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Let's break down the problem step by step to find the speed of the pedestrian.

### Problem Restatement:
- A pedestrian and a cyclist start at the same time from two different places (Aurangabad and Paithan) which are 40 km apart.
- They meet after 2 hours.
- After resuming their trips, the cyclist reaches Aurangabad 7 hours 30 minutes earlier than the pedestrian reaches Paithan.
- We need to determine the speed of the pedestrian.

### Step 1: Define Variables
- Let the speed of the pedestrian be \( x \) km/h.
- Let the speed of the cyclist be \( y \) km/h.
- They meet after 2 hours, so in 2 hours, they cover the entire 40 km distance. Therefore:
  \[
  2x + 2y = 40 \implies x + y = 20 \quad \text{(1)}
  \]

### Step 2: Time difference in travel
After meeting, both continue towards their destinations. The cyclist arrives 7 hours 30 minutes (or 7.5 hours) earlier than the pedestrian.

- The pedestrian has already covered part of the distance when they met, and the remaining distance for the pedestrian is covered at speed \( x \). The remaining distance for the pedestrian after meeting is:
  \[
  \text{Remaining distance for pedestrian} = 40 - 2x
  \]
  The time for the pedestrian to reach Paithan is:
  \[
  \text{Time for pedestrian} = \frac{40 - 2x}{x} \quad \text{(2)}
  \]

- The cyclist has a remaining distance of \( 40 - 2y \) to Aurangabad, and the time taken for the cyclist is:
  \[
  \text{Time for cyclist} = \frac{40 - 2y}{y} \quad \text{(3)}
  \]

### Step 3: Set up time equation
We know the cyclist reaches 7.5 hours earlier, so:
\[
\frac{40 - 2x}{x} - \frac{40 - 2y}{y} = 7.5 \quad \text{(4)}
\]
We already know from equation (1) that \( x + y = 20 \). This equation can be solved to determine \( x \), the speed of the pedestrian.

Let's now solve these equations.The solution to the system of equations gives us two possible sets of values for the speeds:

1. The speed of the pedestrian \( x = 4 \) km/h and the speed of the cyclist \( y = 16 \) km/h.
2. The second solution gives unrealistic values (negative speed for the cyclist), so we discard it.

Thus, the speed of the pedestrian is **4 km/h**.

Would you like a detailed breakdown of the solution, or do you have any questions? Here are 5 related questions to explore further:

1. What would the speed of the cyclist be if the pedestrian's speed changed?
2. How do relative speeds affect the meeting point in such problems?
3. What happens if the time difference after meeting changes?
4. How can this problem be solved using only algebraic methods without symbolic software?
5. How do such problems apply in real-life scenarios like logistics?

**Tip:** When solving distance-time problems, always start by defining the variables and setting up the equations systematically.

A pedestrian and a cyclist start simultaneously towards each other from Aurangabad and Paithan which are 40 km apart and meet 2 hours after the start. Then they resumed their trips and the cyclist arrives at Aurangabad 7 hours 30 minutes earlier than the pedestrian arrives at Paithan. Which of these could be the speed of the pedestrian?

Learn how to solve a speed-distance-time problem where a cyclist and a pedestrian meet after traveling towards each other. The problem covers algebraic concepts and the use of linear equations to calculate speeds. Ideal for students in Grades 8-10.

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