A person goes to his office by scooter at the speed of 50 km/hr and reaches 10 minutes earlier. If he goes at the speed of 40 km/hr, then he reaches 20 minutes late. What will be the speed (in km/hr) of the scooter to reach on time?

Options

A. `47 (4)/(13)`
B. `46 (2)/(13)`
C. `48 (2)/(13)`
D. `47 (2)/(13)`

Answer

To solve the problem step by step, we need to determine the speed of the scooter that allows the person to reach his office on time.

### Step 1: Define the variables
Let the distance to the office be \( D \) km. The time taken to reach the office on time is \( T \) hours.

### Step 2: Set up the equations based on the given speeds
1. When the person travels at 50 km/hr, he reaches 10 minutes early:
\[ \text{Time taken} = T - \frac{10}{60} = T - \frac{1}{6} \text{ hours} \]
Therefore, the equation for this scenario is:
\[ D = 50 \left(T - \frac{1}{6}\right) \]

2. When the person travels at 40 km/hr, he reaches 20 minutes late:
\[ \text{Time taken} = T + \frac{20}{60} = T + \frac{1}{3} \text{ hours} \]
Therefore, the equation for this scenario is:
\[ D = 40 \left(T + \frac{1}{3}\right) \]

### Step 3: Set the two equations for distance equal to each other
From the two equations we have:
\[ 50 \left(T - \frac{1}{6}\right) = 40 \left(T + \frac{1}{3}\right) \]

### Step 4: Expand and simplify the equation
Expanding both sides:
\[ 50T - \frac{50}{6} = 40T + \frac{40}{3} \]
To simplify, convert \(\frac{50}{6}\) and \(\frac{40}{3}\) to have a common denominator:
\[ \frac{50}{6} = \frac{25}{3} \]
So the equation becomes:
\[ 50T - \frac{25}{3} = 40T + \frac{40}{3} \]

### Step 5: Move all terms involving \( T \) to one side and constant terms to the other
\[ 50T - 40T = \frac{40}{3} + \frac{25}{3} \]
\[ 10T = \frac{65}{3} \]
\[ T = \frac{65}{30} = \frac{13}{6} \text{ hours} \]

### Step 6: Substitute \( T \) back to find \( D \)
Using \( D = 50 \left(T - \frac{1}{6}\right) \):
\[ D = 50 \left(\frac{13}{6} - \frac{1}{6}\right) = 50 \left(\frac{12}{6}\right) = 50 \times 2 = 100 \text{ km} \]

### Step 7: Calculate the speed to reach on time
To find the speed \( S \) to reach on time:
\[ S = \frac{D}{T} = \frac{100}{\frac{13}{6}} = 100 \times \frac{6}{13} = \frac{600}{13} \approx 46.15 \text{ km/hr} \]

### Final Answer
The speed of the scooter to reach on time is approximately \( 46.15 \) km/hr.