On selling a television at a 6% profit and a frtidge at a 9% profit, a shopkeeper gains Rs. 5,100. However, if the shopkeeper sells the television at a 9% profit and the fridge at a 6% loss, be gains Rs 1,800 on the whole transaction. The cost price (in Rs) of the fridge is:

Options

A. 35000
B. 37500
C. 30000
D. 40000

Answer

To solve the problem step by step, we will define variables for the cost prices of the television and the fridge, set up equations based on the information provided, and then solve those equations to find the cost price of the fridge.

### Step 1: Define Variables
Let:
- \( X \) = Cost price of the television
- \( Y \) = Cost price of the fridge

### Step 2: Set Up the First Equation
According to the problem, when the shopkeeper sells the television at a 6% profit and the fridge at a 9% profit, he gains Rs. 5,100. This can be expressed as:
\[ 0.06X + 0.09Y = 5100 \]
Multiplying through by 100 to eliminate the decimals gives:
\[ 6X + 9Y = 510000 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
In the second scenario, the television is sold at a 9% profit and the fridge at a 6% loss, resulting in a gain of Rs. 1,800. This can be expressed as:
\[ 0.09X - 0.06Y = 1800 \]
Again, multiplying through by 100 gives:
\[ 9X - 6Y = 180000 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( 6X + 9Y = 510000 \)
2. \( 9X - 6Y = 180000 \)

We can solve these equations using the elimination or substitution method. Let's use the elimination method.

First, we can multiply Equation 1 by 2 to align the coefficients of \( Y \):
\[ 12X + 18Y = 1020000 \quad \text{(Equation 3)} \]

Now, multiply Equation 2 by 3:
\[ 27X - 18Y = 540000 \quad \text{(Equation 4)} \]

### Step 5: Add the Equations
Now, we can add Equations 3 and 4:
\[ (12X + 18Y) + (27X - 18Y) = 1020000 + 540000 \]
This simplifies to:
\[ 39X = 1560000 \]
Dividing both sides by 39 gives:
\[ X = 40000 \]

### Step 6: Substitute to Find \( Y \)
Now that we have \( X \), we can substitute it back into one of the original equations to find \( Y \). Using Equation 1:
\[ 6(40000) + 9Y = 510000 \]
This simplifies to:
\[ 240000 + 9Y = 510000 \]
Subtracting 240000 from both sides gives:
\[ 9Y = 270000 \]
Dividing both sides by 9 gives:
\[ Y = 30000 \]

### Conclusion
The cost price of the fridge is Rs. 30,000.