Question

Given a percent of decrease and an initial amount, explain how to find the final amount after the decrease. A \(30 \%\) discount is to be applied to a dress with an initial price of \(\$ 65.99 .\) What are the discount amount and final price?

Short answer

Answer: The discount amount is $19.80, and the final price of the dress after the discount is $46.19.

Step-by-step solution

Identifying the initial price and percentage of decrease

In this problem, we are given the initial price of the dress as $65.99 and the percentage of decrease (discount) as 30%. We will use these values to calculate the discount amount.

Calculate the discount amount

To find the discount amount, we need to multiply the initial price by the percentage of decrease. We can write the percentage as a decimal (0.30) by dividing by 100. So, the discount amount can be calculated as follows: Discount Amount = Initial Price × Percentage of Decrease (in decimal) = \(65.99 × 0.30\)

Calculate the final price

After finding the discount amount, we can subtract it from the initial price to find the final price. Final Price = Initial Price - Discount Amount Now let's calculate the discount amount and final price using the initial price and percentage of decrease.

Calculation

Discount Amount = \(65.99 × 0.30 = \)19.797\(. To represent the discount amount in dollars, we can round it to two decimal places: \)19.80$ Final Price = \(65.99 - 19.80 = \$ 46.19\)

Answer the question

The discount amount is \(19.80\), and the final price of the dress after the discount is \(46.19\).

Key concepts

Initial Price

Understanding the concept of initial price is the foundation of calculating discounts and final prices. The initial price, also known as the original or list price, is the starting point of any transaction.

This is the price on the tag before any sales or discounts are applied and doesn't include additional fees, such as tax. In our given exercise, the initial price of the dress is clearly stated as \( \$65.99 \). It's essential to accurately identify this value, as it determines the base amount from which all discounts are calculated.

When we talk about the percent of a decrease, like a discount, it is applied to this initial price. Therefore, establishing the correct initial amount ensures that subsequent calculations will lead to an accurate final price. It’s always a smart move to double-check this first step; an error here would carry through all other calculations.

Discount Calculation

The discount calculation involves figuring out how much money is to be subtracted from the initial price to determine the sale price. In the given problem, the discount is expressed as a percent of the initial price.

To calculate the discount amount, you convert the percentage to a decimal by dividing by 100, which turns our 30% into 0.30. You then multiply this decimal by the initial price of the item.

This process is illustrated as:

• Discount Amount = Initial Price × Percentage of Decrease (in decimal)
• Discount Amount = \( \$65.99 \times 0.30 \)

After performing this multiplication, we find the discount amount which in this case is approximately \( \$19.80 \). This step is crucial because it shows how much the original price is being reduced by. Making sure your percentage-to-decimal conversion is accurate and that your multiplication is correct is key to getting the right discount amount.

Final Price Determination

Once the discount is calculated, the next step is to determine the final price. This is done by subtracting the discount amount from the initial price.

In mathematical terms, the formula for the final price is:

• Final Price = Initial Price - Discount Amount

This leads us to the final step in our problem where we subtract the rounded discount amount \( \$19.80 \) from the initial price \( \$65.99 \), yielding the final price of the dress.

Calculation of Final Price:

Final Price = Initial Price - Discount Amount
Final Price = \( \$65.99 - \$19.80 = \$46.19 \)

The calculated final price reflects the amount the customer will pay after the discount is applied. It is vital to round the discount amount carefully to match currency standards, as incorrect rounding can lead to a slightly inaccurate final price. Always ensure to follow through with proper rounding when presenting financial figures to maintain precision.