If I sell a horse for Rs 6200 and a cow for Rs 2600 i will earn 10% profit. If I sell a horse for Rs 6000 and a cow at its C.P. and earn `12(1)/(2)%` profit. Find C.P. of each?

Answer

To solve the problem step by step, we will first establish the relationships between the selling prices (SP) and the cost prices (CP) based on the given profit percentages.

### Step 1: Establish the equations from the first scenario
We know:
- Selling price of horse (SP_H) = Rs 6200
- Selling price of cow (SP_C) = Rs 2600
- Total selling price = SP_H + SP_C = 6200 + 2600 = Rs 8800

Given that this results in a 10% profit, we can express the relationship between CP and SP:
- Let the cost price of the horse be CP_H and the cost price of the cow be CP_C.
- The profit percentage formula is given by:
\[ \text{Profit} = \text{Selling Price} - \text{Cost Price} \]
Therefore, for a 10% profit:
\[ \text{Total CP} = \text{Total SP} - \text{Profit} \]
\[ \text{Profit} = 10\% \text{ of Total CP} = 0.1 \times (CP_H + CP_C) \]
Thus, we can write:
\[ 8800 = (CP_H + CP_C) + 0.1 \times (CP_H + CP_C) \]
This simplifies to:
\[ 8800 = 1.1 \times (CP_H + CP_C) \]
Therefore:
\[ CP_H + CP_C = \frac{8800}{1.1} = 8000 \]

### Step 2: Establish the equations from the second scenario
In the second scenario:
- Selling price of horse (SP_H) = Rs 6000
- Selling price of cow = Cost Price of cow (CP_C)
- Total selling price = SP_H + CP_C = 6000 + CP_C

Given that this results in a 12.5% profit (which is 12.5% = 12.5/100 = 0.125), we can express the relationship:
\[ \text{Total CP} = \text{Total SP} - \text{Profit} \]
The profit can also be expressed as:
\[ \text{Profit} = 12.5\% \text{ of Total CP} = 0.125 \times (CP_H + CP_C) \]
Thus:
\[ 6000 + CP_C = (CP_H + CP_C) + 0.125 \times (CP_H + CP_C) \]
This simplifies to:
\[ 6000 + CP_C = 1.125 \times (CP_H + CP_C) \]

### Step 3: Solve the equations
Now we have two equations:
1. \( CP_H + CP_C = 8000 \) (Equation 1)
2. \( 6000 + CP_C = 1.125 \times (CP_H + CP_C) \) (Equation 2)

Substituting Equation 1 into Equation 2:
\[ 6000 + CP_C = 1.125 \times 8000 \]
Calculating \( 1.125 \times 8000 \):
\[ 1.125 \times 8000 = 9000 \]
Now we have:
\[ 6000 + CP_C = 9000 \]
Solving for \( CP_C \):
\[ CP_C = 9000 - 6000 = 3000 \]

### Step 4: Find CP_H using Equation 1
Now substituting \( CP_C = 3000 \) back into Equation 1:
\[ CP_H + 3000 = 8000 \]
Thus:
\[ CP_H = 8000 - 3000 = 5000 \]

### Final Result
The cost prices are:
- Cost Price of Horse (CP_H) = Rs 5000
- Cost Price of Cow (CP_C) = Rs 3000