Problem 52 - Elementary Algebra

Question

Some people use the following formula for determining the selling price of an item when the profit is based on a percent of the selling price: $$ \text { Selling price }=\frac{\text { Cost }}{100 \%-\text { Percent of profit }} $$ Show how to develop this formula.

Step-by-Step Solution

Answer

The derived formula for selling price is \( S = \frac{C}{1 - P} \), where \( C \) is cost, and \( P \) is profit percentage as a decimal.

Step 1: Understand the Components

Identify the variables involved in creating the formula. We have the cost of the item (let's call it \( C \)), the percent of profit ( the percentage by which we want to increase the cost), and the selling price ( what we need to calculate).

Step 2: Express the Selling Price

When profit is a percent of the selling price, we denote the percent of profit as \( P \). The selling price \( S \) should then cover the original cost and include the profit, mathematically represented as:\[ S = C + PS \]

Step 3: Rearrange the Equation

Rearrange the equation from Step 2 to express \( S \) purely in terms of \( C \) and \( P \). First move the \( PS \) term:\[ S - PS = C \]

Step 4: Factor Out the Selling Price

Factor out \( S \) from the left-side expression: \[ S(1 - P) = C \]

Step 5: Solve for the Selling Price

To solve for \( S \), divide both sides of the equation by \( 1 - P \):\[ S = \frac{C}{1 - P} \]

Step 6: Convert Percent to Decimal

Remember that in equations involving percentages, \( P \) must be converted to a decimal. If \( P \) is given as a percentage, you must divide \( P \) by 100 when using it in the formula.

Key Concepts

Profit Calculation

Understanding profit calculation is crucial in business and everyday transactions
because it determines how much gain is achieved over the cost of an item.
Profit, in simple terms, is the extra money earned above the original cost.
Profit can be expressed as a percentage of the selling price.
For example, if you sell a product for more than its cost, the difference is your profit.
To calculate profit:

• Identify the cost of the item, which is how much you spent to acquire or produce it.
• Determine the selling price, which is the amount at which you intend to sell the item.
• Subtract the cost from the selling price: \[ \text{Profit} = \text{Selling Price} - \text{Cost} \]

This basic understanding is essential when evaluating your pricing strategy or when deciding how deeply to discount a product.

Selling Price Calculation

Selling price calculation is integral in understanding how to set prices for goods or services.
When profit is based on a percentage of the selling price, the formula becomes crucial.
This formula incorporates the cost and desired profit percentage.
Here's how selling price is calculated using the given formula:

• Identify the item's cost \( C \), which is the baseline price.
• Determine the desired profit percentage \( P \), expressed in decimal form.
• Apply the formula: \[ S = \frac{C}{1 - P} \] where \( S \) is the selling price.

This selling price will cover both the cost and the added profit desired.
Knowing how to manipulate this formula allows for efficient price setting.

Percent to Decimal Conversion

Converting a percent to a decimal is a basic yet vital part of many mathematical calculations
as it ensures accuracy when percentages are used in equations.
Percentages represent a number out of 100, a common way to express a proportion or part of a whole.
To convert a percentage to a decimal:

• Take the percentage, say 25%.
• Divide the percentage by 100 to convert it to decimal form, \[ 25\% \rightarrow 0.25 \]
• Use this decimal in calculations.

This conversion is crucial when plugging percentage values into algebraic equations
as it ensures the correct value is used to maintain balance and solve for unknowns.