A shopkeeper sells a pair of sunglasses at a profit of 25%. If he had bought it at 25% less and sold it for Re 10 less, then be would have gained 40%. The cost price of the pair of sunglasses is

Options

A. 25
B. 50
C. 60
D. 70

Answer

To solve the problem step by step, we will denote the cost price of the sunglasses as \( x \).

### Step 1: Determine the Selling Price with 25% Profit
The shopkeeper sells the sunglasses at a profit of 25%. Therefore, the selling price (SP) can be calculated as:
\[ SP = CP + 25\% \text{ of } CP = x + \frac{25}{100} \cdot x = x + \frac{25x}{100} = \frac{125x}{100} = \frac{5x}{4} \]

### Step 2: Calculate the New Cost Price
If the shopkeeper had bought the sunglasses at 25% less, the new cost price (CP') would be:
\[ CP' = x - 25\% \text{ of } x = x - \frac{25}{100} \cdot x = x - \frac{25x}{100} = x - \frac{25x}{100} = \frac{75x}{100} = \frac{3x}{4} \]

### Step 3: Determine the New Selling Price
According to the problem, if he sold it for Re 10 less than the original selling price, the new selling price (SP') would be:
\[ SP' = SP - 10 = \frac{5x}{4} - 10 \]

### Step 4: Set Up the Equation for 40% Gain
The problem states that this new selling price results in a 40% gain on the new cost price. Thus, we can express this as:
\[ SP' = CP' + 40\% \text{ of } CP' = CP' + \frac{40}{100} \cdot CP' = CP' + \frac{40}{100} \cdot \frac{3x}{4} \]
\[ SP' = \frac{3x}{4} + \frac{40 \cdot 3x}{100 \cdot 4} = \frac{3x}{4} + \frac{12x}{100} = \frac{3x}{4} + \frac{3x}{25} \]

### Step 5: Find a Common Denominator
To combine the fractions, we need a common denominator. The least common multiple of 4 and 25 is 100. Thus, we rewrite the fractions:
\[ \frac{3x}{4} = \frac{75x}{100} \quad \text{and} \quad \frac{3x}{25} = \frac{12x}{100} \]
So,
\[ SP' = \frac{75x}{100} + \frac{12x}{100} = \frac{87x}{100} \]

### Step 6: Set the Two Expressions for SP' Equal
Now we can set the two expressions for \( SP' \) equal to each other:
\[ \frac{5x}{4} - 10 = \frac{87x}{100} \]

### Step 7: Solve for x
To eliminate the fractions, multiply through by 100:
\[ 100 \left( \frac{5x}{4} - 10 \right) = 87x \]
\[ 125x - 1000 = 87x \]
Rearranging gives:
\[ 125x - 87x = 1000 \]
\[ 38x = 1000 \]
\[ x = \frac{1000}{38} = 50 \]

### Final Answer
The cost price of the pair of sunglasses is \( \text{Rs } 50 \).

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