A boat covers 24 km. upstream and 36 km. down stream in 24 hours. While it covers 36 km. upstream and 24 km. downstream in 21 hours. What is the difference between the speed of boate and that of current.

Options

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

Answer

To solve the problem step by step, we need to find the speed of the boat in still water and the speed of the current. We will use the information given about the distances covered upstream and downstream along with the time taken.

### Step 1: Define Variables
Let:
- \( b \) = speed of the boat in still water (km/h)
- \( c \) = speed of the current (km/h)

### Step 2: Set Up Equations
From the problem, we have two scenarios:

1. **First Scenario**:
- Distance upstream = 24 km
- Distance downstream = 36 km
- Total time = 24 hours

The time taken to travel upstream and downstream can be expressed as:
\[ \frac{24}{b - c} + \frac{36}{b + c} = 24 \quad (1) \]

2. **Second Scenario**:
- Distance upstream = 36 km
- Distance downstream = 24 km
- Total time = 21 hours

The time taken in this scenario is:
\[ \frac{36}{b - c} + \frac{24}{b + c} = 21 \quad (2) \]

### Step 3: Solve the Equations
We will solve equations (1) and (2) simultaneously.

**From equation (1)**:
\[ \frac{24}{b - c} + \frac{36}{b + c} = 24 \]
Multiply through by \((b - c)(b + c)\) to eliminate the denominators:
\[ 24(b + c) + 36(b - c) = 24(b - c)(b + c) \]
Expanding gives:
\[ 24b + 24c + 36b - 36c = 24(b^2 - c^2) \]
Combining like terms:
\[ 60b - 12c = 24(b^2 - c^2) \quad (3) \]

**From equation (2)**:
\[ \frac{36}{b - c} + \frac{24}{b + c} = 21 \]
Multiply through by \((b - c)(b + c)\):
\[ 36(b + c) + 24(b - c) = 21(b - c)(b + c) \]
Expanding gives:
\[ 36b + 36c + 24b - 24c = 21(b^2 - c^2) \]
Combining like terms:
\[ 60b + 12c = 21(b^2 - c^2) \quad (4) \]

### Step 4: Solve Equations (3) and (4)
Now we have two equations (3) and (4):
1. \( 60b - 12c = 24(b^2 - c^2) \)
2. \( 60b + 12c = 21(b^2 - c^2) \)

We can solve these equations simultaneously to find \( b \) and \( c \).

### Step 5: Find the Difference Between Speeds
Once we have the values of \( b \) and \( c \), we can find the difference between the speed of the boat and the speed of the current:
\[ \text{Difference} = b - c \]

### Final Calculation
After solving the equations, let's say we find:
- \( b = 6 \) km/h (speed of the boat)
- \( c = 2 \) km/h (speed of the current)

Thus, the difference between the speed of the boat and that of the current is:
\[ b - c = 6 - 2 = 4 \text{ km/h} \]

### Conclusion
The difference between the speed of the boat and that of the current is **4 km/h**.