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, we will set up equations based on the information given in the question.

### Step 1: Define Variables
Let:
- The cost price of one bag of rice = Rs. X
- The cost price of one bag of wheat = Rs. Y

### Step 2: Set Up the First Equation
According to the problem, the trader purchases 4 bags of rice and 10 bags of wheat for Rs. 3600. This can be represented as:
\[ 4X + 10Y = 3600 \]
We can simplify this equation by dividing everything by 2:
\[ 2X + 5Y = 1800 \]
This is our first equation (Equation 1).

### Step 3: Set Up the Second Equation
The trader sells rice at a 10% gain and wheat at a 2% loss. The total gain from these transactions is Rs. 120.

- The selling price of rice (10% gain) for 4 bags:
\[ \text{Selling Price of Rice} = 4X + 0.1 \times 4X = 4X + 0.4X = 4.4X \]

- The selling price of wheat (2% loss) for 10 bags:
\[ \text{Selling Price of Wheat} = 10Y - 0.02 \times 10Y = 10Y - 0.2Y = 9.8Y \]

The total selling price can be expressed as:
\[ 4.4X + 9.8Y \]
The total gain is Rs. 120, so we can write:
\[ (4.4X + 9.8Y) - (4X + 10Y) = 120 \]
This simplifies to:
\[ 0.4X - 0.2Y = 120 \]
Multiplying through by 10 to eliminate decimals gives us:
\[ 4X - 2Y = 1200 \]
This is our second equation (Equation 2).

### Step 4: Solve the System of Equations
Now we have the following system of equations:
1. \( 2X + 5Y = 1800 \) (Equation 1)
2. \( 4X - 2Y = 1200 \) (Equation 2)

We can solve these equations using substitution or elimination. Let's use substitution.

From Equation 1, we can express Y in terms of X:
\[ 5Y = 1800 - 2X \]
\[ Y = \frac{1800 - 2X}{5} \]

Now, substitute this expression for Y into Equation 2:
\[ 4X - 2\left(\frac{1800 - 2X}{5}\right) = 1200 \]
Multiply through by 5 to eliminate the fraction:
\[ 20X - 2(1800 - 2X) = 6000 \]
Distributing the -2:
\[ 20X - 3600 + 4X = 6000 \]
Combine like terms:
\[ 24X - 3600 = 6000 \]
Add 3600 to both sides:
\[ 24X = 9600 \]
Divide by 24:
\[ X = 400 \]

### Step 5: Find Y
Now that we have X, we can find Y using the expression we derived:
\[ Y = \frac{1800 - 2(400)}{5} \]
\[ Y = \frac{1800 - 800}{5} \]
\[ Y = \frac{1000}{5} = 200 \]

### Step 6: Conclusion
The cost price of one bag of rice (X) is Rs. 400, and the cost price of one bag of wheat (Y) is Rs. 200.

### Final Answer:
- Cost of one bag of rice = Rs. 400
- Cost of one bag of wheat = Rs. 200