A shopkeeper lists the price of an article as Rs. 500. But he gives a certain discount which allows the buyer to pay Rs. 500 for the article including 10% sales tax. The rate of discount is

Options

A. `10%`
B. `10(1)/(11)%`
C. `9(1)/(11)%`
D. 0.11

Answer

To solve the problem step by step, we will follow the logical deductions made in the video transcript.

### Step 1: Understand the problem
The shopkeeper lists the price of an article as Rs. 500. After applying a discount, the buyer pays Rs. 500, which includes a 10% sales tax.

### Step 2: Define variables
Let the discount given by the shopkeeper be Rs. X. The marked price (MP) of the article is Rs. 500.

### Step 3: Calculate the selling price (SP)
The selling price after applying the discount can be expressed as:
\[ \text{SP} = \text{MP} - \text{Discount} = 500 - X \]

### Step 4: Include sales tax in the selling price
Since the selling price includes a 10% sales tax, we can express the relationship as:
\[ \text{Selling Price} = \text{Cost Price} + \text{Sales Tax} \]
Where the sales tax is 10% of the cost price. Therefore, if the selling price is Rs. 500, we can write:
\[ \text{SP} = \text{Cost Price} + 0.10 \times \text{Cost Price} \]
This can be simplified to:
\[ \text{SP} = 1.10 \times \text{Cost Price} \]

### Step 5: Set up the equation
From the previous steps, we have:
\[ 500 = 1.10 \times (500 - X) \]

### Step 6: Solve for X
Now, we will solve for X:
1. Expand the equation:
\[ 500 = 1.10 \times 500 - 1.10 \times X \]
\[ 500 = 550 - 1.10X \]

2. Rearranging gives:
\[ 1.10X = 550 - 500 \]
\[ 1.10X = 50 \]

3. Divide both sides by 1.10:
\[ X = \frac{50}{1.10} \]
\[ X = \frac{5000}{110} \]
\[ X = \frac{500}{11} \]

### Step 7: Calculate the discount percentage
To find the discount percentage, we use the formula:
\[ \text{Discount Percentage} = \left( \frac{\text{Discount}}{\text{Marked Price}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Discount Percentage} = \left( \frac{500/11}{500} \right) \times 100 \]
\[ \text{Discount Percentage} = \left( \frac{1}{11} \right) \times 100 \]
\[ \text{Discount Percentage} = \frac{100}{11} \]
This can be approximated to:
\[ \text{Discount Percentage} \approx 9.09\% \]

### Final Answer
The rate of discount is approximately **9.09%**.

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