Ramesh and Satish started from two towns P and Q and travelled towards each other simultaneously/ They met after 2 h. After the meeting, Ramesh took 3 h less to reach Q than what Satish took to reach P. Find the ratio fo the speeds of Ramesh and Satish

Options

A. `3 : 1 `
B. `2 : 1`
C. `3 : 2`
D. `4 : 3`

Answer

To solve the problem step by step, we will use the information given about Ramesh and Satish's travel towards each other and their speeds.

### Step 1: Define Variables
Let:
- R = speed of Ramesh (in km/h)
- S = speed of Satish (in km/h)

### Step 2: Distance Travelled Before Meeting
Since Ramesh and Satish meet after 2 hours, the distances they travel can be expressed as:
- Distance travelled by Ramesh = 2R (since distance = speed × time)
- Distance travelled by Satish = 2S

### Step 3: Distance After Meeting
After they meet, Ramesh takes 3 hours less than Satish to reach their respective destinations. Let:
- Time taken by Satish to reach P after meeting = T hours
- Therefore, time taken by Ramesh to reach Q after meeting = T - 3 hours

### Step 4: Distance After Meeting
The distances remaining for Ramesh and Satish after they meet are:
- Distance remaining for Ramesh to reach Q = 2S (the distance Satish travelled before meeting)
- Distance remaining for Satish to reach P = 2R (the distance Ramesh travelled before meeting)

### Step 5: Set Up Equations
Using the formula Distance = Speed × Time, we can set up the following equations based on the distances remaining after they meet:
1. For Ramesh:
\[ 2S = R \times (T - 3) \]
2. For Satish:
\[ 2R = S \times T \]

### Step 6: Solve for T
From the second equation, we can express T in terms of R and S:
\[ T = \frac{2R}{S} \]

### Step 7: Substitute T in First Equation
Substituting T in the first equation:
\[ 2S = R \left(\frac{2R}{S} - 3\right) \]
Expanding this gives:
\[ 2S = \frac{2R^2}{S} - 3R \]

### Step 8: Multiply through by S to eliminate the fraction
Multiplying through by S:
\[ 2S^2 = 2R^2 - 3RS \]

### Step 9: Rearranging the Equation
Rearranging gives:
\[ 2S^2 + 3RS - 2R^2 = 0 \]

### Step 10: Factor the Quadratic Equation
This is a quadratic equation in S. We can factor it:
\[ (2S - R)(S + 2R) = 0 \]

### Step 11: Solve for S
Setting each factor to zero gives us:
1. \(2S - R = 0 \Rightarrow S = \frac{R}{2}\)
2. \(S + 2R = 0\) (not valid since speeds cannot be negative)

### Step 12: Find the Ratio of Speeds
From \(S = \frac{R}{2}\), we can find the ratio of Ramesh's speed to Satish's speed:
\[ \frac{R}{S} = \frac{R}{\frac{R}{2}} = 2 \]
Thus, the ratio of the speeds of Ramesh to Satish is:
\[ \text{Ratio of Ramesh to Satish} = 2:1 \]

### Final Answer
The ratio of the speeds of Ramesh and Satish is **2:1**.

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