A starts from a place P to go to a place Q. At the same time B starts from Q to P. If after meeting each other A and B took 4 and 9 hours more respectively to reach their destinations, the ratio of their speeds is

Options

A. `3:2`
B. `5:2`
C. `9:4`
D. `9:13`

Answer

To solve the problem, we need to find the ratio of the speeds of A and B based on the information provided. Here’s a step-by-step breakdown of the solution:

### Step 1: Define Variables
Let the speed of A be \( x \) km/h and the speed of B be \( y \) km/h.

### Step 2: Meeting Point
Assume A and B meet after \( t \) hours. At this point, A has traveled a distance of \( x \cdot t \) and B has traveled a distance of \( y \cdot t \).

### Step 3: Remaining Distances After Meeting
After they meet:
- A takes 4 hours to reach Q, so the remaining distance for A is \( y \cdot t \) (the distance B has covered).
- B takes 9 hours to reach P, so the remaining distance for B is \( x \cdot t \) (the distance A has covered).

### Step 4: Set Up Equations
From the information given:
1. The remaining distance for A can be expressed as:
\[ \text{Remaining distance for A} = y \cdot t = x \cdot 4 \]
Thus, we have:
\[ y \cdot t = 4x \quad \text{(Equation 1)} \]

2. The remaining distance for B can be expressed as:
\[ \text{Remaining distance for B} = x \cdot t = y \cdot 9 \]
Thus, we have:
\[ x \cdot t = 9y \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations
Now we have two equations:
1. \( y \cdot t = 4x \)
2. \( x \cdot t = 9y \)

We can express \( t \) from both equations:
From Equation 1:
\[ t = \frac{4x}{y} \]

From Equation 2:
\[ t = \frac{9y}{x} \]

### Step 6: Set the Two Expressions for \( t \) Equal
Setting the two expressions for \( t \) equal gives:
\[ \frac{4x}{y} = \frac{9y}{x} \]

### Step 7: Cross Multiply
Cross multiplying yields:
\[ 4x^2 = 9y^2 \]

### Step 8: Rearrange to Find the Ratio
Rearranging gives:
\[ \frac{x^2}{y^2} = \frac{9}{4} \]
Taking the square root of both sides:
\[ \frac{x}{y} = \frac{3}{2} \]

### Step 9: Conclusion
Thus, the ratio of the speeds of A and B is:
\[ \text{Speed Ratio} = \frac{x}{y} = 3:2 \]