If Sohail walks from his home to office at 16 km/h, he is late by 5 min. If he walks at 20 km/h, he reaches 10 min before the office time. Find the distance of his office from his house

Options

A. a. 22 km
B. b. 20 km
C. c. 18 km
D. d. 16 km

Answer

To solve the problem step by step, let's denote the distance from Sohail's home to his office as \( d \) kilometers. We know the following:

1. When Sohail walks at a speed of 16 km/h, he is late by 5 minutes.
2. When he walks at a speed of 20 km/h, he arrives 10 minutes early.

### Step 1: Convert minutes to hours
Since speed is given in km/h, we need to convert the time from minutes to hours:
- 5 minutes = \( \frac{5}{60} \) hours = \( \frac{1}{12} \) hours
- 10 minutes = \( \frac{10}{60} \) hours = \( \frac{1}{6} \) hours

### Step 2: Set up the equations based on the information provided
Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

1. When walking at 16 km/h:
\[ \text{Time taken} = \frac{d}{16} \quad \text{(he is late by 5 minutes)} \]
Therefore, the actual time he should take is:
\[ \frac{d}{16} = T + \frac{1}{12} \quad \text{(where T is the scheduled time)} \]

2. When walking at 20 km/h:
\[ \text{Time taken} = \frac{d}{20} \quad \text{(he reaches 10 minutes early)} \]
Therefore, the actual time he should take is:
\[ \frac{d}{20} = T - \frac{1}{6} \]

### Step 3: Set up the equations
From the two equations, we can express them as:
1. \( \frac{d}{16} - T = \frac{1}{12} \) (1)
2. \( \frac{d}{20} - T = -\frac{1}{6} \) (2)

### Step 4: Rearranging the equations
From equation (1):
\[ T = \frac{d}{16} - \frac{1}{12} \]
From equation (2):
\[ T = \frac{d}{20} + \frac{1}{6} \]

### Step 5: Equate the two expressions for T
\[ \frac{d}{16} - \frac{1}{12} = \frac{d}{20} + \frac{1}{6} \]

### Step 6: Clear the fractions
To eliminate the fractions, find a common denominator, which is 60:
\[ \frac{60d}{16} - \frac{60}{12} = \frac{60d}{20} + \frac{60}{6} \]
This simplifies to:
\[ \frac{15d}{4} - 5 = 3d + 10 \]

### Step 7: Solve for d
Multiply through by 4 to eliminate the fraction:
\[ 15d - 20 = 12d + 40 \]
Rearranging gives:
\[ 15d - 12d = 40 + 20 \]
\[ 3d = 60 \]
\[ d = 20 \]

### Conclusion
The distance from Sohail's home to his office is **20 kilometers**.