When a child goes to school at the speed of 5 km/h reaches 6 minutes late and when he goes at the speed of 6 km/ he reaches 6 minutes early. Find the distance between his home and school.

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the variables
Let the distance between the child's home and school be \( x \) kilometers.

### Step 2: Calculate the time taken at different speeds
- When the child travels at a speed of 5 km/h, the time taken to reach school is:
\[ \text{Time}_1 = \frac{x}{5} \text{ hours} \]
- When the child travels at a speed of 6 km/h, the time taken to reach school is:
\[ \text{Time}_2 = \frac{x}{6} \text{ hours} \]

### Step 3: Understand the time difference
The child is 6 minutes late when traveling at 5 km/h and 6 minutes early when traveling at 6 km/h. Therefore, the total difference in time between the two scenarios is:
\[ \text{Total time difference} = 6 \text{ minutes (late)} + 6 \text{ minutes (early)} = 12 \text{ minutes} \]
Convert 12 minutes into hours:
\[ 12 \text{ minutes} = \frac{12}{60} \text{ hours} = \frac{1}{5} \text{ hours} \]

### Step 4: Set up the equation
The difference in time taken at the two speeds can be expressed as:
\[ \text{Time}_1 - \text{Time}_2 = \frac{1}{5} \]
Substituting the expressions for time:
\[ \frac{x}{5} - \frac{x}{6} = \frac{1}{5} \]

### Step 5: Solve the equation
To solve the equation, first find a common denominator for the left side. The least common multiple of 5 and 6 is 30:
\[ \frac{6x}{30} - \frac{5x}{30} = \frac{1}{5} \]
This simplifies to:
\[ \frac{6x - 5x}{30} = \frac{1}{5} \]
\[ \frac{x}{30} = \frac{1}{5} \]

### Step 6: Cross-multiply to find \( x \)
Cross-multiplying gives:
\[ x = 30 \cdot \frac{1}{5} \]
\[ x = 6 \]

### Conclusion
The distance between the child's home and school is \( 6 \) kilometers.

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