Understanding Amit's Distance Problem

This problem involves calculating the actual distance Amit traveled based on a hypothetical change in his walking speed. We need to use the relationship between distance, speed, and time to solve it.

Key Information Provided:

• Original Speed: 15 km/hr
• Hypothetical Speed: 20 km/hr
• Difference in Distance Covered: 25 km more (if speed was 20 km/hr)
• Goal: Find the actual distance traveled.

Step-by-Step Solution

1. Define Variables

Let's define the variables we'll use:

• Let $D$ be the actual distance Amit traveled (in km).
• Let $T$ be the actual time Amit took to travel the distance $D$ (in hours).
• Let $S_1$ be the original speed = 15 km/hr.
• Let $S_2$ be the hypothetical speed = 20 km/hr.

2. Formulate Equations based on Speed, Distance, Time

We know the fundamental formula: Distance = Speed × Time.

Using this, we can write the equations for the two scenarios:

• Actual Scenario: The actual distance $D$ is covered at speed $S_1$ in time $T$.

  $D = S_1 \times T$

  $D = 15 \times T$ ... (Equation 1)

• Hypothetical Scenario: If Amit walked at speed $S_2$ for the *same time* $T$, the distance covered would be $D_{new}$.

  $D_{new} = S_2 \times T$

  $D_{new} = 20 \times T$ ... (Equation 2)

3. Relate the Distances

The problem states that if Amit walked at 20 km/hr (instead of 15 km/hr) for the same duration, he would have walked 25 km more. This means:

$D_{new} = D + 25$

4. Solve for Time (T)

Now, substitute the expressions for $D_{new}$ and $D$ from Equations 1 and 2 into the relationship above:

$20 \times T = (15 \times T) + 25$

To find $T$, we need to isolate it:

$20T - 15T = 25$

$5T = 25$

Divide both sides by 5:

$T = \frac{25}{5}$

$T = 5$ hours

So, Amit actually took 5 hours to travel.

5. Calculate the Actual Distance (D)

Now that we know the time $T$, we can find the actual distance $D$ using Equation 1:

$D = 15 \times T$

$D = 15 \times 5$

$D = 75$ km

Verification

Let's check if this distance fits the problem statement:

• Actual distance (15 km/hr): 75 km. Time taken = $75 \div 15 = 5$ hours.
• Hypothetical distance (20 km/hr for 5 hours): $20 \times 5 = 100$ km.
• Difference: $100 \text{ km} - 75 \text{ km} = 25 \text{ km}$.

This matches the information given in the question, confirming our calculation.

Final Answer

The actual distance travelled by Amit is 75 km.