Abhishek and Ayush start travelling in same direction at 8 km/hr and 13 km/hr respectively. After 4 hours, Abhishek doubled his speed and Ayush reduced his speed by 1 km/hr and reached the destination together. How long the entire journey last ?

Options

A. A) 9 hr
B. B)8 hr
C. C)4hr
D. D)6 hr

Answer

To solve the problem step by step, we will use the information provided about the speeds of Abhishek and Ayush, the time they traveled, and their changes in speed.

### Step 1: Define the variables
Let \( T \) be the total time taken for the journey.

### Step 2: Calculate the distance covered by Abhishek in the first 4 hours
Abhishek's initial speed is 8 km/hr. In the first 4 hours, the distance covered by Abhishek is:
\[ \text{Distance}_{\text{Abhishek}} = \text{Speed} \times \text{Time} = 8 \, \text{km/hr} \times 4 \, \text{hr} = 32 \, \text{km} \]

### Step 3: Determine the remaining time for Abhishek
After 4 hours, Abhishek doubles his speed to 16 km/hr. The remaining time for Abhishek is:
\[ \text{Remaining time} = T - 4 \, \text{hours} \]

### Step 4: Calculate the distance covered by Abhishek after 4 hours
The distance covered by Abhishek after the first 4 hours is:
\[ \text{Distance}_{\text{Abhishek}} = \text{Speed} \times \text{Time} = 16 \, \text{km/hr} \times (T - 4) \, \text{hr} \]

### Step 5: Total distance covered by Abhishek
The total distance covered by Abhishek is:
\[ \text{Total Distance}_{\text{Abhishek}} = 32 + 16(T - 4) \]

### Step 6: Calculate the distance covered by Ayush in the first 4 hours
Ayush's initial speed is 13 km/hr. In the first 4 hours, the distance covered by Ayush is:
\[ \text{Distance}_{\text{Ayush}} = 13 \, \text{km/hr} \times 4 \, \text{hr} = 52 \, \text{km} \]

### Step 7: Determine the remaining time for Ayush
After 4 hours, Ayush reduces his speed by 1 km/hr, making his new speed 12 km/hr. The remaining time for Ayush is also:
\[ \text{Remaining time} = T - 4 \, \text{hours} \]

### Step 8: Calculate the distance covered by Ayush after 4 hours
The distance covered by Ayush after the first 4 hours is:
\[ \text{Distance}_{\text{Ayush}} = 12 \, \text{km/hr} \times (T - 4) \, \text{hr} \]

### Step 9: Total distance covered by Ayush
The total distance covered by Ayush is:
\[ \text{Total Distance}_{\text{Ayush}} = 52 + 12(T - 4) \]

### Step 10: Set the distances equal to each other
Since both reach the destination together, we can set the total distances equal:
\[ 32 + 16(T - 4) = 52 + 12(T - 4) \]

### Step 11: Simplify the equation
Expanding both sides:
\[ 32 + 16T - 64 = 52 + 12T - 48 \]
This simplifies to:
\[ 16T - 12T = 52 - 48 + 64 - 32 \]
\[ 4T = 36 \]

### Step 12: Solve for \( T \)
\[ T = \frac{36}{4} = 9 \, \text{hours} \]

### Final Answer
The entire journey lasts **9 hours**.

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