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

Options

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

Answer

To solve the problem step by step, we need to analyze the swimming speeds of the person in still water and in the river, both downstream and upstream.

### Step 1: Define the given values
- Speed of the person in still water (V_person) = 5 m/s
- Speed of the river (V_river) = 3 m/s

### Step 2: Calculate the effective speed downstream
When the person swims downstream, the effective speed (V_downstream) is the sum of the person's speed and the river's speed:
\[ V_{downstream} = V_{person} + V_{river} \]
\[ V_{downstream} = 5 \, \text{m/s} + 3 \, \text{m/s} = 8 \, \text{m/s} \]

### Step 3: Calculate the effective speed upstream
When the person swims upstream, the effective speed (V_upstream) is the difference between the person's speed and the river's speed:
\[ V_{upstream} = V_{person} - V_{river} \]
\[ V_{upstream} = 5 \, \text{m/s} - 3 \, \text{m/s} = 2 \, \text{m/s} \]

### Step 4: Define the distance
Let the distance the person swims downstream and upstream be \(d\).

### Step 5: Calculate the time taken for each journey
Using the formula for time, \(t = \frac{d}{v}\):
- Time taken to swim downstream (\(t_1\)):
\[ t_1 = \frac{d}{V_{downstream}} = \frac{d}{8} \]
- Time taken to swim upstream (\(t_2\)):
\[ t_2 = \frac{d}{V_{upstream}} = \frac{d}{2} \]

### Step 6: Find the ratio of times taken
Now, we find the ratio of the time taken upstream to the time taken downstream:
\[ \frac{t_2}{t_1} = \frac{\frac{d}{2}}{\frac{d}{8}} \]
Cancelling \(d\) from the numerator and denominator:
\[ \frac{t_2}{t_1} = \frac{1/2}{1/8} = \frac{8}{2} = 4 \]

### Step 7: Express the ratio in the required format
Thus, the ratio of times taken \(t_1 : t_2\) is:
\[ t_1 : t_2 = 1 : 4 \]

### Final Answer
The ratio of times taken to swim the same distance downstream and upstream is \(1 : 4\).

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