A man bought a buffalo and a cow for Rs. `2600`. He sold the buffalo at a profit of` 25%` and the cow at a profit of `20%`. In this way, his total profit was .`23 1/13 %` Find the cost price of the buffalo and cow.

Answer

To solve the problem step by step, we will follow the given information and apply the necessary calculations.

### Step 1: Define Variables
Let the cost price of the buffalo be \( X \) and the cost price of the cow be \( Y \).

### Step 2: Set Up the Equation
According to the problem, the total cost price of the buffalo and cow is:
\[ X + Y = 2600 \quad \text{(Equation 1)} \]

### Step 3: Calculate Selling Prices
The buffalo is sold at a profit of 25%, so the selling price of the buffalo is:
\[ \text{Selling Price of Buffalo} = X + 0.25X = 1.25X \]

The cow is sold at a profit of 20%, so the selling price of the cow is:
\[ \text{Selling Price of Cow} = Y + 0.20Y = 1.20Y \]

### Step 4: Calculate Total Selling Price
The total selling price of both animals is:
\[ \text{Total Selling Price} = 1.25X + 1.20Y \]

### Step 5: Calculate Total Profit
The total profit percentage is given as \( 23 \frac{1}{13} \% \). We can convert this mixed fraction into an improper fraction:
\[ 23 \frac{1}{13} = \frac{23 \times 13 + 1}{13} = \frac{300}{13} \]

The total profit can also be expressed as:
\[ \text{Total Profit} = \text{Total Selling Price} - \text{Total Cost Price} = (1.25X + 1.20Y) - 2600 \]

### Step 6: Set Up the Profit Equation
The total profit can also be calculated using the profit percentage:
\[ \text{Total Profit} = \frac{300}{13} \% \text{ of } 2600 = \frac{300}{13} \times \frac{2600}{100} = \frac{300 \times 26}{13} = 600 \]

### Step 7: Equate Total Profit
Now we can set up the equation:
\[ (1.25X + 1.20Y) - 2600 = 600 \]
Simplifying this gives:
\[ 1.25X + 1.20Y = 3200 \quad \text{(Equation 2)} \]

### Step 8: Solve the System of Equations
We now have two equations:
1. \( X + Y = 2600 \)
2. \( 1.25X + 1.20Y = 3200 \)

From Equation 1, we can express \( Y \) in terms of \( X \):
\[ Y = 2600 - X \]

Substituting \( Y \) in Equation 2:
\[ 1.25X + 1.20(2600 - X) = 3200 \]
Expanding this gives:
\[ 1.25X + 3120 - 1.20X = 3200 \]
Combining like terms:
\[ 0.05X + 3120 = 3200 \]
Subtracting 3120 from both sides:
\[ 0.05X = 80 \]
Dividing by 0.05:
\[ X = \frac{80}{0.05} = 1600 \]

### Step 9: Find Y
Substituting \( X \) back into Equation 1 to find \( Y \):
\[ Y = 2600 - 1600 = 1000 \]

### Final Result
The cost price of the buffalo is Rs. 1600 and the cost price of the cow is Rs. 1000.