A man rows upstream 13 km and downstream 28 km talking 5 hrs each time. What is the velocity in (km/hr) of the current?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define Variables
Let:
- \( x \) = speed of the boat in still water (in km/hr)
- \( y \) = speed of the current (in km/hr)

### Step 2: Determine Upstream and Downstream Speeds
- The speed of the boat when rowing upstream (against the current) is \( x - y \).
- The speed of the boat when rowing downstream (with the current) is \( x + y \).

### Step 3: Use the Distance, Speed, and Time Relationship
We know that:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
From the problem, we have:
- Upstream distance = 13 km, time = 5 hours
- Downstream distance = 28 km, time = 5 hours

### Step 4: Set Up the Equations
Using the formula for speed, we can set up the following equations:

1. For upstream:
\[ x - y = \frac{13 \text{ km}}{5 \text{ hr}} \quad \text{(1)} \]

2. For downstream:
\[ x + y = \frac{28 \text{ km}}{5 \text{ hr}} \quad \text{(2)} \]

### Step 5: Simplify the Equations
Calculating the right side of the equations:
1. From equation (1):
\[ x - y = 2.6 \quad \text{(since } \frac{13}{5} = 2.6\text{)} \]

2. From equation (2):
\[ x + y = 5.6 \quad \text{(since } \frac{28}{5} = 5.6\text{)} \]

### Step 6: Solve the System of Equations
Now we have the system of equations:
1. \( x - y = 2.6 \) (Equation 1)
2. \( x + y = 5.6 \) (Equation 2)

To eliminate \( y \), we can add both equations:
\[ (x - y) + (x + y) = 2.6 + 5.6 \]
This simplifies to:
\[ 2x = 8.2 \]
Thus:
\[ x = \frac{8.2}{2} = 4.1 \text{ km/hr} \]

Now, substitute \( x \) back into one of the original equations to find \( y \):
Using Equation 1:
\[ 4.1 - y = 2.6 \]
So,
\[ y = 4.1 - 2.6 = 1.5 \text{ km/hr} \]

### Final Answer
The speed of the current \( y \) is **1.5 km/hr**.

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