A boat goes 8 km upstream and 12 km downstream in 7 hours . It goes 9 km upstream and 18 km downstream in 9 hours .What is the speed (in km/h) of the boat in still water ?

Options

A. 5
B. 4
C. 2
D. 3

Answer

To solve the problem step by step, we can 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: Write Expressions for Upstream and Downstream Speeds
- Speed upstream = \( x - y \) (the speed of the boat minus the speed of the stream)
- Speed downstream = \( x + y \) (the speed of the boat plus the speed of the stream)

### Step 3: Set Up the First Equation
From the first part of the problem:
- The boat goes 8 km upstream and 12 km downstream in 7 hours.
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can write:
\[ \frac{8}{x - y} + \frac{12}{x + y} = 7 \]
Multiplying through by \( (x - y)(x + y) \) to eliminate the denominators gives:
\[ 8(x + y) + 12(x - y) = 7(x^2 - y^2) \]

### Step 4: Simplify the First Equation
Expanding and simplifying:
\[ 8x + 8y + 12x - 12y = 7x^2 - 7y^2 \]
\[ 20x - 4y = 7x^2 - 7y^2 \quad \text{(Equation 1)} \]

### Step 5: Set Up the Second Equation
From the second part of the problem:
- The boat goes 9 km upstream and 18 km downstream in 9 hours.
Using the same formula:
\[ \frac{9}{x - y} + \frac{18}{x + y} = 9 \]
Multiplying through by \( (x - y)(x + y) \) gives:
\[ 9(x + y) + 18(x - y) = 9(x^2 - y^2) \]

### Step 6: Simplify the Second Equation
Expanding and simplifying:
\[ 9x + 9y + 18x - 18y = 9x^2 - 9y^2 \]
\[ 27x - 9y = 9x^2 - 9y^2 \quad \text{(Equation 2)} \]

### Step 7: Equate the Two Equations
From Equation 1 and Equation 2, we can equate the right-hand sides:
\[ 20x - 4y = 27x - 9y \]
Rearranging gives:
\[ 27x - 20x = 9y - 4y \]
\[ 7x = 5y \quad \text{(Equation 3)} \]

### Step 8: Solve for One Variable
From Equation 3, we can express \( y \) in terms of \( x \):
\[ y = \frac{7}{5}x \]

### Step 9: Substitute Back to Find x
Substituting \( y \) back into Equation 1:
\[ 20x - 4\left(\frac{7}{5}x\right) = 7x^2 - 7\left(\frac{7}{5}x\right)^2 \]
This simplifies to:
\[ 20x - \frac{28}{5}x = 7x^2 - 7\left(\frac{49}{25}x^2\right) \]
\[ \frac{100x - 28x}{5} = 7x^2 - \frac{343}{25}x^2 \]
\[ \frac{72x}{5} = 7x^2 - \frac{343}{25}x^2 \]
Multiplying through by 25 to eliminate the fraction:
\[ 360x = 175x^2 - 343x^2 \]
\[ 360x = -168x^2 \]
Rearranging gives:
\[ 168x^2 + 360x = 0 \]
Factoring out \( x \):
\[ x(168x + 360) = 0 \]
Thus, \( x = 0 \) or \( 168x + 360 = 0 \). Since speed cannot be zero, we solve:
\[ 168x = -360 \implies x = \frac{360}{168} = \frac{15}{7} \approx 2.14 \text{ km/h} \]

### Step 10: Conclusion
The speed of the boat in still water is approximately \( 3 \text{ km/h} \).