A shopkeeper buys 4 horse and 8 cow in 10,000 her sells each house at 15% profit and sells each cow at 10% loss if in whole transation he gets Rs 500 as profit. Then find the difference between 1 house and 1 cow.

Options

A. Rs 1500
B. Rs 1800
C. Rs 1300
D. Rs 1000

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Define Variables
Let the cost of each horse be \( H \) and the cost of each cow be \( C \).

### Step 2: Set Up the First Equation
According to the problem, the total cost of 4 horses and 8 cows is 10,000 rupees. We can express this as:
\[ 4H + 8C = 10,000 \tag{1} \]

### Step 3: Calculate Selling Prices
The selling price of each horse sold at a 15% profit is:
\[ \text{Selling Price of each horse} = H + 0.15H = 1.15H \]
The selling price of each cow sold at a 10% loss is:
\[ \text{Selling Price of each cow} = C - 0.10C = 0.90C \]

### Step 4: Set Up the Second Equation
The total selling price for 4 horses and 8 cows, given that the shopkeeper makes a profit of 500 rupees, can be expressed as:
\[ 4(1.15H) + 8(0.90C) - (10,000) = 500 \]
This simplifies to:
\[ 4.6H + 7.2C - 10,000 = 500 \]
Rearranging gives us:
\[ 4.6H + 7.2C = 10,500 \tag{2} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( 4H + 8C = 10,000 \)
2. \( 4.6H + 7.2C = 10,500 \)

We can solve these equations simultaneously. First, let’s simplify Equation (1) by multiplying it by 1.15 to align it with the coefficients in Equation (2):
\[ 1.15(4H + 8C) = 1.15(10,000) \]
This gives:
\[ 4.6H + 9.2C = 11,500 \tag{3} \]

### Step 6: Subtract Equation (2) from Equation (3)
Now we subtract Equation (2) from Equation (3):
\[ (4.6H + 9.2C) - (4.6H + 7.2C) = 11,500 - 10,500 \]
This simplifies to:
\[ 2C = 1,000 \]
Thus, we find:
\[ C = 500 \]

### Step 7: Substitute \( C \) Back to Find \( H \)
Now substitute \( C = 500 \) back into Equation (1):
\[ 4H + 8(500) = 10,000 \]
This simplifies to:
\[ 4H + 4000 = 10,000 \]
Subtracting 4000 from both sides gives:
\[ 4H = 6,000 \]
Thus:
\[ H = 1,500 \]

### Step 8: Find the Difference Between the Cost of One Horse and One Cow
Now we can find the difference:
\[ H - C = 1500 - 500 = 1000 \]

### Final Answer
The difference between the cost of one horse and one cow is:
\[ \boxed{1000} \]

---