Let's solve the problem step by step:

1. Initial Setup: Abhi and Gita start from points P and Q respectively, and these points are 81 km apart. Both start at the same time towards each other.

2. Abhi's Speed: Abhi walks at a constant speed of 4 km/h.

3. Gita's Speed: Gita starts walking at 4 km/h, and she increases her speed by 0.25 km/h at the end of each hour.

Step 1: Find the time and position of meeting.

We will calculate when and where they meet.

• Let $t$ be the time (in hours) at which they meet.

• Abhi's distance covered by the time of meeting is $4t$ km.

• Gita's speed changes each hour:

  • 1st hour: 4 km/h
  • 2nd hour: 4.25 km/h
  • 3rd hour: 4.5 km/h
  • And so on...

• Gita's distance covered in time $t$ is the sum of the series of distances covered per hour.

Gita's distance after $t$ hours is given by: $\text{Distance} = 4 + (4 + 0.25) + (4 + 2 \times 0.25) + \ldots + (4 + (t-1) \times 0.25)$

Simplifying, this is: $\text{Distance} = t \cdot 4 + 0.25 \cdot (0 + 1 + 2 + \ldots + (t-1))$

The sum of the series $0 + 1 + 2 + \ldots + (t-1)$ is $\frac{(t-1) \cdot t}{2}$. So, Gita's distance becomes: $\text{Distance} = 4t + 0.25 \cdot \frac{(t-1) \cdot t}{2}$

Now, since they meet when their combined distances are 81 km: $4t + \left(4t + 0.25 \cdot \frac{(t-1) \cdot t}{2}\right) = 81$

Simplifying this equation: $8t + 0.125 \cdot (t-1) \cdot t = 81$

Let's solve this equation to find $t$.The possible values for the time $t$ when they meet are $-72$ hours and $9$ hours. Since time cannot be negative, the valid solution is $t = 9$ hours.

Step 2: Find the time for Abhi to reach point Q after meeting Gita.

Since Abhi meets Gita after 9 hours, he still needs to cover the remaining distance to point Q:

• Abhi's total journey from P to Q is 81 km.
• By the time they meet, Abhi has already covered $4 \times 9 = 36$ km.
• The remaining distance for Abhi is $81 - 36 = 45$ km.

Abhi walks at a constant speed of 4 km/h. So, the time required for Abhi to cover the remaining 45 km is:

$\text{Time} = \frac{45 \text{ km}}{4 \text{ km/h}} = 11.25 \text{ hours}$

Therefore, after meeting Gita, Abhi will take 11.25 hours to reach point Q.

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Would you like more details or have any questions?

Here are some related questions:

1. How far has Gita walked by the time they meet?
2. What is Gita's speed at the moment they meet?
3. How much distance does Gita cover in the first 5 hours?
4. How long would it take Gita to reach point P?
5. What if Gita increased her speed by 0.5 km/h instead of 0.25 km/h each hour?

Tip: For problems involving variable speeds, breaking down the motion into intervals and summing distances covered in each interval is key to finding solutions accurately.