Amit intended to travel a certain distance at a certain uniform speed. But after one hour, he increased his speed by 25%. As a result, in the remaining part of the time that he originally planned for the journey, he could now cover as much distance as he initially thought he would be able to cover.
After Amit increased his speed, if he decided to terminate his journey after covering the distance he initially intended to cover and not cover the extra distance as given in the data, what is the total time taken for the journey?

Options

A. A)4 hr 12 min.
B. B)5 hr 24 min
C. C)3 hr 36 min.
D. D)4 hr 36 min.

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define Variables
Let:
- \( T \) = total time Amit originally planned for the journey (in hours)
- \( S \) = original speed (in km/h)

### Step 2: Understand the Journey
Amit travels for 1 hour at speed \( S \):
- Distance covered in the first hour = \( S \times 1 = S \) km

After 1 hour, he increases his speed by 25%. Thus, his new speed becomes:
- New speed = \( S + 0.25S = 1.25S \)

### Step 3: Remaining Time
The remaining time after the first hour is:
- Remaining time = \( T - 1 \) hours

### Step 4: Distance Calculation
In the remaining time, he covers the same distance he initially planned to cover in total time \( T \):
- Total distance planned = \( S \times T \) km
- Distance remaining after the first hour = \( S \times T - S = S(T - 1) \) km

### Step 5: Distance Covered at Increased Speed
Using the new speed for the remaining distance:
- Distance covered in remaining time = New speed × Remaining time
- \( S(T - 1) = 1.25S \times (T - 1) \)

### Step 6: Setting Up the Equation
Since the distance covered in the remaining time equals the remaining distance:
\[ S(T - 1) = 1.25S \times (T - 1) \]

### Step 7: Simplifying the Equation
Dividing both sides by \( S \) (assuming \( S \neq 0 \)):
\[ T - 1 = 1.25(T - 1) \]
Expanding the right side:
\[ T - 1 = 1.25T - 1.25 \]
Rearranging gives:
\[ T - 1.25T = -1.25 + 1 \]
\[ -0.25T = -0.25 \]
Dividing both sides by -0.25:
\[ T = 5 \text{ hours} \]

### Step 8: Total Distance Calculation
Now, we can calculate the total distance:
- Total distance = \( S \times T = S \times 5 \)

### Step 9: Distance Covered in the First Hour
Distance covered in the first hour:
- Distance = \( S \) km

### Step 10: Remaining Distance
Remaining distance after the first hour:
- Remaining distance = \( S \times 5 - S = 4S \)

### Step 11: Time to Cover Remaining Distance
Time taken to cover the remaining distance at the new speed:
- Time = Distance / Speed = \( \frac{4S}{1.25S} = \frac{4}{1.25} = 3.2 \text{ hours} \)

### Step 12: Total Time Taken
Total time taken for the journey:
- Total time = Time for first hour + Time for remaining distance
- Total time = \( 1 + 3.2 = 4.2 \text{ hours} \)

### Step 13: Convert to Hours and Minutes
Convert 4.2 hours into hours and minutes:
- 0.2 hours = \( 0.2 \times 60 = 12 \) minutes
- Total time = 4 hours and 12 minutes

### Final Answer
The total time taken for the journey is **4 hours and 12 minutes**.

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