A railways half ticket costs half the full fare and the resrvation charge is the same of half ticket as on full ticket. One reserved first class ticket form Mumbai to Delhi cost ₹4150 while one full and one half reserved first class ticket cost ₹6255. What is the basic first class full fare and what is the reservation charges?

Explanation

To solve the problem, we need to set up equations based on the information provided. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( X \) = Basic first class full fare (in ₹)
- \( Y \) = Reservation charges (in ₹)

### Step 2: Write the First Equation
From the information given, one reserved first class ticket costs ₹4150. This ticket includes the full fare and the reservation charge. Therefore, we can write the first equation as:
\[ X + Y = 4150 \tag{1} \]

### Step 3: Write the Second Equation
We also know that the cost of one full ticket and one half ticket (with reservation charges) is ₹6255. The half ticket fare is half of the full fare, which is \( \frac{X}{2} \). Thus, the second equation can be written as:
\[ X + Y + \frac{X}{2} + Y = 6255 \]
This simplifies to:
\[ X + \frac{X}{2} + 2Y = 6255 \]
Combining \( X \) and \( \frac{X}{2} \) gives us:
\[ \frac{3X}{2} + 2Y = 6255 \tag{2} \]

### Step 4: Simplify Equation (2)
To eliminate the fraction, we can multiply the entire equation (2) by 2:
\[ 3X + 4Y = 12510 \tag{3} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( X + Y = 4150 \) (Equation 1)
2. \( 3X + 4Y = 12510 \) (Equation 3)

We will use the elimination method. First, let's multiply Equation (1) by 3:
\[ 3X + 3Y = 12450 \tag{4} \]

### Step 6: Subtract Equation (4) from Equation (3)
Now, we subtract Equation (4) from Equation (3):
\[ (3X + 4Y) - (3X + 3Y) = 12510 - 12450 \]
This simplifies to:
\[ Y = 60 \]

### Step 7: Substitute \( Y \) Back to Find \( X \)
Now that we have \( Y \), we can substitute it back into Equation (1) to find \( X \):
\[ X + 60 = 4150 \]
Subtracting 60 from both sides gives:
\[ X = 4150 - 60 \]
\[ X = 4090 \]

### Final Answer
- The basic first class full fare \( X \) is ₹4090.
- The reservation charges \( Y \) are ₹60.