To solve the problem step by step, let's break it down:
### Step 1: Calculate the distance Ranjeet travels.
Ranjeet drives his car at an average speed of 50 km/hr for 8 hours.
To find the distance, we use the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Substituting the values:
\[ \text{Distance} = 50 \, \text{km/hr} \times 8 \, \text{hours} = 400 \, \text{km} \]
### Step 2: Calculate Rahman's speed.
Rahman covers the same distance of 400 km in 5 hours.
To find Rahman's speed, we use the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Substituting the values:
\[ \text{Speed} = \frac{400 \, \text{km}}{5 \, \text{hours}} = 80 \, \text{km/hr} \]
### Step 3: Determine the new speeds after the increase.
Ranjeet increases his speed by 10 km/hr:
\[ \text{New Speed of Ranjeet} = 50 \, \text{km/hr} + 10 \, \text{km/hr} = 60 \, \text{km/hr} \]
Rahman increases his speed by 20 km/hr:
\[ \text{New Speed of Rahman} = 80 \, \text{km/hr} + 20 \, \text{km/hr} = 100 \, \text{km/hr} \]
### Step 4: Calculate the time taken by Ranjeet with the new speed.
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For Ranjeet:
\[ \text{Time} = \frac{400 \, \text{km}}{60 \, \text{km/hr}} = \frac{400}{60} = \frac{20}{3} \, \text{hours} \]
This is approximately 6 hours and 40 minutes.
### Step 5: Calculate the time taken by Rahman with the new speed.
For Rahman:
\[ \text{Time} = \frac{400 \, \text{km}}{100 \, \text{km/hr}} = 4 \, \text{hours} \]
### Step 6: Find the difference in time taken by Ranjeet and Rahman.
To find the difference:
\[ \text{Difference} = \text{Time taken by Ranjeet} - \text{Time taken by Rahman} \]
\[ \text{Difference} = \left(6 \, \text{hours} + 40 \, \text{minutes}\right) - 4 \, \text{hours} \]
Convert 40 minutes to hours:
\[ 40 \, \text{minutes} = \frac{40}{60} \, \text{hours} = \frac{2}{3} \, \text{hours} \]
So,
\[ \text{Difference} = \left(6 + \frac{2}{3}\right) - 4 = 2 + \frac{2}{3} = 2 \, \text{hours} + 40 \, \text{minutes} \]
### Final Answer:
The difference in time taken by Ranjeet and Rahman to reach their destination is **2 hours and 40 minutes**.
---
