ \frac{t_1}{t_2} = \frac{\frac{d}{8}}{\frac{d}{2}} = \frac{d}{8} \times \frac{2}{d} = \frac{2}{8} = \frac{1}{4} \]

### Step 5: Conclusion
Thus, the ratio of the times taken to swim downstream and upstream is:
\[ t_1 : t_2 = 1 : 4 \]

### Final Answer
The ratio of times taken is \(1 : 4\).

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