In covering a distance of 30 km, Abhay takes 2 hours more than Sameer. If Abhay doubled his speed, then he would have take 1 hour less than Sameer. Find the speed of Abhay and Sameer speed (in km/hr).

Answer

To solve the problem step by step, we will denote the speeds of Abhay and Sameer as \( x \) km/hr and \( y \) km/hr respectively.

### Step 1: Set up the equations based on the given information.
1. **Abhay's time to cover 30 km:**
The time taken by Abhay to cover 30 km is given by:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{30}{x} \]
According to the problem, Abhay takes 2 hours more than Sameer, so:
\[ \frac{30}{x} = \frac{30}{y} + 2 \quad \text{(Equation 1)} \]

2. **Abhay's time if he doubles his speed:**
If Abhay doubles his speed, his new speed is \( 2x \). The time taken by him at this speed is:
\[ \text{Time} = \frac{30}{2x} \]
According to the problem, this time is 1 hour less than Sameer's time:
\[ \frac{30}{2x} = \frac{30}{y} - 1 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations.
1. **From Equation 1:**
\[ \frac{30}{x} - \frac{30}{y} = 2 \]
Multiply through by \( xy \) to eliminate the denominators:
\[ 30y - 30x = 2xy \]
Rearranging gives:
\[ 2xy - 30y + 30x = 0 \quad \text{(Equation 3)} \]

2. **From Equation 2:**
\[ \frac{30}{2x} - \frac{30}{y} = -1 \]
Multiply through by \( 2xy \):
\[ 30y - 60x = -2xy \]
Rearranging gives:
\[ 2xy + 30y - 60x = 0 \quad \text{(Equation 4)} \]

### Step 3: Solve Equations 3 and 4 simultaneously.
1. **From Equation 3:**
\[ 2xy - 30y + 30x = 0 \]
Rearranging gives:
\[ 2xy = 30y - 30x \]

2. **From Equation 4:**
\[ 2xy + 30y - 60x = 0 \]
Rearranging gives:
\[ 2xy = 60x - 30y \]

### Step 4: Set the two expressions for \( 2xy \) equal to each other.
\[ 30y - 30x = 60x - 30y \]
Combine like terms:
\[ 60y = 90x \]
Thus:
\[ y = \frac{3}{2}x \quad \text{(Equation 5)} \]

### Step 5: Substitute Equation 5 back into one of the original equations.
Using Equation 1:
\[ \frac{30}{x} = \frac{30}{\frac{3}{2}x} + 2 \]
This simplifies to:
\[ \frac{30}{x} = \frac{20}{x} + 2 \]
Multiply through by \( x \):
\[ 30 = 20 + 2x \]
Solving for \( x \):
\[ 10 = 2x \implies x = 5 \text{ km/hr} \]

### Step 6: Find \( y \) using Equation 5.
Substituting \( x \) back into Equation 5:
\[ y = \frac{3}{2} \cdot 5 = 7.5 \text{ km/hr} \]

### Final Answer:
- Speed of Abhay: \( 5 \) km/hr
- Speed of Sameer: \( 7.5 \) km/hr