The total railway fare for 5 members in 3-tier and 3 members in 2-tier is Rs 2,050 whereas, the total railway fare for 8 members in 3-tier and 5 members in 2-tier is Rs 3,350. Find the fare to be paid by a couple travelling through 2-tier.

Answer

To solve the problem step by step, we will define the variables, set up the equations, solve them, and then find the fare for a couple traveling in 2-tier.

### Step 1: Define the Variables
Let:
- \( x \) = fare for one member traveling in 3-tier
- \( y \) = fare for one member traveling in 2-tier

### Step 2: Set Up the Equations
From the problem statement, we can form two equations based on the given information:

1. For 5 members in 3-tier and 3 members in 2-tier:
\[ 5x + 3y = 2050 \quad \text{(Equation 1)} \]

2. For 8 members in 3-tier and 5 members in 2-tier:
\[ 8x + 5y = 3350 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
To eliminate one of the variables, we can multiply both equations by suitable numbers so that the coefficients of \( y \) are the same.

- Multiply Equation 1 by 5:
\[ 25x + 15y = 10250 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 3:
\[ 24x + 15y = 10050 \quad \text{(Equation 4)} \]

### Step 4: Subtract the Equations
Now, we can subtract Equation 4 from Equation 3 to eliminate \( y \):
\[ (25x + 15y) - (24x + 15y) = 10250 - 10050 \]
This simplifies to:
\[ x = 200 \]

### Step 5: Substitute to Find \( y \)
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We will use Equation 1:
\[ 5(200) + 3y = 2050 \]
This simplifies to:
\[ 1000 + 3y = 2050 \]
Now, isolate \( 3y \):
\[ 3y = 2050 - 1000 \]
\[ 3y = 1050 \]
Now, divide by 3:
\[ y = 350 \]

### Step 6: Calculate the Fare for a Couple in 2-tier
The fare for a couple traveling in 2-tier is:
\[ 2y = 2(350) = 700 \]

### Final Answer
The fare to be paid by a couple traveling through 2-tier is **Rs 700**.

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