Per cent profit earned when an article is sold for ₹806, is double the per cent profit earned when the same article is sold for ₹728. If the marked price of the article is `40%` above the cost price, what is the marked price of the article?

Options

A. ₹945
B. ₹952
C. ₹882
D. ₹910

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Define Variables
Let the Cost Price (CP) of the article be \( CP \).

### Step 2: Set Up the Selling Price Equations
1. When the article is sold for ₹728, the profit percentage is \( x\% \).
- The selling price (SP) can be expressed as:
\[ SP = CP + \frac{x}{100} \times CP = CP \left(1 + \frac{x}{100}\right) \]
- Therefore, we have:
\[ 728 = CP \left(1 + \frac{x}{100}\right) \]

2. When the article is sold for ₹806, the profit percentage is \( 2x\% \).
- The selling price can be expressed as:
\[ SP = CP + \frac{2x}{100} \times CP = CP \left(1 + \frac{2x}{100}\right) \]
- Therefore, we have:
\[ 806 = CP \left(1 + \frac{2x}{100}\right) \]

### Step 3: Rearrange the Equations
From the first equation:
\[ CP = \frac{728 \times 100}{100 + x} \tag{1} \]

From the second equation:
\[ CP = \frac{806 \times 100}{100 + 2x} \tag{2} \]

### Step 4: Set the Two Expressions for CP Equal
Since both expressions represent the same CP, we can set them equal to each other:
\[ \frac{728 \times 100}{100 + x} = \frac{806 \times 100}{100 + 2x} \]

### Step 5: Cross Multiply
Cross-multiplying gives us:
\[ 728 \times (100 + 2x) = 806 \times (100 + x) \]

### Step 6: Expand Both Sides
Expanding both sides:
\[ 72800 + 1456x = 80600 + 806x \]

### Step 7: Rearrange to Solve for x
Rearranging gives:
\[ 1456x - 806x = 80600 - 72800 \]
\[ 650x = 7800 \]
\[ x = \frac{7800}{650} = 12 \]

### Step 8: Calculate the Cost Price (CP)
Substituting \( x = 12 \) back into equation (1):
\[ CP = \frac{728 \times 100}{100 + 12} = \frac{72800}{112} = 650 \]

### Step 9: Calculate the Marked Price (MP)
The marked price is 40% above the cost price:
\[ MP = CP + 0.4 \times CP = 1.4 \times CP = 1.4 \times 650 = 910 \]

### Final Answer
The marked price of the article is ₹910.

---