A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km down-stream in 6 hours and 30 minutes. The speed of the boat in still water is

Options

A. 10 km/hr
B. 4 km/hr
C. 14 km/hr
D. 6 km/hr

Answer

To solve the problem of finding the speed of the boat in still water, we will follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = speed of the boat in still water (in km/h)
- \( y \) = speed of the stream (in km/h)

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

1. **First Condition:**
- Distance upstream = 24 km
- Distance downstream = 28 km
- Total time = 6 hours

The speed upstream is \( x - y \) and the speed downstream is \( x + y \). The time taken for each part can be expressed as:
\[ \frac{24}{x - y} + \frac{28}{x + y} = 6 \quad \text{(Equation 1)} \]

2. **Second Condition:**
- Distance upstream = 30 km
- Distance downstream = 21 km
- Total time = 6 hours 30 minutes = 6.5 hours

The time taken for this scenario can be expressed as:
\[ \frac{30}{x - y} + \frac{21}{x + y} = 6.5 \quad \text{(Equation 2)} \]

### Step 3: Simplify the Equations
To simplify calculations, we can let:
- \( u = \frac{1}{x - y} \)
- \( v = \frac{1}{x + y} \)

Now, we can rewrite the equations:
1. From Equation 1:
\[ 24u + 28v = 6 \quad \text{(Equation 1)} \]

2. From Equation 2:
\[ 30u + 21v = 6.5 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Simultaneously
To eliminate one variable, we can multiply Equation 1 by 3 and Equation 2 by 4:
1. \( 72u + 84v = 18 \) (from Equation 1)
2. \( 120u + 84v = 26 \) (from Equation 2)

Now, subtract the first modified equation from the second:
\[ (120u + 84v) - (72u + 84v) = 26 - 18 \]
This simplifies to:
\[ 48u = 8 \implies u = \frac{1}{6} \]

### Step 5: Substitute Back to Find \( v \)
Now substitute \( u \) back into Equation 1:
\[ 24\left(\frac{1}{6}\right) + 28v = 6 \]
This simplifies to:
\[ 4 + 28v = 6 \implies 28v = 2 \implies v = \frac{1}{14} \]

### Step 6: Find \( x \) and \( y \)
Now we can find \( x \) and \( y \):
1. From \( u = \frac{1}{x - y} \):
\[ x - y = 6 \quad \text{(Equation 3)} \]

2. From \( v = \frac{1}{x + y} \):
\[ x + y = 14 \quad \text{(Equation 4)} \]

### Step 7: Solve for \( x \) and \( y \)
Now we can add Equation 3 and Equation 4:
\[ (x - y) + (x + y) = 6 + 14 \]
This simplifies to:
\[ 2x = 20 \implies x = 10 \]

Now substitute \( x \) back into Equation 3:
\[ 10 - y = 6 \implies y = 4 \]

### Final Answer
The speed of the boat in still water is:
\[ \boxed{10 \text{ km/h}} \]