A and B row on a river starting simultaneously from the same point A downstream and B upstream in 15 minutes they are 2.5 km apart. A then turns to follow B. After 30 minutes from the beginning the boats have together rowed 3.5 km. If speed of A and B is constant. Find the speed of current speed.

Answer

To solve the problem step by step, we will break down the information given and apply the relevant formulas.

### Step 1: Understand the Problem
A and B start rowing from the same point, A, with A rowing downstream and B rowing upstream. After 15 minutes, they are 2.5 km apart. We need to find the speed of the current.

### Step 2: Convert Time to Hours
Since the distance is given in kilometers, we should convert the time from minutes to hours for consistency.
- 15 minutes = 15/60 hours = 1/4 hours

### Step 3: Set Up the Equation for Distance
The distance between A and B after 15 minutes is given as 2.5 km. The formula for distance is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Let the speed of A in still water be \( S_A \) km/h and the speed of B in still water be \( S_B \) km/h. The speed of the current is \( V \) km/h.

- Speed of A downstream = \( S_A + V \)
- Speed of B upstream = \( S_B - V \)

The total distance covered by both A and B in 15 minutes can be expressed as:
\[ (S_A + V) \times \frac{1}{4} + (S_B - V) \times \frac{1}{4} = 2.5 \]

### Step 4: Simplify the Equation
Multiplying through by 4 to eliminate the fraction:
\[ S_A + V + S_B - V = 10 \]
This simplifies to:
\[ S_A + S_B = 10 \quad \text{(Equation 1)} \]

### Step 5: Analyze the Second Condition
After 30 minutes (which is 30/60 = 1/2 hours), the total distance covered by both A and B is 3.5 km. Since they have already covered 2.5 km in the first 15 minutes, they cover an additional:
\[ 3.5 - 2.5 = 1 \text{ km in the next 15 minutes} \]

### Step 6: Set Up the Equation for the Second Distance
In the next 15 minutes (1/4 hours), the distance covered can be expressed as:
\[ (S_A + V) \times \frac{1}{4} + (S_B - V) \times \frac{1}{4} = 1 \]

### Step 7: Simplify the Second Equation
Multiplying through by 4:
\[ S_A + V + S_B - V = 4 \]
This simplifies to:
\[ S_A + S_B = 4 \quad \text{(Equation 2)} \]

### Step 8: Solve the Equations
Now we have two equations:
1. \( S_A + S_B = 10 \) (Equation 1)
2. \( S_A + S_B = 4 \) (Equation 2)

From these equations, we can see that there is a contradiction. We need to reconsider the second condition.

### Step 9: Correct Understanding of Speeds
The relative speed when A turns to follow B is:
\[ (S_A + V) + (S_B - V) \]
This is the combined speed when both are moving towards each other.

### Step 10: Calculate the Current Speed
From the first condition, we can express:
\[ S_A + S_B = 10 \]
From the second condition, we can express the distance covered in the second 15 minutes:
\[ (S_A + V) + (S_B - V) = 4 \]
This can be rearranged to:
\[ S_A + S_B = 4 + 2V \]

Setting the two equations equal:
\[ 10 = 4 + 2V \]
\[ 2V = 6 \]
\[ V = 3 \text{ km/h} \]

### Final Answer
The speed of the current is **3 km/h**.

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