Osaka walks from his house at 5 km/h and reaches his office 10 minutes late. If this speed had been 6 km/h he would have reached 15 minutes early. The distance of his office from his house is :

Options

A. 15 km
B. 12.5 km
C. 10.5 km
D. 18 km

Answer

To solve the problem, we need to set up equations based on the information given about Osaka's walking speeds and the time differences.

### Step 1: Define Variables
Let the distance from Osaka's house to his office be \( D \) kilometers.

### Step 2: Calculate Time Taken at 5 km/h
At a speed of 5 km/h, the time taken to reach the office is given by:
\[ \text{Time}_{5} = \frac{D}{5} \text{ hours} \]
Since he is 10 minutes late, we can express this time in relation to the scheduled time \( T \):
\[ \text{Time}_{5} = T + \frac{10}{60} \text{ hours} \quad \text{(10 minutes converted to hours)} \]
Thus, we have:
\[ \frac{D}{5} = T + \frac{1}{6} \]

### Step 3: Calculate Time Taken at 6 km/h
At a speed of 6 km/h, the time taken to reach the office is:
\[ \text{Time}_{6} = \frac{D}{6} \text{ hours} \]
Since he arrives 15 minutes early, we can express this time as:
\[ \text{Time}_{6} = T - \frac{15}{60} \text{ hours} \quad \text{(15 minutes converted to hours)} \]
Thus, we have:
\[ \frac{D}{6} = T - \frac{1}{4} \]

### Step 4: Set Up the Equations
Now we have two equations:
1. \( \frac{D}{5} = T + \frac{1}{6} \)
2. \( \frac{D}{6} = T - \frac{1}{4} \)

### Step 5: Solve for \( T \)
From the first equation, we can express \( T \):
\[ T = \frac{D}{5} - \frac{1}{6} \]

### Step 6: Substitute \( T \) into the Second Equation
Substituting \( T \) into the second equation:
\[ \frac{D}{6} = \left(\frac{D}{5} - \frac{1}{6}\right) - \frac{1}{4} \]

### Step 7: Simplify the Equation
Now we simplify the right-hand side:
\[ \frac{D}{6} = \frac{D}{5} - \frac{1}{6} - \frac{1}{4} \]
To combine the fractions on the right, we need a common denominator. The least common multiple of 6 and 4 is 12:
\[ \frac{D}{6} = \frac{D}{5} - \frac{2}{12} - \frac{3}{12} \]
\[ \frac{D}{6} = \frac{D}{5} - \frac{5}{12} \]

### Step 8: Clear the Fractions
To eliminate the fractions, multiply the entire equation by 60 (the least common multiple of 5, 6, and 12):
\[ 10D = 12D - 25 \]

### Step 9: Solve for \( D \)
Rearranging gives:
\[ 10D - 12D = -25 \]
\[ -2D = -25 \]
\[ D = \frac{25}{2} = 12.5 \text{ km} \]

### Final Answer
The distance of Osaka's office from his house is **12.5 kilometers**.

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