A man bought an article and sold it at a gain of 10%. If he had bought it at 20% less and sold it for Rs 10 more, he would have made a profit of 40%. Find the C.P. of the article?

Answer

To find the cost price (C.P.) of the article, we can follow these steps:

### Step 1: Assume the Cost Price
Let the cost price of the article be \( C.P. = 100x \) rupees.

### Step 2: Calculate the Selling Price with 10% Gain
Since the man sold the article at a gain of 10%, the selling price (S.P.) can be calculated as:
\[ S.P. = C.P. + 10\% \text{ of } C.P. = 100x + 0.1 \times 100x = 110x \text{ rupees} \]

### Step 3: Calculate the New Cost Price
If he had bought it at 20% less, the new cost price would be:
\[ \text{New C.P.} = C.P. - 20\% \text{ of } C.P. = 100x - 0.2 \times 100x = 80x \text{ rupees} \]

### Step 4: Calculate the New Selling Price
If he sold it for Rs 10 more than the old selling price, the new selling price would be:
\[ \text{New S.P.} = \text{Old S.P.} + 10 = 110x + 10 \text{ rupees} \]

### Step 5: Set Up the Profit Equation
According to the problem, if he sold it at the new selling price, he would have made a profit of 40%. The profit can be expressed as:
\[ \text{Profit} = \text{New S.P.} - \text{New C.P.} = (110x + 10) - 80x = 30x + 10 \]

### Step 6: Set Up the Profit Percentage Equation
The profit percentage is given as 40%, which can be expressed as:
\[ \frac{\text{Profit}}{\text{New C.P.}} \times 100 = 40 \]
Substituting the values we have:
\[ \frac{30x + 10}{80x} \times 100 = 40 \]

### Step 7: Simplify the Equation
Now, simplify the equation:
\[ 30x + 10 = 40 \times \frac{80x}{100} \]
\[ 30x + 10 = 32x \]

### Step 8: Solve for x
Rearranging gives:
\[ 10 = 32x - 30x \]
\[ 10 = 2x \]
\[ x = 5 \]

### Step 9: Calculate the Cost Price
Now, substituting \( x \) back to find the cost price:
\[ C.P. = 100x = 100 \times 5 = 500 \text{ rupees} \]

Thus, the cost price of the article is **Rs 500**.

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