A man car row - third of a kiometre downstream in 5 min and return to the starting point in another 10 min. Find the speed of the man in still water

Answer

To solve the problem step-by-step, we will determine the speed of the man in still water based on the information provided.

### Step 1: Understand the Problem
The man rows 1/3 km downstream in 5 minutes and returns to the starting point in another 10 minutes. We need to find his speed in still water (U) and the speed of the current (V).

### Step 2: Convert Time to Hours
First, we convert the time taken from minutes to hours since speed is usually expressed in kilometers per hour (km/h).

- Downstream time = 5 minutes = 5/60 hours = 1/12 hours
- Upstream time = 10 minutes = 10/60 hours = 1/6 hours

### Step 3: Calculate Downstream Speed
When rowing downstream, the effective speed is the sum of the man's speed in still water (U) and the speed of the current (V).

Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]

For downstream:
\[ \frac{1}{3} = (U + V) \times \frac{1}{12} \]

Multiplying both sides by 12:
\[ 12 \times \frac{1}{3} = U + V \]
\[ 4 = U + V \]
(Equation 1)

### Step 4: Calculate Upstream Speed
When rowing upstream, the effective speed is the man's speed in still water minus the speed of the current.

Using the same formula for upstream:
\[ \frac{1}{3} = (U - V) \times \frac{1}{6} \]

Multiplying both sides by 6:
\[ 6 \times \frac{1}{3} = U - V \]
\[ 2 = U - V \]
(Equation 2)

### Step 5: Solve the Equations
Now, we have two equations:
1. \( U + V = 4 \) (Equation 1)
2. \( U - V = 2 \) (Equation 2)

We can add these two equations to eliminate V:
\[ (U + V) + (U - V) = 4 + 2 \]
\[ 2U = 6 \]
\[ U = 3 \text{ km/h} \]

### Step 6: Find the Speed of the Current
Now, we can substitute U back into either equation to find V. Using Equation 1:
\[ 3 + V = 4 \]
\[ V = 4 - 3 \]
\[ V = 1 \text{ km/h} \]

### Final Answer
The speed of the man in still water is **3 km/h**.

---