Two cities A and B are at a distance of 60 km from each other . Two persons P and Q start from First city at a speed of 10km/hr and 5km/hr respectively . P reached the second city B and returns back and meets Q at Y . Find the distance between A and Y .

Options

A. 30 km
B. 40 km
C. 50 km
D. 55km

Explanation

To solve the problem step by step, we will follow the logical sequence of events and calculations based on the information provided.

### Step 1: Understand the Problem
We have two cities, A and B, which are 60 km apart. Two persons, P and Q, start from city A towards city B at speeds of 10 km/hr and 5 km/hr, respectively. P reaches city B, returns, and meets Q at point Y. We need to find the distance between city A and point Y.

### Step 2: Calculate the Time Taken by P to Reach B
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For P:
- Distance from A to B = 60 km
- Speed of P = 10 km/hr

Calculating the time taken by P to reach B:
\[ T_P = \frac{60 \text{ km}}{10 \text{ km/hr}} = 6 \text{ hours} \]

### Step 3: Calculate the Distance Covered by Q in the Same Time
In the same time (6 hours), we calculate how far Q travels:
- Speed of Q = 5 km/hr

Distance covered by Q:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 5 \text{ km/hr} \times 6 \text{ hours} = 30 \text{ km} \]

### Step 4: Determine the Position of Q When P Returns
When P reaches B and starts to return, Q has already traveled 30 km towards B. Therefore, the distance from Q to B is:
\[ \text{Distance from Q to B} = 60 \text{ km} - 30 \text{ km} = 30 \text{ km} \]

### Step 5: Calculate the Time Taken by P to Return to Y
Let the distance from B to Y be \(d\). Since the total distance from A to B is 60 km, the distance from A to Y will be \(60 - d\).

The time taken by P to return to Y is:
\[ T_{PY} = \frac{d}{10} \]

### Step 6: Calculate the Time Taken by Q to Reach Y
The distance from A to Y is \(60 - d\). The time taken by Q to reach Y is:
\[ T_{QY} = \frac{60 - d}{5} \]

### Step 7: Set the Times Equal
Since both P and Q meet at Y, the time taken by P to return to Y equals the time taken by Q to reach Y:
\[ \frac{d}{10} = \frac{60 - d}{5} \]

### Step 8: Solve the Equation
Cross-multiplying gives:
\[ 5d = 10(60 - d) \]
\[ 5d = 600 - 10d \]
Combining like terms:
\[ 15d = 600 \]
\[ d = 40 \text{ km} \]

### Step 9: Calculate the Distance from A to Y
Now that we have \(d\), we can find the distance from A to Y:
\[ \text{Distance from A to Y} = 60 - d = 60 - 40 = 20 \text{ km} \]

### Final Answer
The distance between city A and point Y is **20 km**.