Problem 57 - College Algebra

Question

When hired at a new job selling electronics, you are given two pay options: \(\cdot\) Option \(\mathrm{A} :\) Base salary of \(\$ 20,000\) a year with a commission of 12\(\%\) of your sales \(\cdot\) Option \(\mathrm{B}\) : Base salary of \(\$ 26,000\) a year with a commission of 3\(\%\) of your sales How much electronics would you need to sell for option \(\mathrm{A}\) to produce a larger income?

Step-by-Step Solution

Answer

Sell more than \$66,666.67 in electronics for Option A to be better.

Step 1: Identify Variables for Both Options

Let \( x \) represent the total sales you make in a year. For Option A, the total income \( I_A \) can be calculated as the base salary plus 12% of the sales, so \( I_A = 20000 + 0.12x \). For Option B, the total income \( I_B \) is the base salary plus 3% of the sales, given by \( I_B = 26000 + 0.03x \).

Step 2: Set Up the Inequality

To find the sales \( x \) where Option A produces a larger income than Option B, we need \( I_A > I_B \). Substitute the expressions for \( I_A \) and \( I_B \) into this inequality: \( 20000 + 0.12x > 26000 + 0.03x \).

Step 3: Simplify the Inequality

Subtract \( 0.03x \) from both sides to get \( 20000 + 0.09x > 26000 \). Then subtract \( 20000 \) from both sides: \( 0.09x > 6000 \).

Step 4: Solve for \( x \)

Divide both sides of the inequality by 0.09 to solve for \( x \): \( x > \frac{6000}{0.09} = 66666.67 \).

Step 5: Conclusion

You need to sell more than \( \$66,666.67 \) worth of electronics for Option A to provide a larger income than Option B.

Key Concepts

Inequality solving

Inequality solving is a fundamental concept in algebra that involves determining the set of values for a variable that satisfy an inequality statement. In our exercise, we want to know when Option A's income surpasses that of Option B. To do this, we set up an inequality involving the total incomes of both payment options: \(I_A > I_B\). This translates to the expression \(20000 + 0.12x > 26000 + 0.03x\).

• Begin by simplifying both sides of the inequality.
• Subtract any similar terms on both sides.
• After simplification, solve the inequality for \(x\).

Remember, solving inequalities is similar to solving equations until you multiply or divide by a negative number - which reverses the inequality sign. In this exercise, no reversal is needed since we only divide by positive numbers.

Linear equations

Linear equations are equations of the first order, meaning they involve only the first power of the variable. They appear in the format of \( y = mx + b \), with \(m\) being the slope and \(b\) the y-intercept. However, in our task, each option's total income can be described using a linear equation. For example:- Option A: \(I_A = 20000 + 0.12x\)- Option B: \(I_B = 26000 + 0.03x\) Linear equations are straightforward to handle because they lead directly to the solution without generating complex numbers.

• A quick trick to understanding linear equations is identifying how the slope \(m\) affects the rate of increase or decrease.
• In our example, the slopes (\(0.12\) and \(0.03\)) tell how quickly income rises with each additional sale.

Commission calculation

Commission calculations are essential for anyone working in a sales-driven job. Commissions act as a reward for sales made. In our scenario, Option A provides a 12\% commission rate, while Option B offers a 3\% rate. Calculating the commission is quite simple:- **Formula:** For a commission of a percentage of sales, multiply the total sales by the commission rate. For Option A: \(Commission = 0.12 \, \times \, Sales\)- **Understanding Commissions:**

• Higher commission rates provide more income when sales are high.
• But if the base salary is higher, it might initially outweigh the benefits of a higher commission rate.

By comprehending commission calculations, you better understand how potential earnings fluctuate with sales performance.

Income comparison

Income comparison is a key step in choosing the best payment scheme in work scenarios. This involves calculating and comparing the total annual income from each option—such as base salary plus commissions in our case. Here’s how you make an informed decision:1. Calculate the total income for each option. Use the formula for income, considering base salary and commissions.2. Compare the results by setting an inequality \(I_A > I_B\) to know under what sales conditions one option becomes superior.

• Use income comparison to determine break-even points. In our problem, you find out how much you need to sell to make Option A worthwhile.
• Recognize factors like job stability with higher base pay or potential higher earnings with a better commission rate.

Through careful analysis, you can select the setup that best aligns with your expected performance and financial goals.