Write an equation for the relationship between the discount price of an item, \( d \), and its original price, \( p \). Equation:

Certainly! To establish the relationship between the **discounted price** of an item (\( d \)) and its **original price** (\( p \)), we need to consider the **discount rate** (\( r \)), which represents the percentage reduction applied to the original price. ### Equation: \[ d = p \times (1 - r) \] ### Where: - \( d \) = Discounted price of the item - \( p \) = Original price of the item - \( r \) = Discount rate (expressed as a decimal) ### Explanation: 1. **Discount Rate (\( r \))**: This is the percentage by which the original price is reduced. For example, a 20% discount would mean \( r = 0.20 \). 2. **Calculating the Discounted Price (\( d \))**: - Multiply the original price (\( p \)) by the discount rate (\( r \)) to find the amount of the discount: \( p \times r \). - Subtract this discount amount from the original price to get the discounted price: \( d = p - (p \times r) \). - This simplifies to: \( d = p \times (1 - r) \). ### **Example:** Suppose an item has an original price (\( p \)) of \$50 and there's a **15% discount** (\( r = 0.15 \)). \[ d = 50 \times (1 - 0.15) = 50 \times 0.85 = \$42.50 \] So, the discounted price (\( d \)) would be \$42.50. ### Alternative Representation: If you prefer to express the equation without factoring: \[ d = p - (p \times r) \] Both forms convey the same relationship and can be used based on your preference or the context of the problem.