A certain amount is charged for 1 km for a scooter fare and lesser amount for each additional km. If one day Aliya paid Rs. 24 for 5 km and on another day she paid Rs. 35.40 for 8 km. Find the fare for the `1^(st)` km and for the additional kms.

Answer

To solve the problem, we need to set up equations based on the information given about Aliya's scooter fares.

### Step 1: Define Variables
Let:
- \( x \) = fare for the first kilometer (in Rs.)
- \( y \) = fare for each additional kilometer (in Rs.)

### Step 2: Set Up Equations
From the problem, we have two scenarios:

1. For 5 km, Aliya paid Rs. 24:
- The fare for the first kilometer is \( x \).
- The fare for the additional 4 kilometers is \( 4y \).
- Therefore, we can write the equation:
\[ x + 4y = 24 \quad \text{(Equation 1)} \]

2. For 8 km, Aliya paid Rs. 35.40:
- The fare for the first kilometer is \( x \).
- The fare for the additional 7 kilometers is \( 7y \).
- Therefore, we can write the equation:
\[ x + 7y = 35.40 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
Now, we will solve these two equations simultaneously.

**Subtract Equation 1 from Equation 2:**
\[ (x + 7y) - (x + 4y) = 35.40 - 24 \]
This simplifies to:
\[ 3y = 11.40 \]

**Step 4: Solve for \( y \)**
Now, divide both sides by 3:
\[ y = \frac{11.40}{3} = 3.80 \]

### Step 5: Substitute \( y \) back into Equation 1
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[ x + 4(3.80) = 24 \]
This simplifies to:
\[ x + 15.20 = 24 \]

### Step 6: Solve for \( x \)
Now, subtract 15.20 from both sides:
\[ x = 24 - 15.20 = 8.80 \]

### Final Answer
The fare for the first kilometer is Rs. 8.80, and the fare for each additional kilometer is Rs. 3.80.

### Summary
- Fare for the first kilometer (\( x \)): Rs. 8.80
- Fare for each additional kilometer (\( y \)): Rs. 3.80

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