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

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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.
The relationship between the discounted price $$ d $$ and the original price $$ p $$ is given by the equation: $$ d = p \times (1 - r) $$ where $$ r $$ is the discount rate (expressed as a decimal).

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

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