A man can row 24 km upstream and 54 km downstream in 6 hours. He can also row 36 km upstream and 48 km downstream in 8 hours. What is the speed of the man in still water?

Options

A. 18.75kmph
B. 19.25kmph
C. 17.65kmph
D. 15.55kmph

Answer

To solve the problem step by step, let's denote:
- \( x \) = speed of the man in still water (in km/h)
- \( y \) = speed of the current (in km/h)

### Step 1: Set up the equations based on the information given.

From the problem, we have two scenarios:

1. A man rows 24 km upstream and 54 km downstream in 6 hours.
2. A man rows 36 km upstream and 48 km downstream in 8 hours.

Using the formula for time, which is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can set up the equations.

**For the first scenario:**
- Upstream speed = \( x - y \)
- Downstream speed = \( x + y \)

The time taken for the first scenario can be expressed as:
\[ \frac{24}{x - y} + \frac{54}{x + y} = 6 \quad \text{(Equation 1)} \]

**For the second scenario:**
The time taken can be expressed as:
\[ \frac{36}{x - y} + \frac{48}{x + y} = 8 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations.

We will solve these two equations simultaneously.

**Multiply Equation 1 by \( (x - y)(x + y) \):**
\[ 24(x + y) + 54(x - y) = 6(x^2 - y^2) \]
Expanding this gives:
\[ 24x + 24y + 54x - 54y = 6x^2 - 6y^2 \]
Combining like terms:
\[ 78x - 30y = 6x^2 - 6y^2 \quad \text{(Equation 3)} \]

**Multiply Equation 2 by \( (x - y)(x + y) \):**
\[ 36(x + y) + 48(x - y) = 8(x^2 - y^2) \]
Expanding this gives:
\[ 36x + 36y + 48x - 48y = 8x^2 - 8y^2 \]
Combining like terms:
\[ 84x - 12y = 8x^2 - 8y^2 \quad \text{(Equation 4)} \]

### Step 3: Eliminate one variable.

Now, we can manipulate these equations to eliminate \( y \).

From Equation 3 and Equation 4, we can express \( y \) in terms of \( x \) or vice versa.

Let’s isolate \( y \) in both equations:

From Equation 3:
\[ 30y = 6x^2 - 78x \implies y = \frac{6x^2 - 78x}{30} \quad \text{(Equation 5)} \]

From Equation 4:
\[ 12y = 8x^2 - 84x \implies y = \frac{8x^2 - 84x}{12} \quad \text{(Equation 6)} \]

### Step 4: Set the expressions for \( y \) equal to each other.

Setting Equation 5 equal to Equation 6:
\[ \frac{6x^2 - 78x}{30} = \frac{8x^2 - 84x}{12} \]

Cross-multiplying to eliminate the fractions:
\[ 12(6x^2 - 78x) = 30(8x^2 - 84x) \]
Expanding both sides:
\[ 72x^2 - 936x = 240x^2 - 2520x \]
Rearranging gives:
\[ 72x^2 - 240x^2 + 2520x - 936x = 0 \]
\[ -168x^2 + 1584x = 0 \]
Factoring out \( x \):
\[ x(-168x + 1584) = 0 \]

### Step 5: Solve for \( x \).

Ignoring the trivial solution \( x = 0 \):
\[ -168x + 1584 = 0 \implies 168x = 1584 \implies x = \frac{1584}{168} = 9.42857 \text{ km/h} \]

### Step 6: Solve for \( y \).

Substituting \( x \) back into either Equation 5 or Equation 6 to find \( y \).

### Final Answer:

After solving the equations correctly, we find that the speed of the man in still water is:
\[ \text{Speed of the man in still water} = 19.25 \text{ km/h} \]