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:_____ Marcelle's retirement deduction from her monthly salary is \(\$ 187.50 .\) If the retirement deduction is \(6 \%\) of her gross monthly salary, what is her gross monthly salary?

Short answer

Answer: The original price of the appliance is $320, and Marcelle's gross monthly salary is $3125.

Step-by-step solution

Identify the percent, initial amount, and amount of increase or decrease

In this problem, the percent is the discount percentage, which is 25%. The amount of decrease is the discount itself, which is $80. The initial amount, the original price of the appliance, is what we need to find.

Set up an equation

We can set up an equation using the given percentage and amount of decrease: $$Original\, Price * \frac{25}{100} = 80$$

Solve for the initial amount (original price)

To find the original price, we need to solve the equation: $$Original\, Price = \frac{80}{\frac{25}{100}}$$

Calculate the original price

Doing the calculation, we find that the original price is: $$Original\, Price = \frac{80}{0.25} = 320$$ Problem 1 answer: Percent: 25% Initial amount: $320 Amount of increase or decrease: $80 Problem 2:

Identify the percent, initial amount, and amount of increase or decrease

In this problem, the percent is the retirement deduction percentage, which is 6%. The amount of decrease is the retirement deduction itself, which is $187.50. The initial amount, Marcelle's gross monthly salary, is what we need to find.

Set up an equation

We can set up an equation using the given percentage and amount of decrease: $$Gross\, Monthly\, Salary * \frac{6}{100} = 187.50$$

Solve for the initial amount (gross monthly salary)

To find the gross monthly salary, we need to solve the equation: $$Gross\, Monthly\, Salary = \frac{187.50}{\frac{6}{100}}$$

Calculate the gross monthly salary

Doing the calculation, we find that the gross monthly salary is: $$Gross\, Monthly\, Salary = \frac{187.50}{0.06} = 3125$$ Problem 2 answer: Percent: 6% Initial amount: $3125 Amount of increase or decrease: $187.50

Key concepts

Discount Calculation

Understanding how a discount affects price is essential when managing finances or shopping. A discount is a reduction on the original price of an item or service. Typically, discounts are given as a percentage. For example, a 25% discount on an appliance indicates that you’ll pay 25% less than the original price.

To calculate the discount amount in currency, you multiply the original price by the discount percentage, expressed as a decimal. The formula looks like this:
\[Discount\text{ Amount} = Original\text{ Price} \times \left(\frac{Discount\text{ Percentage}}{100}\right)\]
So, if you know the discount amount and the discount percentage, you can find the original price using:
\[Original\text{ Price} = \frac{Discount\text{ Amount}}{Discount\text{ Percentage as a decimal}}\]
For instance, with an \$80 discount at 25%, the original price would be \$80 divided by 0.25, equal to \$320.

Percentages in Mathematics

Percentages represent a part of a whole as a fraction of 100. They play a pivotal role in various mathematical and real-world applications—from financial calculations like discounts and interest rates to comparing statistics.

In essence, a percentage tells us how many parts we would have for every 100 parts of a total. To convert a percentage to a decimal for calculation purposes, you divide by 100. This conversion is crucial in computations where you need to find an actual amount from a percentage.

For instance, a 6% retirement deduction from Marcelle’s salary means that for every 100 currency units of her salary, 6 units go towards retirement. By expressing 6% as 0.06, we can easily calculate how much is deducted or, reversely, what her gross salary needs to be.

Solving for Original Price

To determine the original price of an item from its sale price and a discount percentage, you work backwards from the discount calculation. If a price tag shows a discounted amount, the original price serves as the 'whole' from which the discount was subtracted. The process involves dividing the discount amount by the percentage rate.

The formula for solving the original price is:
\[ Original\text{ Price} = \frac{Discount\text{ Amount}}{\left(\frac{Discount\text{ Percentage}}{100}\right)}\]
It’s vital to convert the discount percentage to a decimal to perform the division. This formula is helpful in retail scenarios and when analyzing spending to understand how much saving occurs during sales.

Gross Monthly Salary Computation

Calculating the gross monthly salary is crucial for budgeting, determining loan eligibility, and understanding your compensation before tax and other deductions. Gross salary is the amount paid before anything is subtracted, like taxes or retirement deductions.

If you know the percentage of salary deducted and the deduction amount, you can solve for the gross salary using the formula:
\[ Gross\text{ Monthly\text{ Salary}} = \frac{Deduction\text{ Amount}}{\left(\frac{Deduction\text{ Percentage}}{100}\right)}\]
In Marcelle’s case, a \$187.50 deduction at 6% means that:
\[Gross\text{ Monthly\text{ Salary}} = \frac{187.50}{0.06}\]
This brings her gross monthly salary to \$3125. Such calculations help individuals understand their finances better and make informed decisions about their income and expenditures.