Problem 55 - College Algebra

Question

When hired at a new job selling jewelry, you are given two pay options: \(\cdot\) Option A: Base salary of \(\$ 17,000\) a year with a commission of 12\(\%\) of your sales \(\cdot\) Option \(\mathrm{B} :\) Base salary of \(\$ 20,000\) a year with a commission of 5\(\%\) of your sales How much jewelry would you need to sell for option A to produce a larger income?

Step-by-Step Solution

Answer

You need to sell more than $42,857 in jewelry for Option A to be better.

Step 1: Define Variables and Formula for Income

Let the sales amount be represented by \( x \). The income for Option A is given by: \( \text{Income}_A = 17000 + 0.12x \). The income for Option B is: \( \text{Income}_B = 20000 + 0.05x \).

Step 2: Set Up Inequality

To find the sales amount \( x \) at which Option A yields a higher income than Option B, we need to set up the inequality: \( 17000 + 0.12x > 20000 + 0.05x \).

Step 3: Solve the Inequality

Subtract \( 20000 \) from both sides of the inequality: \( 17000 + 0.12x - 20000 > 0.05x \) which simplifies to \( -3000 + 0.12x > 0.05x \). Now, subtract \( 0.05x \) from both sides: \( -3000 + 0.12x - 0.05x > 0 \), simplifying further to \( 0.07x > 3000 \).

Step 4: Find the Sales Amount

Divide both sides by \( 0.07 \) to solve for \( x \): \( x > \frac{3000}{0.07} \). Compute: \( x > 42857.14 \).

Step 5: Interpret the Result

Since we're dealing with sales, \( x \) must be a whole number. Therefore, you need to sell more than \( 42857 \) worth of jewelry for Option A to be more profitable.

Key Concepts

Linear Inequalities

Understanding linear inequalities is the key to solving problems where one amount should be greater, less, or equal to another amount. In this situation, we are figuring out when one salary with a commission will be bigger than another. To represent this situation mathematically, we use linear inequalities.

In our jewelry seller example, we are finding the point where Option A's income exceeds Option B's income. We set up an inequality equation from the two income formulas:

• Option A: \( 17000 + 0.12x \)
• Option B: \( 20000 + 0.05x \)

To show Option A's income is greater, we write: \[17000 + 0.12x > 20000 + 0.05x\] This inequality helps us determine how much jewelry needs to be sold to make Option A more lucrative. Linear inequalities like this are useful for comparing values that change based on a variable, like sales in this case.

Variable Substitution

Variable substitution is a powerful technique in algebra that simplifies problems by letting us replace parts of an expression with a variable, often denoted by a letter such as \( x \). This makes complicated statements more manageable and allows us to solve equations step-by-step.

In our exercise, we defined \( x \) as the total sales amount. By using this variable, we could express both incomes in terms of \( x \), like so:

• Option A: \( 17000 + 0.12x \)
• Option B: \( 20000 + 0.05x \)

This helped convert the word problem into math equations, making it easier to set up an inequality. Without variable substitution, the problem would be harder to calculate as we scale up the sales. It's like a shortcut to see how changing one part (the sales) affects the whole (the income).

Income Calculation

Income calculation in this context is about determining how much money you would make under each pay option, based on sales. A commission is an additional payment made as a percentage of sales; it’s crucial to factor into income predictions.

Here’s how we calculated income for each option:

• Option A offers a base salary of \\(17,000 plus 12% of sales as a commission. So, the income formula becomes: \( 17000 + 0.12x \).
• Option B gives a higher base salary of \\)20,000 but with a smaller commission at 5%, leading to the formula: \( 20000 + 0.05x \).

By filling in these formulas with the sales amount \( x \), you can see how both the base salary and commission impact total earnings. This form of income calculation helps evaluate different compensation packages and make informed financial decisions.

Commission Calculation

Understanding how commission works is essential in jobs where part of the income depends on sales. A commission rewards you with a percentage of your sales, encouraging higher sales performance.

For Option A, the commission is 12% of the sales amount \( x \). So, the commission is calculated as: \( 0.12x \). This value adds to the base salary to form the total income.

In Option B, the commission rate is lower, at 5% of \( x \). The commission contribution here is \( 0.05x \). This smaller percentage means the employee earns less from the same sales compared to Option A.

Being able to calculate and understand commissions is vital for comparing job offers, setting sales goals, and maximizing your earnings. Always remember, the percentage rate and the actual sales amount both play significant roles in determining overall commission.