Two people A and B start from P and Q (distance = D) at the same time towards each other. They meet at a point R, which is at a distance 0.4 D from P. They continue to move to and fro between the two points. Find the distance from point P at which the fourth meeting takes place

Options

A. 0.8 D
B. 0.6 D
C. 0.3 D
D. 0.4 D

Answer

To solve the problem step by step, we can follow this structured approach:

### Step 1: Understand the Initial Conditions
Two people, A and B, start from points P and Q, respectively, which are D distance apart. They meet at point R, which is 0.4D from P.

**Hint:** Identify the distances and initial positions of A and B.

### Step 2: Calculate the Distance from Q to R
Since the total distance between P and Q is D, the distance from Q to R can be calculated as:
\[ \text{Distance from Q to R} = D - 0.4D = 0.6D \]

**Hint:** Use the total distance to find the distance from the other point.

### Step 3: Determine the Ratio of Distances Covered
The distance covered by A is 0.4D and by B is 0.6D. The ratio of distances covered by A and B is:
\[ \text{Ratio of A to B} = \frac{0.4D}{0.6D} = \frac{2}{3} \]

**Hint:** Simplify the distances to find the ratio.

### Step 4: Relate Distance to Speed
Since A and B start at the same time and meet at R, the ratio of the distances they cover is equal to the ratio of their speeds. Therefore:
\[ \text{Speed of A : Speed of B} = 2 : 3 \]

**Hint:** Remember that time is constant when they meet.

### Step 5: Calculate Total Distance Covered by Both Until the Fourth Meeting
When two people move towards each other, the total distance they cover until the nth meeting is given by:
\[ \text{Total Distance} = (2n - 1)D \]
For the fourth meeting (n=4):
\[ \text{Total Distance} = (2 \times 4 - 1)D = 7D \]

**Hint:** Use the formula for total distance covered to find the distance for the nth meeting.

### Step 6: Calculate the Distance Covered by A
Using the ratio of their speeds, we can find the distance covered by A when they meet for the fourth time:
\[ \text{Distance covered by A} = \frac{2}{5} \times 7D = \frac{14}{5}D = 2.8D \]

**Hint:** Apply the ratio of speeds to the total distance to find the distance covered by A.

### Step 7: Calculate the Distance from P at the Fourth Meeting
The distance from point P at which the fourth meeting occurs is:
\[ \text{Distance from P} = 0.4D + (2.8D - 0.4D) = 2.8D \]

However, since we need to find the distance from P, we can also consider the total distance covered:
\[ \text{Distance from P} = 0.8D \]

**Hint:** Ensure to account for the initial distance and the additional distance covered.

### Conclusion
The fourth meeting takes place at a distance of **0.8D from point P**.

### Final Answer
The distance from point P at which the fourth meeting takes place is **0.8D**.