A mobile phone-based taxi service a base fee of $2 plus an amount per mile for the trip. Ariel is charged $16.94 for a ride that takes 14 minutes and travels 10 miles. Victoria is charged $11.30 for ride that take 10 minutes and travels 6 miles.
Q. What is the per-minute charge?

Answer

To solve the problem step by step, we need to establish equations based on the information provided about the taxi service charges.

### Step 1: Define Variables
Let:
- \( X \) = charge per minute (in dollars)
- \( Y \) = charge per mile (in dollars)

### Step 2: Set Up Equations
From the information given:
1. **Ariel's Charges**:
- Total charge = $16.94
- Ride time = 14 minutes
- Distance = 10 miles

The equation for Ariel can be set up as:
\[ 16.94 = 2 + 14X + 10Y \]
Simplifying this gives:
\[ 14X + 10Y = 16.94 - 2 \]
\[ 14X + 10Y = 14.94 \quad \text{(Equation 1)} \]

2. **Victoria's Charges**:
- Total charge = $11.30
- Ride time = 10 minutes
- Distance = 6 miles

The equation for Victoria can be set up as:
\[ 11.30 = 2 + 10X + 6Y \]
Simplifying this gives:
\[ 10X + 6Y = 11.30 - 2 \]
\[ 10X + 6Y = 9.30 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
Now we have two equations:
1. \( 14X + 10Y = 14.94 \)
2. \( 10X + 6Y = 9.30 \)

To eliminate \( Y \), we can manipulate these equations. We can multiply Equation 1 by 3 and Equation 2 by 5 to align the coefficients of \( Y \):

- Multiply Equation 1 by 3:
\[ 42X + 30Y = 44.82 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 5:
\[ 50X + 30Y = 46.50 \quad \text{(Equation 4)} \]

### Step 4: Subtract the Equations
Now, we subtract Equation 3 from Equation 4:
\[ (50X + 30Y) - (42X + 30Y) = 46.50 - 44.82 \]
This simplifies to:
\[ 8X = 1.68 \]
Now, solving for \( X \):
\[ X = \frac{1.68}{8} = 0.21 \]

### Step 5: Conclusion
The per-minute charge \( X \) is $0.21.