Problem 318 - Elementary Algebra

Question

In the following exercises, solve. Kimuyen needs to earn \(\$ 4,150\) per month in order to pay all her expenses. Her job pays her \(\$ 3,475\) per month plus \(4 \%\) of her total sales. What is the minimum Kimuyen's total sales must be in order for her to pay all her expenses?

Step-by-Step Solution

Answer

Kimuyen's total sales must be at least \(\$ 16,875\).

Step 1: Identify the Known Values

Kimuyen needs to earn \(\$ 4,150\) per month. Her job pays her \(\$ 3,475\) per month plus \(4 \%\) of her total sales.

Step 2: Set Up the Equation for Total Earnings

Let \(S\) be the total sales Kimuyen needs to make. Her total earnings can be calculated as \(\text{Earnings} = 3,475 + 0.04S\).

Step 3: Set Up the Inequality

Kimuyen's total earnings need to be at least \(\$ 4,150\). Therefore, we set up the inequality: \(3,475 + 0.04S \geq 4,150\).

Step 4: Solve the Inequality

Subtract \(3,475\) from both sides to isolate the term with \(S\): \(0.04S \geq 675\).

Step 5: Isolate the Variable S

Divide both sides by \(0.04\) to solve for \(S\): \(S \geq 675 / 0.04 = 16,875\).

Key Concepts

solving inequalities

When solving inequalities, the goal is to isolate the variable on one side of the inequality sign, just as you would in an equality (equation).
Here are the basics you need to understand:

• Isolate the variable: Perform operations on both sides of the inequality to get the variable by itself.
• Keep the inequality balanced: Whatever you do to one side, you must do to the other, just like in an equation.
• Direction of the Inequality: If you multiply or divide both sides by a negative number, reverse the direction of the inequality sign.

In Kimuyen's case, we began with the inequality: \[3,475 + 0.04S \geq \4,150\]
The first step was to subtract \$3,475\ from both sides:
\[0.04S \geq \4,150 \- 3,475\]
Then, solve for \$S\ by dividing both sides by \$0.04\:
\[S \geq 675 / 0.04 \approx \ 16,875\]
This tells us that Kimuyen’s total sales must be at least \$16,875\ per month to meet her financial goal.

percentage calculations

Percentage calculations help determine how much one quantity is in relation to another. Here is a simple way to understand them:

• Percentage of a Number: This is found by multiplying the number by the percentage expressed as a decimal.
• For example, to find 4% of \$S\, multiply \$S\ by \$0.04\.
• Simplifying percentages helps in understanding contributions to total values.

In Kimuyen's problem, she earns 4% of her total sales. If her total sales are \$S\, then her earnings from sales are:
\[0.04S\]
We used this to set up the initial equation:
\[3,475 + 0.04S\]
This value represented her total earnings, which needed to meet or exceed \$4,150\.

linear equations

Linear equations represent relationships where the variable is raised to the power of one. They form straight lines when graphed and have a general format:
\[ax + b = c\]
In this format, \$x\ is the variable, \$a\ is the coefficient, \$b\ is the constant term, and \$c\ is the result.
For Kimuyen, we set up a linear equation to reflect her total earnings:
\[3,475 + 0.04S = 4,150\]
Then, rearrange to isolate \$S\:
Subtract \$3,475\ from both sides:
\[0.04S = 675\]
Finally, solve for \$S\ by dividing both sides by \$0.04\:
\[S = 16,875\]
This process of forming and solving the linear equation helped determine the minimum sales required for Kimuyen to meet her expense needs.