Paulson belongs to Town A and Robson belongs to Town B . They start their journey towards each other's town following the same time . They meet somewhere on the way and continue with their journey . After meeting Robison , Paulson takes another 2 hours to reach the destination while Robson takes another 4.5 hours to reach Paulson's town . if Robson travelled at the speed of 30 km/h , find's Paulson's speed in km/h.

Options

A. 45
B. 42.5
C. 47.5
D. 40

Answer

To find Paulson's speed, we can use the relationship between their speeds and the time taken after they meet. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
Paulson and Robson start their journeys at the same time towards each other's towns. After meeting, Paulson takes 2 hours to reach Robson's town, while Robson takes 4.5 hours to reach Paulson's town. We know Robson's speed is 30 km/h, and we need to find Paulson's speed.

### Step 2: Set Up the Relationship
Let:
- \( S_P \) = Paulson's speed (in km/h)
- \( S_R \) = Robson's speed = 30 km/h
- \( T_P \) = Time taken by Paulson after meeting = 2 hours
- \( T_R \) = Time taken by Robson after meeting = 4.5 hours

According to the relationship between their speeds and times after meeting:
\[ \frac{S_R}{S_P} = \frac{T_P}{T_R} \]

### Step 3: Substitute the Known Values
Substituting the known values into the equation:
\[ \frac{30}{S_P} = \frac{2}{4.5} \]

### Step 4: Simplify the Right Side
To simplify \(\frac{2}{4.5}\), we can convert 4.5 into a fraction:
\[ \frac{2}{4.5} = \frac{2}{\frac{9}{2}} = \frac{2 \times 2}{9} = \frac{4}{9} \]
So now we have:
\[ \frac{30}{S_P} = \frac{4}{9} \]

### Step 5: Cross-Multiply
Cross-multiplying gives us:
\[ 30 \times 9 = 4 \times S_P \]
\[ 270 = 4 S_P \]

### Step 6: Solve for \( S_P \)
Now, divide both sides by 4 to find \( S_P \):
\[ S_P = \frac{270}{4} = 67.5 \text{ km/h} \]

### Step 7: Conclusion
Thus, Paulson's speed is \( 67.5 \) km/h.