A shopkeeper faced 16% loss by selling an item at 30% discount. What will be the loss or profit if the item is sold at 10% discount?

To find the loss or profit when the item is sold at a 10% discount, we first need to establish a relationship between the cost price (C), selling price (S), and discount. Here’s the step-by-step process:

### Step 1: Define Variables
- Let the **marked price** of the item be $ M $.
- Let the **cost price** of the item be $ C $.

### Step 2: Information Given
- The shopkeeper gives a **30% discount** and incurs a **16% loss**.
  - The selling price after 30% discount is:  
    \[
    S_{30\%} = 0.70M \quad \text{(since selling price is 70% of the marked price)}
    \]
  - The loss incurred is 16%, meaning the selling price is 84% of the cost price:
    \[
    S_{30\%} = 0.84C
    \]

### Step 3: Establish Equation
Equating the two expressions for $ S_{30\%} $:
\[
0.70M = 0.84C
\]
This implies:
\[
M = \frac{0.84C}{0.70} = 1.2C
\]
So, the **marked price** is 120% of the cost price.

### Step 4: Find Selling Price with 10% Discount
With a 10% discount, the selling price becomes:
\[
S_{10\%} = 0.90M
\]
Substitute $ M = 1.2C $ into the equation:
\[
S_{10\%} = 0.90 \times 1.2C = 1.08C
\]
Thus, the selling price with a 10% discount is 108% of the cost price.

### Step 5: Profit or Loss Calculation
Since the selling price is 108% of the cost price, the shopkeeper makes a **profit** of 8%.

### Conclusion:
If the shopkeeper sells the item at a 10% discount, they will make a **profit of 8%**.

Do you have any questions or need further details?

Here are some related questions for practice:
1. What would be the profit or loss if the item is sold at 20% discount?
2. If the cost price was \$100, what would the selling price be at 10% discount?
3. How much discount would result in no profit or loss for the shopkeeper?
4. What is the relation between marked price and cost price in this scenario?
5. How would a 50% discount affect the profit or loss?

**Tip:** Always calculate the marked price in relation to the cost price before determining the effect of discounts.

Find out the profit or loss when a shopkeeper sells an item at a 10% discount after facing a 16% loss at 30% discount. This solution covers key concepts in profit and loss, percentage calculations, and discount strategies, suitable for Grades 9-10.

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