Two items A and B are sold at a profit of 10% and 15% respectively. If the amount of profit received is the same, then the cost price of A and B may be?

Answer

To solve the problem step by step, we will denote the cost prices of items A and B as CP_A and CP_B respectively.

### Step 1: Understand the Profit Percentages
- Item A is sold at a profit of 10%.
- Item B is sold at a profit of 15%.

### Step 2: Express Selling Prices in Terms of Cost Prices
- The selling price (SP) of item A can be expressed as:
\[ SP_A = CP_A + 0.10 \times CP_A = 1.10 \times CP_A \]
- The selling price (SP) of item B can be expressed as:
\[ SP_B = CP_B + 0.15 \times CP_B = 1.15 \times CP_B \]

### Step 3: Set the Profits Equal
Since the profit from both items is the same, we can set up the equation:
\[ SP_A - CP_A = SP_B - CP_B \]
Substituting the expressions from Step 2:
\[ (1.10 \times CP_A - CP_A) = (1.15 \times CP_B - CP_B) \]
This simplifies to:
\[ 0.10 \times CP_A = 0.15 \times CP_B \]

### Step 4: Rearranging the Equation
Rearranging the equation gives us:
\[ \frac{CP_A}{CP_B} = \frac{0.15}{0.10} = \frac{15}{10} = \frac{3}{2} \]
This means that the cost price of A is to the cost price of B in the ratio of 3:2.

### Step 5: Assign Values to Cost Prices
Let’s assign a value to CP_B. If we let:
\[ CP_B = 2x \quad \text{(for some value } x\text{)} \]
Then:
\[ CP_A = 3x \]

### Step 6: Possible Values of Cost Prices
Now, we can express the cost prices of A and B as:
- Cost Price of A (CP_A) = 3x
- Cost Price of B (CP_B) = 2x

If we choose a specific value for \(x\), we can find specific cost prices. For example, if \(x = 20\):
- CP_A = 3 * 20 = 60
- CP_B = 2 * 20 = 40

### Conclusion
Thus, the possible cost prices of A and B are:
- Cost Price of A = 60
- Cost Price of B = 40