Clement and Daniel both drove from Town X to Town Y. Clement started his journey at 10:00 am travelling at an average speed of 75 km/h. Sometime later, Daniel started his journey. At 12:00 noon, Daniel overtook Clement. When Daniel reached Town Y at 2:00pm, Clement was still 50 km from Town Y. Find Daniel's average speed.

Let's define the variables and set up the problem step by step.

### Step 1: Define the variables.
- Let the distance between Town X and Town Y be \( D \) km.
- Clement's average speed is \( 75 \, \text{km/h} \).
- Let Daniel's average speed be \( v_D \, \text{km/h} \).
- Clement starts at 10:00 am, and Daniel overtakes him at 12:00 noon.
- Daniel reaches Town Y at 2:00 pm, while Clement is still 50 km away.

### Step 2: Clement's journey.
Clement starts at 10:00 am and is overtaken by Daniel at 12:00 noon, meaning Clement has been driving for 2 hours by the time Daniel overtakes him.

- In 2 hours, Clement covers a distance of:
  \[
  \text{Distance covered by Clement by 12:00 noon} = 75 \times 2 = 150 \, \text{km}.
  \]

Thus, Clement has driven 150 km by 12:00 noon when Daniel overtakes him.

### Step 3: Daniel's journey.
Daniel overtakes Clement at 12:00 noon and reaches Town Y at 2:00 pm. Therefore, Daniel's total journey from where he overtook Clement to Town Y takes 2 hours.

Let’s calculate the distance remaining for Clement to reach Town Y when Daniel overtakes him.

Since Clement is 50 km away from Town Y at 2:00 pm, we know that by 2:00 pm, Clement must have covered \( D - 50 \, \text{km} \) from the start of the trip.

### Step 4: Determine the time taken by Clement.
Clement's average speed is 75 km/h. The distance covered by Clement by 2:00 pm is \( D - 50 \, \text{km} \).

From 10:00 am to 2:00 pm, Clement has driven for 4 hours. So the total distance Clement covers in 4 hours is:
\[
\text{Distance covered by Clement in 4 hours} = 75 \times 4 = 300 \, \text{km}.
\]
Thus, \( D - 50 = 300 \), which means:
\[
D = 350 \, \text{km}.
\]

### Step 5: Find Daniel's average speed.
Daniel reaches Town Y at 2:00 pm, so the total time Daniel takes from Town X to Town Y is from the time he starts driving to 2:00 pm.

Daniel overtakes Clement at 12:00 noon, and by then, Clement has driven 150 km. Since Daniel also covered 150 km by 12:00 noon, the remaining distance for Daniel to cover is \( 350 - 150 = 200 \, \text{km} \).

Since Daniel takes 2 hours (from 12:00 noon to 2:00 pm) to cover this 200 km, his average speed is:
\[
v_D = \frac{200}{2} = 100 \, \text{km/h}.
\]

### Final Answer:
Daniel's average speed is \( \boxed{100 \, \text{km/h}} \).

Would you like any further clarification or details on this solution? Here are five related questions:

1. How would the result change if Daniel had overtaken Clement at a different time?
2. What if Clement's speed was 80 km/h instead of 75 km/h?
3. How can we verify the distance between Town X and Town Y using Daniel's speed?
4. What would be Daniel's travel time if his average speed were reduced by 10 km/h?
5. How can we calculate Clement's position when Daniel reaches Town Y?

**Tip:** When solving relative speed problems, carefully break down time intervals and distances for each person involved to simplify the solution.

Solve the problem of finding Daniel's average speed as he drives from Town X to Town Y. This detailed solution involves speed, distance, and time concepts, covering relative motion and uniform motion theorems. Suitable for Grades 9-11.

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