A ship sails out an island at the rate of 15km/hr and sails back to the starting point at 10km/hr . Find the average sailing speed an average speed for the whole journey.

Explanation

To find the average sailing speed for the whole journey, we can follow these steps:

### Step 1: Define the Distance
Let the distance from the starting point to the island be \( x \) kilometers.

### Step 2: Calculate the Time for Each Leg of the Journey
- **Time to sail to the island**:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{x}{15} \text{ hours} \]

- **Time to sail back**:
\[ \text{Time} = \frac{x}{10} \text{ hours} \]

### Step 3: Calculate the Total Time for the Journey
The total time taken for the round trip is the sum of the time taken to go to the island and the time taken to return:
\[ \text{Total Time} = \frac{x}{15} + \frac{x}{10} \]

To add these fractions, we need a common denominator. The least common multiple of 15 and 10 is 30:
\[ \text{Total Time} = \frac{2x}{30} + \frac{3x}{30} = \frac{5x}{30} = \frac{x}{6} \text{ hours} \]

### Step 4: Calculate the Total Distance
The total distance for the round trip is:
\[ \text{Total Distance} = x + x = 2x \text{ kilometers} \]

### Step 5: Calculate the Average Speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{\frac{x}{6}} \]

### Step 6: Simplify the Average Speed Calculation
To simplify:
\[ \text{Average Speed} = 2x \times \frac{6}{x} = 12 \text{ km/hr} \]

### Final Answer
The average sailing speed for the whole journey is **12 km/hr**.

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