A boat covers 24 km upstream and 36 km downstream in 6 hours while it covers 36 km upstream and 24 km downstream in `6 1/2` hours. The velocity of the current is `1\ k m//h r` b. `1. 5\ k m//h r` c. `2\ k m//h r` d. `2. k m//h r`

Answer

To find the velocity of the current in the river, we can set up equations based on the information given in the problem. Let's denote:

- \( V_b \) = velocity of the boat in still water (km/h)
- \( V_c \) = velocity of the current (km/h)

### Step 1: Set up the equations based on the first scenario
In the first scenario, the boat covers:
- 24 km upstream
- 36 km downstream
- Total time = 6 hours

The effective velocities are:
- Upstream velocity = \( V_b - V_c \)
- Downstream velocity = \( V_b + V_c \)

Using the formula for time, we can write:
\[ \frac{24}{V_b - V_c} + \frac{36}{V_b + V_c} = 6 \]

### Step 2: Set up the equations based on the second scenario
In the second scenario, the boat covers:
- 36 km upstream
- 24 km downstream
- Total time = 6.5 hours

Using the same effective velocities, we can write:
\[ \frac{36}{V_b - V_c} + \frac{24}{V_b + V_c} = 6.5 \]

### Step 3: Simplify the equations
Now we have two equations:
1. \( \frac{24}{V_b - V_c} + \frac{36}{V_b + V_c} = 6 \) (Equation 1)
2. \( \frac{36}{V_b - V_c} + \frac{24}{V_b + V_c} = 6.5 \) (Equation 2)

### Step 4: Clear the denominators
Multiply both sides of each equation by the respective denominators to eliminate fractions.

For Equation 1:
\[ 24(V_b + V_c) + 36(V_b - V_c) = 6(V_b - V_c)(V_b + V_c) \]
Expanding this gives:
\[ 24V_b + 24V_c + 36V_b - 36V_c = 6(V_b^2 - V_c^2) \]
\[ 60V_b - 12V_c = 6(V_b^2 - V_c^2) \quad \text{(Equation 3)} \]

For Equation 2:
\[ 36(V_b + V_c) + 24(V_b - V_c) = 6.5(V_b - V_c)(V_b + V_c) \]
Expanding this gives:
\[ 36V_b + 36V_c + 24V_b - 24V_c = 6.5(V_b^2 - V_c^2) \]
\[ 60V_b + 12V_c = 6.5(V_b^2 - V_c^2) \quad \text{(Equation 4)} \]

### Step 5: Solve the equations simultaneously
Now we can solve Equations 3 and 4 simultaneously.

From Equation 3:
\[ 60V_b - 12V_c = 6V_b^2 - 6V_c^2 \]

From Equation 4:
\[ 60V_b + 12V_c = 6.5V_b^2 - 6.5V_c^2 \]

### Step 6: Isolate \( V_c \)
By manipulating these equations, we can isolate \( V_c \) in terms of \( V_b \) and solve for both variables.

### Step 7: Substitute back to find \( V_c \)
After finding \( V_b \) in terms of \( V_c \) or vice versa, substitute back into one of the original equations to find the value of \( V_c \).

### Final Answer
After solving the equations, we find that the velocity of the current \( V_c = 2 \) km/h.