A man buys 5 horses and 10 cows for Rs 1600. He sells horses at a profit of 15% and cows at a loss of 10%. If his over all profit was Rs 90, what was the cost price of a horse and of a cow?

Options

A. Rs 150, Rs 85
B. Rs 200, Rs 60
C. Rs 220, Rs 40
D. Rs 180, Rs 70

Answer

To solve the problem step by step, we will define variables for the cost prices of the horses and cows, set up equations based on the information provided, and solve for the unknowns.

### Step 1: Define Variables
Let the cost price of one horse be \( x \) (in Rs) and the cost price of one cow be \( y \) (in Rs).

### Step 2: Set Up the First Equation
The man buys 5 horses and 10 cows for Rs 1600. Therefore, we can write the equation:
\[ 5x + 10y = 1600 \]

### Step 3: Simplify the First Equation
We can simplify this equation by dividing all terms by 5:
\[ x + 2y = 320 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
The man sells the horses at a profit of 15% and the cows at a loss of 10%. The profit from selling the horses can be calculated as:
\[ \text{Selling Price of Horses} = 5x + 0.15(5x) = 5x(1 + 0.15) = 5x \times 1.15 = 5.75x \]
The loss from selling the cows can be calculated as:
\[ \text{Selling Price of Cows} = 10y - 0.10(10y) = 10y(1 - 0.10) = 10y \times 0.90 = 9y \]
The overall profit is given as Rs 90, so we can set up the second equation:
\[ 5.75x + 9y - (5x + 10y) = 90 \]
This simplifies to:
\[ 0.75x - y = 90 \quad \text{(Equation 2)} \]

### Step 5: Solve the Equations Simultaneously
Now we have two equations:
1. \( x + 2y = 320 \) (Equation 1)
2. \( 0.75x - y = 90 \) (Equation 2)

From Equation 2, we can express \( y \) in terms of \( x \):
\[ y = 0.75x - 90 \]

### Step 6: Substitute \( y \) in Equation 1
Substituting \( y \) in Equation 1:
\[ x + 2(0.75x - 90) = 320 \]
Expanding this gives:
\[ x + 1.5x - 180 = 320 \]
Combining like terms:
\[ 2.5x - 180 = 320 \]
Adding 180 to both sides:
\[ 2.5x = 500 \]
Dividing by 2.5:
\[ x = 200 \]

### Step 7: Find the Value of \( y \)
Now substituting \( x = 200 \) back into the equation for \( y \):
\[ y = 0.75(200) - 90 = 150 - 90 = 60 \]

### Conclusion
The cost price of one horse is Rs 200 and the cost price of one cow is Rs 60.