Problem 44 - Intermediate Algebra

Question

Solve each formula for the specified variable. \(P=\frac{R-C}{n}\) for \(C\) (from business)

Step-by-Step Solution

Answer

The solution for \(C\) is \(C = R - nP\).

Step 1: Understand the Formula

The formula given is \(P=\frac{R-C}{n}\), where \(P\) is related to \(R\), \(C\), and \(n\). We need to solve this formula for \(C\). This means we must express \(C\) in terms of \(P\), \(R\), and \(n\).

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by \(n\). This will give us \(nP = R - C\).

Step 3: Isolate the Variable

We now have \(nP = R - C\). To solve for \(C\), we need to isolate it on one side. So, add \(C\) to both sides to get \(nP + C = R\).

Step 4: Solve for C

Now, subtract \(nP\) from both sides to isolate \(C\). This gives us \(C = R - nP\).

Step 5: Solution Confirmation

The final expression for \(C\) in terms of \(P\), \(R\), and \(n\) is \(C = R - nP\). Check this by substituting back into the original equation to verify correctness.

Key Concepts

Solving Equations

The art of solving equations is all about finding the value of a variable that makes an equation true. Equations are like puzzles that need solving, with each component representing different parts of the puzzle. In our given problem, the equation is \(P=\frac{R-C}{n}\). Our task is to solve this for the variable \(C\). This process involves rearranging other parts of the equation to isolate \(C\).

Solving equations typically involves a few key steps:

• Understand the equation: Recognize what you need to find and which terms you have at your disposal.
• Eliminate fractions: If the equation includes fractions, simplify them first by multiplying all terms by a common denominator.
• Isolate the variable: Perform operations to keep the variable you’re solving for on one side of the equation.

In this specific scenario, we started by eliminating the fraction by multiplying both sides by \(n\), giving us \(nP = R - C\). From there, isolating \(C\) means adding \(C\) to both sides and finally subtracting \(nP\) from \(R\) to get \(C = R - nP\).

Formula Manipulation

Formula manipulation is the process of rewriting equations to make them more useful or easier to work with, depending on the context you are considering. It allows you to rearrange the components of a formula to solve for a specific variable while maintaining the equation's integrity.

In our example where \(P = \frac{R-C}{n}\), you want \(C\) as the subject of the formula. Here is how formula manipulation works in steps:

• 1. Clear fractions and denominators: Multiply through by any denominators to remove fractions from the equation.
• 2. Simplify operations: Add or subtract terms to group similar elements, thus simplifying the math.
• 3. Rearrange terms: Move terms around to achieve the desired form.

These steps help convert the original equation into \(nP = R - C\), then further to \(C = R - nP\). The power of formula manipulation is that once you identify the variable you need to express, you can proceed systematically to solve for it.

Business Mathematics

In business mathematics, understanding and applying formulas is pivotal to solving real-world problems. This branch of mathematics uses algebraic techniques to process complex information related to both costs and revenues as represented by variables in equations.

Let's explore what makes the given formula \(P=\frac{R-C}{n}\) relevant in business contexts:

• Profit analysis: \(P\) often represents profit margins where \(R\) is revenue, \(C\) is cost, and \(n\) is a scaling factor, such as the number of items sold.
• Cost management: Solving the equation for \(C\) as \(C = R - nP\) provides insights into understanding how different elements affect total costs.
• Operational decisions: By isolating \(C\), businesses can identify how changing profit expectations or sales volume impacts their overall cost.

Therefore, mastering basic algebra and formula manipulation is essential for anyone working in business, as it aids in making informed financial decisions.