Speed of a boat in still water is 12 kmph and speed of stream in y kmph. If in travelling 230 km upstream boat takes `66 2/3%` more time than travelling 230 km downstream, then find the average of the speed of boat and stream.

Options

A. 3 kmph
B. `8 1/2` kmph
C. `7 1/2` kmph
D. 7 kmph

Answer

To solve the problem step by step, we will follow the given information and derive the necessary equations.

### Step 1: Define the Variables
- Let the speed of the boat in still water = 12 km/h
- Let the speed of the stream = y km/h
- Distance travelled = 230 km

### Step 2: Calculate the Speed Downstream and Upstream
- Speed downstream = Speed of boat + Speed of stream = \(12 + y\) km/h
- Speed upstream = Speed of boat - Speed of stream = \(12 - y\) km/h

### Step 3: Calculate the Time Taken for Downstream and Upstream
- Time taken to travel downstream = Distance / Speed downstream = \(\frac{230}{12 + y}\) hours
- Time taken to travel upstream = Distance / Speed upstream = \(\frac{230}{12 - y}\) hours

### Step 4: Set Up the Equation Based on Time Difference
According to the problem, the time taken to travel upstream is \(66 \frac{2}{3}\%\) more than the time taken to travel downstream. This can be expressed as:
\[ \text{Time upstream} = \text{Time downstream} + \frac{200}{3} \times \text{Time downstream} \]
This simplifies to:
\[ \text{Time upstream} = \frac{500}{300} \times \text{Time downstream} = \frac{5}{3} \times \text{Time downstream} \]

### Step 5: Substitute the Time Expressions
Substituting the expressions for time:
\[ \frac{230}{12 - y} = \frac{5}{3} \times \frac{230}{12 + y} \]

### Step 6: Cancel Out the Distance
Since both sides have the same distance (230 km), we can cancel it out:
\[ \frac{1}{12 - y} = \frac{5}{3(12 + y)} \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ 3(12 + y) = 5(12 - y) \]

### Step 8: Expand and Rearrange the Equation
Expanding both sides:
\[ 36 + 3y = 60 - 5y \]
Rearranging gives:
\[ 3y + 5y = 60 - 36 \]
\[ 8y = 24 \]

### Step 9: Solve for y
Dividing both sides by 8:
\[ y = 3 \text{ km/h} \]

### Step 10: Calculate the Average Speed
Now, we find the average speed of the boat and the stream:
\[ \text{Average speed} = \frac{\text{Speed of boat} + \text{Speed of stream}}{2} = \frac{12 + 3}{2} = \frac{15}{2} = 7.5 \text{ km/h} \]

### Final Answer
The average of the speed of the boat and the stream is \(7.5\) km/h.

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