You arrive at your school 5 minutes late if you walk with a speed of 4 km/h, but you arrive 10 minutes before the schedule time if you walk with a speed of 5 km/h. The distance of your school from your house (in km) is :

Options

A. a)4
B. b)5
C. c)10
D. d)2

Answer

To solve the problem step by step, we will denote the distance to the school as \( D \) km and the usual time taken to reach the school as \( t \) hours.

### Step 1: Set up the equations based on the given information.
1. When walking at a speed of 4 km/h, you arrive 5 minutes late:
\[ \text{Time taken} = t + \frac{5}{60} \text{ hours} \]
Using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \):
\[ D = 4 \left( t + \frac{5}{60} \right) \]

2. When walking at a speed of 5 km/h, you arrive 10 minutes early:
\[ \text{Time taken} = t - \frac{10}{60} \text{ hours} \]
Again using the distance formula:
\[ D = 5 \left( t - \frac{10}{60} \right) \]

### Step 2: Set the two equations for distance equal to each other.
Since both expressions equal \( D \):
\[ 4 \left( t + \frac{5}{60} \right) = 5 \left( t - \frac{10}{60} \right) \]

### Step 3: Simplify the equation.
Expanding both sides:
\[ 4t + \frac{20}{60} = 5t - \frac{50}{60} \]
This simplifies to:
\[ 4t + \frac{1}{3} = 5t - \frac{5}{6} \]

### Step 4: Combine like terms.
Rearranging gives:
\[ 4t - 5t = -\frac{5}{6} - \frac{1}{3} \]
Converting \(\frac{1}{3}\) to sixths:
\[ 4t - 5t = -\frac{5}{6} - \frac{2}{6} \]
This simplifies to:
\[ -t = -\frac{7}{6} \]
Thus:
\[ t = \frac{7}{6} \text{ hours} \]

### Step 5: Calculate the distance \( D \).
Using the value of \( t \) in one of the distance equations:
\[ D = 4 \left( \frac{7}{6} + \frac{5}{60} \right) \]
Convert \(\frac{5}{60}\) to sixths:
\[ \frac{5}{60} = \frac{1}{12} \]
Thus:
\[ D = 4 \left( \frac{7}{6} + \frac{1}{12} \right) \]
Finding a common denominator (12):
\[ D = 4 \left( \frac{14}{12} + \frac{1}{12} \right) = 4 \left( \frac{15}{12} \right) \]
Calculating:
\[ D = \frac{60}{12} = 5 \text{ km} \]

### Final Answer:
The distance of your school from your house is **5 km**.

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