Problem 121 - Elementary Algebra

Question

Mitchell currently sells stoves for company \(A\) at a salary of \(\$ 12,000\) plus a \(\$ 150\) commission for each stove he sells. Company B offers him a position with a salary of \(\$ 24,000\) plus a \(\$ 50\) commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

Step-by-Step Solution

Answer

Mitchell needs to sell 120 stoves.

Step 1 – Define the Variables

Let the number of stoves Mitchell needs to sell be represented by the variable \(x\).

Step 2 – Write Equations for Each Company's Earnings

For Company A, Mitchell's total earnings can be described by the equation: \[ E_A = 12000 + 150x \] For Company B, Mitchell's total earnings can be described by the equation: \[ E_B = 24000 + 50x \]

Step 3 – Set the Equations Equal to Each Other

To find the number of stoves \(x\) where Mitchell's earnings from both companies are equal, set the two equations equal to each other: \[ 12000 + 150x = 24000 + 50x \]

Step 4 – Solve for \(x\)

First, isolate the variable \(x\) by subtracting \(50x\) from both sides: \[ 12000 + 100x = 24000 \]Next, subtract 12000 from both sides: \[ 100x = 12000 \]Finally, divide both sides by 100: \[ x = 120 \]Mitchell needs to sell 120 stoves for his earnings to be equal with both companies.

Key Concepts

Variable Definition

When solving algebra equations, the first critical step is defining the variables. A variable represents an unknown value that you're trying to find. This lets you create an equation to solve the problem.
In this exercise, the variable we define is the number of stoves Mitchell needs to sell. We represent this unknown number by the variable \(x\). This means whenever you see \(x\) in the equations, it stands for the number of stoves.
Using variables makes it easier to manipulate and solve equations in a structured way. By defining variables, you set a foundation to build your equations and ultimately find the solution.

Earnings Equations

Once the variable is defined, the next step is to write equations that express the total earnings for each company. This involves using the variable in mathematical expressions.
For Company A, Mitchell's total earnings equation is:
\[ E_A = 12000 + 150x \]
Here, \(12000\) is his base salary, and \(150x\) represents the commission earned from selling \(x\) stoves. Similarly, for Company B, his earnings are given by:
\[ E_B = 24000 + 50x \]
In this equation, \(24000\) is the base salary, and \(50x\) is the commission earned from selling \(x\) stoves.
Writing these equations helps in comparing the earnings from the two companies based on the same variable, \(x\).

Solving Equations

The final step is solving the equations to find the value of the variable. To do this, we set the earnings equations equal to each other to find where Mitchell's earnings would be the same for both companies.
Set the equations equal:
\[ 12000 + 150x = 24000 + 50x \]
First, isolate the variable term by subtracting \(50x\) from both sides:
\[ 12000 + 100x = 24000 \]
Next, isolate \(100x\) by subtracting \(12000\) from both sides:
\[ 100x = 12000 \]
Finally, solve for \(x\) by dividing both sides by 100:
\[ x = 120 \]
This means Mitchell needs to sell 120 stoves for his earnings to be equal at both companies.
Solving equations involves manipulating the equation step-by-step to isolate the variable, which reveals the unknown value.