The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is `315rs` and for a journey of 15km, the charge paid is `465rs`. What are the fixed charges and the charge per kilometre ? How much does a person have to pay for travelling a distance of `32km ` ?

Answer

To solve the problem of taxi charges, we will set up a system of equations based on the information provided. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( x \) = fixed charge (in rupees)
- \( y \) = charge per kilometer (in rupees)

### Step 2: Set Up the Equations
From the information given:
1. For a distance of 10 km, the total charge is 315 rupees:
\[ x + 10y = 315 \quad \text{(Equation 1)} \]
2. For a distance of 15 km, the total charge is 465 rupees:
\[ x + 15y = 465 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
We can solve these equations simultaneously. First, we can express \( x \) from Equation 1:
\[ x = 315 - 10y \quad \text{(Substituting into Equation 2)} \]

Now, substitute this expression for \( x \) into Equation 2:
\[ (315 - 10y) + 15y = 465 \]

### Step 4: Simplify and Solve for \( y \)
Combine like terms:
\[ 315 - 10y + 15y = 465 \]
\[ 315 + 5y = 465 \]

Now, isolate \( y \):
\[ 5y = 465 - 315 \]
\[ 5y = 150 \]
\[ y = \frac{150}{5} = 30 \]

### Step 5: Substitute \( y \) Back to Find \( x \)
Now that we have \( y \), we can find \( x \):
\[ x = 315 - 10 \cdot 30 \]
\[ x = 315 - 300 = 15 \]

### Step 6: Summary of Charges
- Fixed charge \( x = 15 \) rupees
- Charge per kilometer \( y = 30 \) rupees

### Step 7: Calculate Charge for 32 km
To find out how much a person has to pay for traveling a distance of 32 km:
\[ \text{Total charge} = x + 32y \]
\[ \text{Total charge} = 15 + 32 \cdot 30 \]
\[ = 15 + 960 = 975 \]

### Final Answer
The fixed charge is 15 rupees, the charge per kilometer is 30 rupees, and the total charge for traveling 32 km is 975 rupees.

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