A boat moves downstream at the rate of one km in 7 1/2 minutes and upstream at the rate of 5 km an hour.What is the velocity of current ?

Options

A. 1.3 km/hr
B. 1.2 km/hr
C. 1.6 km/hr
D. 1.5 km/hr

Answer

To solve the problem of finding the velocity of the current, we can follow these steps:

### Step 1: Convert the downstream speed to km/h
The boat moves downstream at the rate of 1 km in 7.5 minutes. We need to convert this speed into kilometers per hour (km/h).

1. **Convert minutes to hours**:
\[ 7.5 \text{ minutes} = \frac{7.5}{60} \text{ hours} = 0.125 \text{ hours} \]

2. **Calculate the speed downstream (U)**:
\[ U = \frac{\text{Distance}}{\text{Time}} = \frac{1 \text{ km}}{0.125 \text{ hours}} = 8 \text{ km/h} \]

### Step 2: Identify the upstream speed
The boat moves upstream at a speed of 5 km/h. We denote this speed as \( V \).

### Step 3: Set up the equations
Let \( X \) be the speed of the boat in still water and \( Y \) be the speed of the current.

1. **Downstream speed equation**:
\[ X + Y = U \quad \text{(1)} \]
Substituting \( U = 8 \text{ km/h} \):
\[ X + Y = 8 \quad \text{(1)} \]

2. **Upstream speed equation**:
\[ X - Y = V \quad \text{(2)} \]
Substituting \( V = 5 \text{ km/h} \):
\[ X - Y = 5 \quad \text{(2)} \]

### Step 4: Solve the equations
Now we have a system of two equations:
1. \( X + Y = 8 \)
2. \( X - Y = 5 \)

To solve for \( Y \), we can add both equations:
\[ (X + Y) + (X - Y) = 8 + 5 \]
This simplifies to:
\[ 2X = 13 \quad \Rightarrow \quad X = 6.5 \text{ km/h} \]

Now, substitute \( X \) back into equation (1) to find \( Y \):
\[ 6.5 + Y = 8 \quad \Rightarrow \quad Y = 8 - 6.5 = 1.5 \text{ km/h} \]

### Conclusion
The velocity of the current is \( Y = 1.5 \text{ km/h} \).