Question

Prep Exercise 4 Identify the percent, initial amount, and amount of increase or decrease. A \(25 \%\) discount is applied to the price of an appliance, resulting in a discount of \(\$ 80\). What was the original price of the appliance? Percent:_____ Initial amount:_____ Amount of increase or decrease:_____ Patricia purchased a table on sale for \(30 \%\) off. If the discount amount was \(\$ 44.09,\) what was the initial price?

Short answer

Answer: The initial price of the appliance was $320. 2. For the table with a 30% discount and a $44.09 discount amount, what was its initial price? Answer: The initial price of the table was approximately $146.97.

Step-by-step solution

Problem 1: Appliance Discount

In this problem, the percent of discount is given as \(25 \%\) and the discount amount is given as \(\$ 80\). We will use the formula mentioned above to find the initial price of the appliance. Step 1: Set up the equation Discount amount = (Percent of discount) * (Initial price) \(80 = (0.25) * (Initial\:price)\) Step 2: Solve the equation for the initial price Divide both sides of the equation by 0.25: Initial price = \(\frac{80}{0.25} = 320\) So, the initial price of the appliance is \(\$ 320\). Percent: \(25 \%\) Initial amount: \(\$ 320\) Amount of increase or decrease: \(\$ 80\)

Problem 2: Table Discount

In this problem, the percent of discount is given as \(30 \%\) and the discount amount is given as \(\$ 44.09\). We will use the same formula as in the previous problem to find the initial price of the table. Step 1: Set up the equation Discount amount = (Percent of discount) * (Initial price) \(44.09 = (0.30) * (Initial\:price)\) Step 2: Solve the equation for the initial price Divide both sides of the equation by 0.30: Initial price = \(\frac{44.09}{0.30} = 146.97\) So, the initial price of the table is approximately \(\$ 146.97\). Percent: \(30 \%\) Initial amount: \(\$ 146.97\) Amount of increase or decrease: \(\$ 44.09\)

Key concepts

Discount Calculation

Calculating the discount is essential to know how much you save when buying a product on sale. Discounts are usually given as a percentage of the initial price, and understanding how to calculate this percentage as a dollar amount is crucial.

Here's how you do it:

• Identify the percent of discount given. In many cases, this is stated in the offer, like a 25% or 30% discount.
• Use the formula for calculating the dollar value of the discount: \[\text{Discount amount} = \left(\frac{\text{Percent of discount}}{100}\right) \times \text{Initial price}\]
• Plug in the given percent of discount and solve for how much money the discount represents.

For example, if an appliance has a 25% discount leading to a savings of $80, this calculation helps check or compute the initial price.

Initial Price

The initial price is the original price a product had before any discounts were applied. Calculating the initial price when the discount amount and discount percentage are provided is straightforward.

Here's a step-by-step approach:

• Use the equation: \[\text{Discount amount} = \text{Percent of discount} \times \text{Initial price}\]
• Rearrange the equation to solve for the initial price: \[\text{Initial price} = \frac{\text{Discount amount}}{\text{Percent of discount}}\]
• In mathematical terms, divide the known discount amount by the discount percentage (in its decimal form, e.g., 25% as 0.25) to find the initial price.

Using these steps helps buyers understand what the product originally cost and see the true value of the discount.

Percent of Discount

The percent of discount shows how much of the original price you're saving. It's crucial for calculating both the dollar savings and determining how much you'll actually pay.
Here’s what you should know about it:

• The percent of discount is the portion of the full price that is reduced. It is expressed as a percentage, which means 100 represents paying full price, whereas 0 would mean no discount at all.
• To find how the discount relates to the original price, convert the percent of discount into a decimal by dividing it by 100.
• This decimal form is then used to calculate the discount amount in money, as indicated in the formula for discount calculation.

Using a 30% discount example: if a table’s discounted price saves you $44.09, knowing the percent helps accurately calculate the initial price to confirm its value.