Two articles A and B are sold for 1,167 making 5% profit on A and 7% profit on B. If the two articles are sold for 1,165, a profit of 7% is made on A and a profit of 5% is made on B. Find the cost price of each article.

Answer

To solve the problem, we need to set up equations based on the information given about the selling prices and profits of articles A and B.

### Step 1: Define Variables
Let the cost price of article A be \( X \) and the cost price of article B be \( Y \).

### Step 2: Set Up the First Equation
According to the first scenario, when articles A and B are sold for 1167, the profit on A is 5% and the profit on B is 7%. Therefore, we can express the selling prices as:
- Selling price of A = \( X + 0.05X = 1.05X \)
- Selling price of B = \( Y + 0.07Y = 1.07Y \)

The total selling price is:
\[ 1.05X + 1.07Y = 1167 \]
Multiplying through by 100 to eliminate decimals gives:
\[ 105X + 107Y = 116700 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
In the second scenario, articles A and B are sold for 1165, with a profit of 7% on A and 5% on B. Thus, we have:
- Selling price of A = \( X + 0.07X = 1.07X \)
- Selling price of B = \( Y + 0.05Y = 1.05Y \)

The total selling price is:
\[ 1.07X + 1.05Y = 1165 \]
Multiplying through by 100 gives:
\[ 107X + 105Y = 116500 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( 105X + 107Y = 116700 \)
2. \( 107X + 105Y = 116500 \)

#### Step 4.1: Add the Equations
Adding both equations:
\[ (105X + 107Y) + (107X + 105Y) = 116700 + 116500 \]
This simplifies to:
\[ 212X + 212Y = 233200 \]
Dividing the entire equation by 212:
\[ X + Y = 1100 \quad \text{(Equation 3)} \]

#### Step 4.2: Subtract the Equations
Now, subtract Equation 1 from Equation 2:
\[ (107X + 105Y) - (105X + 107Y) = 116500 - 116700 \]
This simplifies to:
\[ 2X - 2Y = -200 \]
Dividing the entire equation by 2:
\[ X - Y = -100 \quad \text{(Equation 4)} \]

### Step 5: Solve Equations 3 and 4
Now we have:
1. \( X + Y = 1100 \) (Equation 3)
2. \( X - Y = -100 \) (Equation 4)

Adding these two equations:
\[ (X + Y) + (X - Y) = 1100 - 100 \]
This simplifies to:
\[ 2X = 1000 \implies X = 500 \]

Substituting \( X = 500 \) back into Equation 3:
\[ 500 + Y = 1100 \implies Y = 600 \]

### Conclusion
The cost price of article A is \( \text{Rs. } 500 \) and the cost price of article B is \( \text{Rs. } 600 \).

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