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Let's break down and solve the given problem step by step.

### Problem Summary:
- **Abhi and Gita** start from points **P** and **Q**, which are 81 km apart.
- Abhi walks at a constant speed of **4 km/h**.
- Gita starts at the same speed (**4 km/h**) but increases her speed by **0.25 km/h** at the end of each hour.

We need to find out how many hours will Abhi take to reach point **Q** after meeting Gita.

### Step-by-Step Solution:

Let:
- **\( t \)** be the time in hours when Abhi and Gita meet.
- **\( d \)** be the total distance between them (81 km).

1. **Abhi's distance**: Since Abhi walks at a constant speed of 4 km/h, the distance he covers in **\( t \)** hours is:

   \[
   \text{Distance by Abhi} = 4 \cdot t
   \]

2. **Gita's distance**: Gita's speed increases every hour. Let's calculate her total distance after **\( t \)** hours.
   - In the first hour, her speed is **4 km/h**.
   - In the second hour, her speed is **4.25 km/h**.
   - In the third hour, her speed is **4.5 km/h**, and so on.

   Therefore, the distance Gita covers can be written as the sum of an arithmetic progression where:
   - First term (\(a_1\)) = 4 km/h,
   - Common difference (\(d\)) = 0.25 km/h,
   - Number of terms = \(t\).

   The distance Gita covers in **\( t \)** hours is the sum of the first **\( t \)** terms of the arithmetic progression:

   \[
   \text{Distance by Gita} = t \cdot \left(4 + \frac{(t-1) \cdot 0.25}{2}\right)
   \]

3. **Total distance** covered by both Abhi and Gita must add up to 81 km:

   \[
   \text{Distance by Abhi} + \text{Distance by Gita} = 81
   \]

   Substituting the expressions for their distances, we get:

   \[
   4t + t \cdot \left(4 + \frac{(t-1) \cdot 0.25}{2}\right) = 81
   \]

   Now, let's solve this equation for **\( t \)**. I'll calculate the value of \( t \).The solution shows that Abhi and Gita meet after **9 hours**.

### Next, we need to calculate the total time Abhi takes to reach point **Q** after meeting Gita.

1. **Abhi's total time to reach Q**: After meeting Gita, Abhi has already walked for 9 hours. Since the total distance from **P to Q** is 81 km, and Abhi walks at 4 km/h, his total time to cover the entire distance is:

\[
\text{Total time} = \frac{81}{4} = 20.25 \text{ hours}
\]

Since Abhi has already walked for 9 hours, the remaining time after meeting Gita is:

\[
\text{Remaining time} = 20.25 - 9 = 11.25 \text{ hours}
\]

Thus, it will take Abhi **11.25 hours** to reach point **Q** after meeting Gita.

### Final Answer: The correct option is **(2) 11.25 hours**.

---

Would you like further details or have any questions? Here are 5 related questions:

1. How would the solution change if Gita increased her speed by 0.5 km/h every hour?
2. What if Abhi’s speed was reduced to 3 km/h?
3. Can this problem be solved using graphs? How would you plot the speeds?
4. What would happen if the total distance was doubled but speeds stayed the same?
5. How would the time change if Gita increased her speed every 30 minutes instead of every hour?

**Tip:** When solving problems with varying speeds, arithmetic progressions are a useful tool to sum up distances effectively.

Abhi and Gita set out at the same time respectively from points P and Q, 81 kms apart towards Q and P, respectively. Abhi walks uniformly at the rate of 4 km/h for his entire journey. Gita also starts walking at the speed of 4 km/h but she increases her speed by 0.25 km/h at the end of every hour. How many hours will Abhi take to reach point Q after meeting Gita?

Solve a complex journey problem involving Abhi and Gita walking towards each other from points P and Q. Learn how to use algebra and arithmetic progressions to find how long Abhi will take to reach point Q after meeting Gita. Suitable for students in Grades 9-11.

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