A man crosses the river perpendicular to river in time t seconds and travels an equal distance down the stream in T seconds. The ratio of man's speed in still water to the river water will be:

Options

A. `(t^(2)-T^(2))/(t^(2)+T^(2))`
B. `(T^(2)-t^(2))/(T^(2)+t^(2))`
C. `(t^(2)+T^(2))/(t^(2)-T^(2))`
D. `(T^(2)+t^(2))/(T^(2)-t^(2))`

Explanation

To solve the problem, we need to find the ratio of the man's speed in still water (v) to the speed of the river water (u).

### Step-by-Step Solution:

1. **Define Variables:**
- Let \( v \) = speed of the man in still water.
- Let \( u \) = speed of the river water.
- The man crosses the river in time \( t \) seconds.
- The man travels an equal distance downstream in time \( T \) seconds.

2. **Determine the Distance:**
- Let the width of the river (the distance crossed perpendicular to the flow) be \( d \).
- The distance traveled downstream (along the flow of the river) is also \( d \).

3. **Calculate Speeds:**
- The speed of the man perpendicular to the river flow is given by:
\[ \text{Speed} = \frac{d}{t} = \frac{d}{t} \]
- The speed of the man downstream (with the flow of the river) is:
\[ \text{Speed} = \frac{d}{T} = v + u \]

4. **Relate Speeds:**
- From the first part, we have:
\[ v = \frac{d}{t} \]
- From the second part, we have:
\[ v + u = \frac{d}{T} \]

5. **Set Up the Equation:**
- We can set up the equation using the two expressions for \( d \):
\[ d = vt \quad \text{and} \quad d = (v + u)T \]
- Thus, we can equate these:
\[ vt = (v + u)T \]

6. **Rearranging the Equation:**
- Rearranging gives:
\[ vt = vT + uT \]
- Rearranging further, we isolate \( u \):
\[ uT = vt - vT \]
\[ uT = v(t - T) \]
- Therefore, we can express \( u \) as:
\[ u = \frac{v(t - T)}{T} \]

7. **Finding the Ratio:**
- Now, we can find the ratio \( \frac{v}{u} \):
\[ \frac{v}{u} = \frac{v}{\frac{v(t - T)}{T}} = \frac{vT}{v(t - T)} = \frac{T}{t - T} \]

8. **Final Result:**
- Thus, the ratio of the man's speed in still water to the speed of the river water is:
\[ \frac{v}{u} = \frac{T}{t - T} \]