Neha walks at a speed of 12 km/h and reaches her school 12 minutes late. If she walks at the speed of 18 km/h, she reaches her school 18 minutes early. Find the distance of her school from her house.

Options

A. 18 km
B. 64 km
C. 75 km
D. 72 km

Answer

To solve the problem step by step, we can use the information given about Neha's walking speeds and the time differences to find the distance to her school.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let the distance from Neha's house to her school be \( D \) kilometers.
- Let the time taken to reach school on time be \( T \) hours.

2. **Convert Time to Hours:**
- Neha is 12 minutes late when walking at 12 km/h. This means she took \( T + \frac{12}{60} \) hours to reach school.
- Neha is 18 minutes early when walking at 18 km/h. This means she took \( T - \frac{18}{60} \) hours to reach school.

3. **Set Up the Equations:**
- From the first scenario (12 km/h):
\[ D = 12 \left( T + \frac{12}{60} \right) \]
- From the second scenario (18 km/h):
\[ D = 18 \left( T - \frac{18}{60} \right) \]

4. **Equate the Two Expressions for Distance:**
\[ 12 \left( T + \frac{12}{60} \right) = 18 \left( T - \frac{18}{60} \right) \]

5. **Simplify the Equation:**
- Convert minutes to hours:
\[ 12 \left( T + \frac{1}{5} \right) = 18 \left( T - \frac{3}{10} \right) \]
- Distributing both sides:
\[ 12T + \frac{12}{5} = 18T - \frac{54}{10} \]
- Convert \(\frac{54}{10}\) to \(\frac{27}{5}\):
\[ 12T + \frac{12}{5} = 18T - \frac{27}{5} \]

6. **Rearranging the Equation:**
- Move all terms involving \( T \) to one side and constant terms to the other:
\[ 12T - 18T = -\frac{27}{5} - \frac{12}{5} \]
\[ -6T = -\frac{39}{5} \]
\[ T = \frac{39}{30} = \frac{13}{10} \text{ hours} \]

7. **Calculate the Distance:**
- Substitute \( T \) back into one of the distance equations. Using \( D = 12 \left( T + \frac{12}{60} \right) \):
\[ D = 12 \left( \frac{13}{10} + \frac{1}{5} \right) \]
- Convert \(\frac{1}{5}\) to \(\frac{2}{10}\):
\[ D = 12 \left( \frac{13}{10} + \frac{2}{10} \right) = 12 \left( \frac{15}{10} \right) = 12 \times 1.5 = 18 \text{ km} \]

### Final Answer:
The distance from Neha's house to her school is **18 km**.