A trader purchases 4 bags of rice and 10 bags of wheat for Rs. 3600. He sells rice at 10% gain and wheat at 2% loss and thus he gained Rs. 120. Find the C.P. of one bag of rice and one bag of wheat.

Answer

To solve the problem step by step, let's denote the cost price of one bag of rice as \( x \) and the cost price of one bag of wheat as \( y \).

### Step 1: Set up the equation for the total cost price
The trader purchases 4 bags of rice and 10 bags of wheat for a total of Rs. 3600. Therefore, we can write the equation:

\[ 4x + 10y = 3600 \]

### Step 2: Simplify the equation
To make the equation simpler, we can divide the entire equation by 2:

\[ 2x + 5y = 1800 \quad \text{(Equation 1)} \]

### Step 3: Set up the equation for the selling prices
The trader sells rice at a 10% gain and wheat at a 2% loss. The selling price of rice can be expressed as:

\[ \text{Selling Price of Rice} = x + 0.1x = 1.1x \]

The selling price of wheat can be expressed as:

\[ \text{Selling Price of Wheat} = y - 0.02y = 0.98y \]

### Step 4: Write the equation for total selling price
The total selling price from selling all bags of rice and wheat is:

\[ \text{Total Selling Price} = 4(1.1x) + 10(0.98y) \]

This simplifies to:

\[ 4.4x + 9.8y \]

### Step 5: Set up the equation for the overall gain
The trader gains Rs. 120 from selling the rice and wheat. Therefore, we can write:

\[ \text{Total Selling Price} - \text{Total Cost Price} = 120 \]

Substituting the values we have:

\[ (4.4x + 9.8y) - 3600 = 120 \]

This simplifies to:

\[ 4.4x + 9.8y = 3720 \quad \text{(Equation 2)} \]

### Step 6: Solve the system of equations
Now we have two equations:

1. \( 2x + 5y = 1800 \) (Equation 1)
2. \( 4.4x + 9.8y = 3720 \) (Equation 2)

To solve these equations, we can multiply Equation 1 by 2.2 to align the coefficients of \( x \):

\[ 2.2(2x + 5y) = 2.2(1800) \]
\[ 4.4x + 11y = 3960 \quad \text{(Equation 3)} \]

### Step 7: Subtract Equation 2 from Equation 3
Now, we can subtract Equation 2 from Equation 3:

\[ (4.4x + 11y) - (4.4x + 9.8y) = 3960 - 3720 \]
\[ 1.2y = 240 \]

### Step 8: Solve for \( y \)
Dividing both sides by 1.2 gives:

\[ y = \frac{240}{1.2} = 200 \]

### Step 9: Substitute \( y \) back to find \( x \)
Now we can substitute \( y = 200 \) back into Equation 1:

\[ 2x + 5(200) = 1800 \]
\[ 2x + 1000 = 1800 \]
\[ 2x = 800 \]
\[ x = 400 \]

### Final Answer
The cost price of one bag of rice is Rs. 400 and the cost price of one bag of wheat is Rs. 200.