Two towns X and Y are some distance apart. A man cycle from X to Y at a speed of 10 km/hr and then back from Y to X at the rate of 15 km/hr. The average speed during the whole journey is

Options

A. `12 cdot 5` km/ hr
B. 12 km/ hr
C. The data is inadequate
D. 13 km/ hr

Answer

To find the average speed during the whole journey from town X to town Y and back, we can use the formula for average speed when the distances are the same but the speeds are different.

### Step 1: Define the distance between towns X and Y.
Let the distance between towns X and Y be \( d \) kilometers.

### Step 2: Calculate the time taken to travel from X to Y.
The speed from X to Y is 10 km/hr. The time taken to travel this distance can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{10} \text{ hours} \]

### Step 3: Calculate the time taken to travel from Y to X.
The speed from Y to X is 15 km/hr. The time taken to travel this distance can be calculated as:
\[ \text{Time} = \frac{d}{15} \text{ hours} \]

### Step 4: Calculate the total time for the whole journey.
The total time for the journey is the sum of the time taken to go from X to Y and from Y to X:
\[ \text{Total Time} = \frac{d}{10} + \frac{d}{15} \]

### Step 5: Find a common denominator and simplify the total time.
The least common multiple of 10 and 15 is 30. Thus, we can write:
\[ \text{Total Time} = \frac{3d}{30} + \frac{2d}{30} = \frac{5d}{30} = \frac{d}{6} \text{ hours} \]

### Step 6: Calculate the total distance traveled.
The total distance traveled is the distance from X to Y plus the distance from Y to X:
\[ \text{Total Distance} = d + d = 2d \text{ kilometers} \]

### Step 7: Calculate the average speed.
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{6}} = 2d \times \frac{6}{d} = 12 \text{ km/hr} \]

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

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