A lives at P and B lives at Q. A usually goes to meet B at Q. He covers the distance in 3 hours at 150 km/h. On a particular day B started moving away from Q and A took total 5 hour to meet B at C. (i) What is the speed of B?
(ii) What is the ratio of speeds of A : B ?

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Calculate the distance between P and Q
A usually covers the distance from P to Q in 3 hours at a speed of 150 km/h.

**Formula:**
Distance = Speed × Time

**Calculation:**
Distance = 150 km/h × 3 hours = 450 km

### Step 2: Understand the scenario when B is moving away
On a particular day, B starts moving away from Q, and A takes a total of 5 hours to meet B at point C.

### Step 3: Set up the equation for relative speed
Let the speed of B be x km/h. Since A is moving at 150 km/h and both are moving in the same direction, the relative speed of A with respect to B will be:

Relative Speed = Speed of A - Speed of B = 150 km/h - x km/h

### Step 4: Use the time taken to meet B
Since A takes 5 hours to meet B, we can set up the equation using the distance covered:

**Equation:**
Relative Speed × Time = Distance
(150 - x) × 5 = 450

### Step 5: Solve for x
Now we can solve the equation:

1. Expand the equation:
5(150 - x) = 450

2. Distribute:
750 - 5x = 450

3. Rearrange to find x:
750 - 450 = 5x
300 = 5x
x = 300 / 5
x = 60 km/h

### Step 6: Conclusion for part (i)
The speed of B is 60 km/h.

### Step 7: Calculate the ratio of speeds A : B
Now we need to find the ratio of the speeds of A and B.

**Speeds:**
Speed of A = 150 km/h
Speed of B = 60 km/h

**Ratio:**
A : B = 150 : 60

### Step 8: Simplify the ratio
To simplify the ratio, divide both sides by 30:

150 ÷ 30 : 60 ÷ 30 = 5 : 2

### Step 9: Conclusion for part (ii)
The ratio of speeds of A to B is 5 : 2.

---