Problem 119 - Elementary Algebra

Question

Jackie has been offered positions by two cable companies. The first company pays a salary of \(\$ 14,000\) plus a commission of \(\$ 100\) for each cable package sold. The second pays a salary of \(\$ 20,000\) plus a commission of \(\$ 25\) for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

Step-by-Step Solution

Answer

Jackie needs to sell 80 cable packages.

Step 1: Understand the problem

Determine the number of cable packages Jackie needs to sell to make the total pay equal between the two job offers.

Step 2: Set up the equations

Let the number of cable packages sold be represented by the variable \(x\). The total pay for each company can be represented by equations:Company 1: \(14000 + 100x\)Company 2: \(20000 + 25x\)

Step 3: Set the equations equal

Equate the expressions for the total pay from both companies to find the value of \(x\): \[ 14000 + 100x = 20000 + 25x \]

Step 4: Solve for x

Isolate \(x\) by first subtracting 25x from both sides of the equation: \[ 14000 + 75x = 20000 \]Then, subtract 14000 from both sides: \[ 75x = 6000 \]Finally, divide both sides by 75: \[ x = \frac{6000}{75} = 80 \]

Step 5: Conclusion

Jackie needs to sell 80 cable packages to make the total pay the same from both companies.

Key Concepts

Commission Calculation

Understanding how to calculate commission is crucial in solving this problem. A commission is an additional payment based on the number of items sold, and it's typically added to a base salary. For example, in the first company's offer, Jackie receives a base salary of \(\$ 14,000\) along with \(\$ 100\) per cable package sold. The formula for her total compensation can be written as: \[ 14000 + 100x \] where \(x\) represents the number of packages sold.
Similarly, in the second company's offer, she gets a base salary of \(\$ 20,000\) plus \(\$ 25\) per package: \[ 20000 + 25x \]Understanding these equations is the first step towards analyzing different salary structures to evaluate offers or compare jobs.

Pay Comparison

Comparing different job offers often requires a detailed understanding of their pay structures. To determine when Jackie’s total compensation would be the same for both offers, we can set the two compensation equations equal to each other:
\[ 14000 + 100x = 20000 + 25x \]
This equation tells us that for some number of packages sold, the total pay from both companies will be equal. By solving this equation, we can find the breakeven point in terms of packages sold. This allows Jackie to make an informed decision about which job offer will benefit her the most under different selling scenarios.

Solving Equations

To find the number of cable packages Jackie needs to sell, we need to solve the equation \[ 14000 + 100x = 20000 + 25x \]
First, we subtract \( 25x \) from both sides: \[ 14000 + 75x = 20000 \]
Next, we subtract \( 14000 \) from both sides: \[ 75x = 6000 \]
Then, we divide both sides by \( 75 \): \[ x = \frac{6000}{75} \]This simplifies to \( x = 80 \).
So, Jackie would need to sell 80 cable packages to make the total pay from both companies the same. Solving equations like this helps in understanding different scenarios and making strategic decisions in real-life situations.