P and Q start from A and B respectively with uniform velocities. P is headed towards B and Q towards A and both cities are 600 km apart. P rests whenever Q is on the move and Q rest whenever P is on the move. The speed of P and Q is 25 km/h and 30 km/ h respectively. If P starts first and reaches B in 36 hours, then find the least time that Q would take to each his destination after P makes a start.

Answer

To solve the problem step by step, we will analyze the movements of P and Q, their speeds, and the distances they cover.

### Step 1: Determine the distance covered by P
P travels from A to B, which is a distance of 600 km. According to the problem, P takes 36 hours to reach B.

**Calculation:**
Distance = Speed × Time
600 km = 25 km/h × Time
Time = 600 km / 25 km/h
Time = 24 hours

### Step 2: Calculate the time Q is moving
Since P rests whenever Q is moving and Q rests whenever P is moving, we need to find out how long Q is moving during the time P is traveling.

P takes 36 hours to reach B, but since P rests for 12 hours (36 hours - 24 hours), Q will be moving for 24 hours (the time P is traveling).

### Step 3: Calculate the distance covered by Q while P is moving
Q's speed is 30 km/h. We can calculate the distance Q covers while P is moving.

**Calculation:**
Distance = Speed × Time
Distance covered by Q = 30 km/h × 24 hours = 720 km

### Step 4: Find the remaining distance for Q to reach A
Initially, Q is at B and needs to reach A, which is 600 km away. However, since Q has already covered 720 km while P was moving, Q has actually passed A.

**Calculation:**
Remaining distance to A = Distance covered by Q - Distance between A and B
Remaining distance = 720 km - 600 km = 120 km

### Step 5: Calculate the time taken by Q to return to A
Now, we need to find out how long it takes Q to cover the remaining 120 km back to A.

**Calculation:**
Time = Distance / Speed
Time = 120 km / 30 km/h = 4 hours

### Final Answer
The least time that Q would take to reach his destination after P makes a start is **4 hours**.

---