Question

Solve. Hannah purchased a curio stand discounted \(35 \%\). If the price after the discount was \(\$ 246.68\), what was the initial price?

Short answer

Answer: The initial price of the curio stand before the discount was approximately $379.51.

Step-by-step solution

- Set up an equation based on given information

Let x be the initial price of the curio stand. Since the price was reduced by 35%, the remaining percentage paid is (100% - 35%) which is equal to 65%. So, the discounted price is 65% of the initial price. This can be formulated as an equation: 65% * x = $246.68

- Convert the percentage to a decimal

To make the equation easier to solve, we need to convert the percentage to a decimal. Divide the percentage value by 100 to get the equivalent decimal: 65% ÷ 100 = 0.65 Now our equation will be: 0.65 * x = $246.68

- Solve the equation for x

To find the initial price (x), we need to solve the equation for x. To do this, we should divide both sides by 0.65: x = $246.68 ÷ 0.65 Calculate the value of x: x ≈ $379.51

- Conclude the answer

Hannah's initial price of the curio stand before the discount was approximately $379.51.

Key concepts

Solving Equations

In mathematical problems, solving equations is a crucial skill. It involves finding the value of an unknown variable that satisfies a given mathematical statement. In our exercise, we are tasked to find the initial price of an item after a discount is applied, which involves an unknown variable, often represented by \( x \).

Here's how you approach it:

• Identify the known values (in this case, the discounted price and the discount percentage).
• Set up an equation with the known values, representing the unknown with a variable like \( x \).
• Solve for \( x \) by performing necessary mathematical operations, such as addition, subtraction, multiplication, or division, to isolate \( x \) on one side of the equation.

In our example, the equation is set up as \( 0.65 \times x = 246.68 \), where \( x \) needs to be calculated. By solving this equation, you determine that Hannah's initial price (\( x \)) of the curio stand was approximately $379.51.

Initial Price Calculation

Calculating the initial price of a discounted item is a common mathematical application. This involves essentially reversing the process of applying a discount. You start with the final amount paid and work backward to find the start price before any deductions.

Here’s how you break it down:

• The initial price is always 100%, whereas the discounted price represents the remaining percentage after the discount.
• In our problem, an item which costs \\(246.68 corresponds to 65% of its initial price.
• The task is to find the full (100%) price by dividing the paid amount by the percentage in decimal form. In simple terms, solve for \( x \) in the equation \( 0.65 \times x = 246.68 \).

By executing the division, you end up with the initial price, which was not directly apparent from the discount scenario. In this case, Hannah's curio stand initially cost about \\)379.51 before the 35% discount was applied.

Percentage to Decimal Conversion

Converting percentages to decimals is a fundamental step in many mathematical problems, especially ones involving discounts or rates. The conversion is straightforward and immensely useful for simplification of equations.

To convert any percentage to a decimal:

• Divide the percentage by 100. For example, to convert 65% to a decimal, you calculate \( 65 \div 100 = 0.65 \).
• This operation moves the decimal point two places to the left, effectively simplifying any calculations involving the percentage.

This conversion allows you to handle percentages in mathematical equations just as you would handle any other number. In our scenario, converting 65% to \( 0.65 \) enabled a direct multiplication with the variable \( x \), leading to the discovery of the initial price by simple arithmetic operations.