A person driving at a speed of 42 km/h reaches office 1 minute early, whereas if driving at a speed of 36 km/h, the person reaches office 3 minutes late. What is the distance (in km) that the person is covering?
A)`15.4`
B)`16.8`
C)`12.9`
D)`18.2`

Options

A. `15.4`
B. `16.8`
C. `12.9`
D. `18.2`

Answer

To solve the problem, we will use the relationship between speed, distance, and time. The formula we will use is:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Let's denote:
- \( d \) = distance to the office (in km)
- \( t \) = time taken to reach the office at the normal speed (in hours)

### Step 1: Set up the equations based on the given information

1. When driving at 42 km/h, the person reaches 1 minute early:
- Time taken = \( t - \frac{1}{60} \) hours
- Distance equation:
\[ d = 42 \left(t - \frac{1}{60}\right) \]

2. When driving at 36 km/h, the person reaches 3 minutes late:
- Time taken = \( t + \frac{3}{60} \) hours
- Distance equation:
\[ d = 36 \left(t + \frac{3}{60}\right) \]

### Step 2: Set the two equations for distance equal to each other

Since both expressions equal \( d \), we can set them equal to each other:
\[ 42 \left(t - \frac{1}{60}\right) = 36 \left(t + \frac{3}{60}\right) \]

### Step 3: Expand both sides

Expanding both sides gives:
\[ 42t - \frac{42}{60} = 36t + \frac{108}{60} \]

### Step 4: Rearrange the equation

Rearranging the equation to isolate \( t \):
\[ 42t - 36t = \frac{42}{60} + \frac{108}{60} \]
\[ 6t = \frac{150}{60} \]
\[ 6t = 2.5 \]
\[ t = \frac{2.5}{6} = \frac{5}{12} \text{ hours} \]

### Step 5: Calculate the distance using one of the speeds

Now, we can substitute \( t \) back into either distance equation. We will use the first equation:
\[ d = 42 \left(t - \frac{1}{60}\right) \]
Substituting \( t \):
\[ d = 42 \left(\frac{5}{12} - \frac{1}{60}\right) \]

### Step 6: Find a common denominator and simplify

Finding a common denominator for \( \frac{5}{12} \) and \( \frac{1}{60} \):
- The least common multiple of 12 and 60 is 60.
- Convert \( \frac{5}{12} \) to have a denominator of 60:
\[ \frac{5}{12} = \frac{25}{60} \]
So,
\[ d = 42 \left(\frac{25}{60} - \frac{1}{60}\right) = 42 \left(\frac{24}{60}\right) \]
\[ d = 42 \times \frac{2}{5} = \frac{84}{5} = 16.8 \text{ km} \]

### Final Answer
The distance that the person is covering is \( \text{16.8 km} \).