A man can row a boat 78 km downstream and 42 km upstream in 6 hours. The speed of the stream is 6 km h. In how many hours can be row the boat a distance of 100 km in still water?

Options

A. 5
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to determine the speed of the man rowing the boat in still water. We can then use that speed to find out how long it will take him to row 100 km in still water.

### Step 1: Define the variables
Let:
- \( b \) = speed of the boat in still water (in km/h)
- \( s \) = speed of the stream = 6 km/h

### Step 2: Calculate the effective speeds
When rowing downstream, the effective speed is:
\[ b + s = b + 6 \text{ km/h} \]
When rowing upstream, the effective speed is:
\[ b - s = b - 6 \text{ km/h} \]

### Step 3: Set up the equations based on the given information
The man rows 78 km downstream and 42 km upstream in a total of 6 hours. We can express this as:
\[ \frac{78}{b + 6} + \frac{42}{b - 6} = 6 \]

### Step 4: Solve the equation
Multiply through by the common denominator \((b + 6)(b - 6)\) to eliminate the fractions:
\[ 78(b - 6) + 42(b + 6) = 6(b^2 - 36) \]
Expanding both sides:
\[ 78b - 468 + 42b + 252 = 6b^2 - 216 \]
Combine like terms:
\[ 120b - 216 = 6b^2 - 216 \]
Rearranging gives:
\[ 6b^2 - 120b = 0 \]
Factoring out \(6b\):
\[ 6b(b - 20) = 0 \]
Thus, \(b = 0\) or \(b = 20\). Since the speed cannot be zero, we have:
\[ b = 20 \text{ km/h} \]

### Step 5: Calculate the time to row 100 km in still water
Now that we know the speed of the boat in still water is 20 km/h, we can find the time to row 100 km:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{100}{20} = 5 \text{ hours} \]

### Final Answer
The man can row the boat a distance of 100 km in still water in **5 hours**.

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