 If the discount offered on an item is 20% of the selling price and the actual discount percent equals the profit percent. Find the ratio of above discount offered to the cost price

Let's define the variables:

- Let \( S \) be the selling price.
- Let \( C \) be the cost price.
- Let \( D \) be the discount offered, which is given as 20% of the selling price. So, \( D = 0.20S \).

It is also given that the discount percent equals the profit percent. This means the percentage of profit (based on cost price) is the same as the percentage of the discount (based on the selling price). 

### Step-by-step solution:

1. **Discount as a percentage of selling price**:
   \[
   D = 0.20S
   \]
   So, the discount offered is 20% of the selling price.

2. **Profit percent**:
   Let the profit percent be \( p \). We know that:
   \[
   p = \frac{S - C}{C} \times 100
   \]
   This equation represents the profit percentage.

3. **Discount percent equals profit percent**:
   According to the problem, the discount percentage is equal to the profit percentage. Therefore:
   \[
   \frac{S - C}{C} \times 100 = 20
   \]
   Simplifying this equation:
   \[
   \frac{S - C}{C} = 0.20
   \]
   \[
   S - C = 0.20C
   \]
   \[
   S = 1.20C
   \]
   This means that the selling price is 1.20 times the cost price.

4. **Find the ratio of discount offered to cost price**:
   The discount offered is \( D = 0.20S \), and \( S = 1.20C \). Substituting for \( S \):
   \[
   D = 0.20 \times 1.20C = 0.24C
   \]
   Therefore, the ratio of the discount offered to the cost price is:
   \[
   \frac{D}{C} = \frac{0.24C}{C} = 0.24
   \]
   So, the ratio is 0.24, or \( \frac{6}{25} \).

Thus, the ratio of the discount offered to the cost price is \( \frac{6}{25} \).

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the ratio change if the discount was 25% instead of 20%?
2. What is the profit percentage if the cost price is known?
3. How do profit margins affect selling price calculations?
4. What is the formula to calculate selling price based on profit and cost price?
5. If the cost price increases by 10%, how would the discount ratio change?

**Tip:** Always relate percentages in business problems to their respective bases (selling price or cost price) to avoid confusion.

If the discount offered on an item is 20% of the selling price and the actual discount percent equals the profit percent, find the ratio of the above discount offered to the cost price.

Discover how to calculate the ratio of the discount offered to the cost price when the discount is 20% of the selling price, and the discount percent equals the profit percent. This detailed solution explores concepts in profit and loss and ratio calculations, suitable for students in Grades 8-10.

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