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:_____ If \(26 \%\) of Gordon's gross monthly salary results in total deductions of \(\$ 674.96,\) what is his gross monthly salary?

Short answer

Answer: The original price of the appliance was $320, and Gordon's gross monthly salary is approximately $2,596.

Step-by-step solution

Identify the percent

In this problem, the percent is the 25% discount applied to the original price. #Step 2: Identify the amount of decrease#

Identify the amount of decrease

The amount of decrease in this problem is the \(\$80\) discount. #Step 3: Form the equation#

Form the equation

We can use the equation: Amount of decrease = Percent × Initial amount. In this case: \(80 = 0.25 \times Initial\ amount\) #Step 4: Solve for the initial amount#

Solve for the initial amount

To find the initial amount, we can simply divide both sides of the equation by 0.25: Initial amount = \(\frac{80}{0.25}\) = \(320\) The original price of the appliance was \(\$320\). #Problem 2: Gross Monthly Salary# #Step 1: Identify the percent#

Identify the percent

In this problem, the percent is 26% representing the deductions from Gordon's gross monthly salary. #Step 2: Identify the amount of decrease#

Identify the amount of decrease

The amount of decrease in this problem is the \(\$674.96\) deducted from Gordon's gross monthly salary. #Step 3: Form the equation#

Form the equation

We can use the equation: Amount of decrease = Percent × Initial amount. In this case: \(674.96 = 0.26 \times Gross\ monthly\ salary\) #Step 4: Solve for the initial amount#

Solve for the initial amount

To find the initial amount, we can simply divide both sides of the equation by 0.26: Gross monthly salary = \(\frac{674.96}{0.26}\) ≈ \(2596\) Gordon's gross monthly salary is approximately \(\$2,596\).

Key concepts

Discount Calculation

Understanding discount calculations is crucial, especially when budgeting or trying to maximize savings when shopping. Discounts are reductions in the original price of a product or service. They are usually expressed as a percentage.
To calculate the discount amount, use the formula:

• Discount Amount = (Discount Percent / 100) × Original Price

For example, if you have a 25% discount, and it results in a reduction of $80, you are essentially saving 25% off the item's original price.
Calculating discounts requires knowing either the discount percentage or the amount of money saved. Once you understand these components, finding the original price becomes straightforward, as seen in our example for the appliance, where the original price was computed by reversing the discount application process.

Initial Amount Determination

Determining the initial amount before any deductions or discounts is important in various financial scenarios. It involves working backwards from what is known–such as the resulting value after a percentage change has occurred.
The formula used is:

• Initial Amount = Discount Amount / (Discount Percent / 100)

This equation easily helps you revert a discounted price back to its original state. In the example with the appliance, knowing the $80 decrease at a 25% discount allowed us to find the original price using simple division.
The same principle applies to any scenario involving percentage reductions or increases. This method is versatile and applicable across many real-world problems, from sales to personal finance, like calculating salaries in a work context.

Gross Salary Calculation

Gross salary calculation is an essential aspect of understanding one's financial standing before deductions like taxes and other withholdings. It's the amount before any deductions.
To calculate the gross salary if certain deductions are given, use the equation:

• Deductions = Deduction Percent × Gross Salary

For instance, if Gordon's deductions total $674.96, which is 26% of his gross salary, reversing this calculation as shown allows us to determine that his gross monthly salary is approximately $2,596. Calculating this correctly is crucial as it helps in budgeting and understanding the full scope of one's income.

Percent Decrease Problem

Solving percent decrease problems involves understanding how a reduction affects an original amount. These types of problems help in real-world scenarios like budgeting and finance.
To solve a percent decrease problem, apply the formula:

• Amount of Decrease = Percent Decrease × Initial Amount

The given problem of a discount on an appliance can be interpreted as a percent decrease issue, where the price reduction (or percent decrease) needs assessing for understanding the original value. Starting with the known values—like a $80 decrease at 25%—helps frame the problem so you can efficiently solve for the initial amount.
Mastering this allows one to tackle problems involving reduced prices or diminished quantities, offering clarity on how decreases affect overall values.