A sailor in river takes a boat from place A to place B, and returns to A. place A and B are 21 km apart. And he takes 10 hours to go and return. The time taken by the boat to row 7 km downstream is equal to the time taken by the boat to row 3 km upstream. Then what is the speed of the current?

Options

A. 1Km/h
B. 2Km/h
C. 3Km/h
D. 4Km/h

Answer

To solve the problem step by step, we will follow the given information and use the relevant formulas.

### Step 1: Understand the Problem
The sailor travels from place A to place B, which are 21 km apart, and takes a total of 10 hours for the round trip. The time taken to row 7 km downstream is equal to the time taken to row 3 km upstream.

### Step 2: Define Variables
Let:
- \( x \) = speed of the boat in still water (km/h)
- \( y \) = speed of the current (km/h)

The downstream speed (when going from A to B) is \( x + y \) and the upstream speed (when returning from B to A) is \( x - y \).

### Step 3: Set Up Equations from the Given Information
1. The time taken to row 7 km downstream:
\[ \text{Time downstream} = \frac{7}{x + y} \]

2. The time taken to row 3 km upstream:
\[ \text{Time upstream} = \frac{3}{x - y} \]

Since these two times are equal:
\[ \frac{7}{x + y} = \frac{3}{x - y} \]

### Step 4: Cross Multiply to Solve for the Ratio
Cross multiplying gives:
\[ 7(x - y) = 3(x + y) \]
Expanding both sides:
\[ 7x - 7y = 3x + 3y \]
Rearranging gives:
\[ 7x - 3x = 7y + 3y \]
\[ 4x = 10y \]
Thus, we have:
\[ \frac{x}{y} = \frac{10}{4} = \frac{5}{2} \]

### Step 5: Use the Total Time for the Round Trip
The total distance for the round trip is \( 21 + 21 = 42 \) km, and the total time is 10 hours. Therefore:
\[ \frac{21}{x + y} + \frac{21}{x - y} = 10 \]

### Step 6: Substitute \( x \) in Terms of \( y \)
From the ratio \( x = \frac{5}{2}y \):
\[ \frac{21}{\frac{5}{2}y + y} + \frac{21}{\frac{5}{2}y - y} = 10 \]
This simplifies to:
\[ \frac{21}{\frac{7}{2}y} + \frac{21}{\frac{3}{2}y} = 10 \]
\[ \frac{21 \cdot 2}{7y} + \frac{21 \cdot 2}{3y} = 10 \]
\[ \frac{42}{7y} + \frac{42}{3y} = 10 \]
\[ \frac{6}{y} + \frac{14}{y} = 10 \]
\[ \frac{20}{y} = 10 \]
Thus, \( y = 2 \) km/h.

### Step 7: Conclusion
The speed of the current is **2 km/h**.