Question

Discount Rate A satellite radio system for your car has been discounted by \(\$ 30\). The sale price is \(\$ 119\). What percent of the original list price is the discount?

Short answer

The discount is 20% of the original price.

Step-by-step solution

Compute the original price

In order to find the original list price, add the sale price and the discount. The formula to calculate the original price is: Original Price = Sale Price + Discount. So, Original Price = \$119 + \$30 = \$149

Compute the percentage of the discount

After finding the original list price, calculate the percentage of the discount from the original price. The formula is: (Discount / Original Price) * 100. So,( \$30 / \$149 ) * 100 = 20.13%. To match common practice we will round this to the nearest whole number resulting in 20%.

Key concepts

Percentages in Algebra

Working with percentages is a fundamental skill in algebra, especially when dealing with financial matters such as discounts. A percentage represents a fraction with a denominator of 100, symbolizing a part of a whole. For example, if a quantity 'x' represents 50% of 'y', it means that 'x' is half of 'y' because 50% is the same as the fraction \frac{1}{2}.

When calculating the percentage of one number relative to another in algebra, the standard formula used is: \
\[\text{Percentage} = \bigg(\frac{\text{Part}}{\text{Whole}}\bigg) \times 100\]
In the context of discounts, the 'part' is the amount of discount given, and the 'whole' is the original price of the item before the discount. By manipulating this algebraic formula, you can find missing variables, such as the original price or the discounted percentage, which is essential for understanding many real-world problems.

Sale Price Calculation

The sale price of an item is what customers pay after a discount is applied to the original price. Calculating the sale price is straightforward once you know the original price and the discount rate.

To find the sale price, you use the formula: \
\[\text{Sale Price} = \text{Original Price} - (\text{Discount Percentage} \times \text{Original Price})\]
However, if the discount is given in a dollar amount and not a percentage, you would simply subtract this amount from the original price:

\
\[\text{Sale Price} = \text{Original Price} - \text{Discount Amount}\]
Calculating the sale price is an essential skill not just in academic exercises but also in everyday shopping and budgeting. By mastering this, consumers can make more informed decisions and potentially save money.

Price Discount Percentage

The price discount percentage is used to determine how much cheaper an item is during a sale relative to its original price. It is expressed as a percentage and reflects the proportion of the reduction.

To calculate the discount percentage, you use the following formula: \
\[\text{Discount Percentage} = \bigg(\frac{\text{Discount Amount}}{\text{Original Price}}\bigg) \times 100\]
This calculation gives you insight into the actual value of the discount in relation to the original price. Retailers often use discount percentages to advertise sales because it provides a clear idea of the savings potential. When consumers understand how to calculate and interpret this percentage, they can compare discounts across different products and retailers to determine which offers are truly the best deals.