Problem 317 - Elementary Algebra

Question

In the following exercises, solve. Keshad gets paid \(\$ 2,400\) per month plus \(6 \%\) of his sales. His brother earns \(\$ 3,300\) per month. For what amount of total sales will Keshad's monthly pay be higher than his brother's monthly pay?

Step-by-Step Solution

Answer

Keshad's total sales must be more than \ $15000 \ to earn more than his brother.

Step 1: Define Variables

Let Keshad's total sales be represented by the variable \( S \).

Step 2: Write Keshad's Monthly Pay Formula

Keshad's monthly pay consists of his base pay and 6% of his sales. Thus, his monthly pay can be represented as \( 2400 + 0.06S \).

Step 3: Set Up the Inequality

For Keshad's monthly pay to be higher than his brother's monthly pay, we need to set up the following inequality: \( 2400 + 0.06S > 3300 \).

Step 4: Solve for S

First, subtract \( 2400 \) from both sides of the inequality: \[ 0.06S > 3300 - 2400 \]Simplify the equations:\[ 0.06S > 900 \]Then, divide each side by \( 0.06 \): \[ S > \frac{900}{0.06}\]Compute the final value:\[ S > 15000 \]

Key Concepts

inequalities

In the exercise, we need to determine when Keshad's monthly pay will exceed his brother's monthly pay. This is where inequalities come into play. An inequality is a mathematical statement indicating that two values are not equal and can involve the symbols <, >, ≤, or ≥. For this problem, we use the 'greater than' symbol (>) because we want Keshad's salary to be higher than his brother's. During the solution, we set up the inequality as: ul>

\(2400 + 0.06S > 3300\) where \(S\) is the total sales. This inequality helps us understand the condition that needs to be fulfilled to ensure Keshad's earnings surpass his brother's.

variable representation

Variable representation plays a crucial role in solving algebra word problems. Here, we define variable \(S\) to represent Keshad's total sales. This helps simplify the problem and allows us to construct mathematical equations or inequalities. By representing unknown quantities with variables, we make the problem more manageable. In the provided solution, \(S\) is used in the monthly pay formula for Keshad. The formula becomes:

• \(2400 + 0.06S\)

representing his total earnings in terms of his sales. This approach makes it easier to manipulate and solve the equation.

percentage calculations

Percentage calculations are fundamental in many algebra word problems, including the one at hand. Here, Keshad receives \(6\text{\textperthousand}\) of his total sales as part of his monthly pay. To represent this, we convert the percentage to a decimal format:

• \(6\text{\textperthousand} = 0.06\)

This allows us to write Keshad's pay as a formula:

• \(2400 + 0.06S\)

where \(0.06S\) represents the 6% of his sales. Percentage calculations help in expressing these relationships accurately and solving the subsequent algebraic equations or inequalities.

linear equations

Linear equations, which are equations of the first degree, are often used to solve algebra problems involving a single variable. In this exercise, we end up with a linear equation when simplifying the inequality:

• \(0.06S > 900\)

To isolate \(S\), we divide both sides by 0.06, yielding:

• \(S > \frac{900}{0.06}\)

This simplifies to:

• \(S > 15000\)

indicating that Keshad's total sales must exceed $15000 for his monthly pay to be higher than his brother's. This example demonstrates how linear equations help in finding the required variable value to meet certain conditions.