There are two places X and Y, 200 km apart from each other. Initially two persons P and Q both are at 'X'. The speed of P is 20 km/h and speed of Q is 30 km/h. Later on they starts to move to and fro between X and Y.
If they meet third time each other at a point C, somewhere between X and Y, then the ratio of distances CX and CY is :

Options

A. `3:2`
B. `1:3`
C. `2:3`
D. `2:5`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have two persons, P and Q, starting from point X and moving towards point Y, which is 200 km apart. P moves at a speed of 20 km/h, and Q moves at a speed of 30 km/h. We need to find the ratio of distances CX and CY when they meet for the third time at point C.

### Step 2: Determine the speeds and ratios
The speeds of P and Q are:
- Speed of P = 20 km/h
- Speed of Q = 30 km/h

The ratio of their speeds is:
\[ \text{Speed ratio of P to Q} = \frac{20}{30} = \frac{2}{3} \]

### Step 3: Calculate the time taken for the first meeting
When both start from X, they are moving towards Y. The relative speed when they are moving towards each other is:
\[ \text{Relative speed} = 20 + 30 = 50 \text{ km/h} \]

The distance between them is 200 km. The time taken to meet for the first time is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{200 \text{ km}}{50 \text{ km/h}} = 4 \text{ hours} \]

### Step 4: Calculate the distances covered by P and Q in that time
In 4 hours:
- Distance covered by P = Speed of P × Time = \( 20 \text{ km/h} \times 4 \text{ hours} = 80 \text{ km} \)
- Distance covered by Q = Speed of Q × Time = \( 30 \text{ km/h} \times 4 \text{ hours} = 120 \text{ km} \)

### Step 5: Determine the position of the first meeting
At the first meeting point (let's call it N):
- CX = 80 km
- CY = 120 km

### Step 6: Calculate the time taken for the second meeting
After meeting, P will continue to Y, and Q will return to X. The time taken for them to meet again will be the same as the time taken to go to Y and return to X:
- Total distance for both to meet again = 200 km (to Y) + 200 km (back to X) = 400 km
- Time for second meeting = \( \frac{400 \text{ km}}{50 \text{ km/h}} = 8 \text{ hours} \)

### Step 7: Calculate the distances covered by P and Q in that time
In 8 hours:
- Distance covered by P = \( 20 \text{ km/h} \times 8 \text{ hours} = 160 \text{ km} \)
- Distance covered by Q = \( 30 \text{ km/h} \times 8 \text{ hours} = 240 \text{ km} \)

### Step 8: Determine the position of the second meeting
After the second meeting, they will again move towards each other. The total distance they cover will be 400 km, and they will meet again after another 8 hours.

### Step 9: Calculate the time taken for the third meeting
The time taken for the third meeting is the same as for the second meeting:
- Time for third meeting = 8 hours

### Step 10: Calculate the distances covered by P and Q in that time
In 8 hours:
- Distance covered by P = \( 20 \text{ km/h} \times 8 \text{ hours} = 160 \text{ km} \)
- Distance covered by Q = \( 30 \text{ km/h} \times 8 \text{ hours} = 240 \text{ km} \)

### Step 11: Calculate total distances covered by P and Q
Total distance covered by P in 24 hours (4 + 8 + 8 = 20 hours):
- Total distance by P = \( 20 \text{ km/h} \times 24 \text{ hours} = 480 \text{ km} \)

### Step 12: Determine distances CX and CY at the third meeting
From the total distance:
- CX = 80 km (from first meeting) + 160 km (from second meeting) + 80 km (from third meeting) = 320 km
- CY = 200 km - CX = 200 km - 80 km = 120 km

### Step 13: Find the ratio of CX and CY
The ratio of distances CX and CY is:
\[ \text{Ratio} = \frac{CX}{CY} = \frac{80}{120} = \frac{2}{3} \]

### Final Answer
The ratio of distances CX and CY is \( 2:3 \).

---