Question

Solve. After deductions, Luther's net pay is \(\$ 1850.40 .\) If the total of the deductions was \(28 \%\) of his gross monthly salary, what was his gross monthly salary?

Short answer

Answer: Luther's gross monthly salary is $2570.

Step-by-step solution

Set up the equation

Let x be the gross monthly salary. Deductions amount to \(28\%\) of x, which is \(0.28x\). Then, Luther's net pay is his gross salary minus the deductions. Therefore, we can write the equation: $$ x - 0.28x = 1850.40 $$

Simplify the equation

Combine the x terms in the equation: $$ x - 0.28x = (1 - 0.28)x = 0.72x $$ So the simplified equation is: $$ 0.72x = 1850.40 $$

Solve for x

Now, let's divide both sides of the equation by \(0.72\) to solve for x: $$ x = \frac{1850.40}{0.72} $$ Now use a calculator to find the value of x: $$ x = 2570 $$

Interpret the result

The value of x is 2570, which represents Luther's gross monthly salary. Therefore, his gross monthly salary is \(\$2570\).

Key concepts

Percentages

In prealgebra, percentages help us express numbers as a part of 100. This is especially useful when dealing with financial calculations, such as discounts, interest rates, or salary deductions. In the given problem, deductions from Luther's salary are expressed as a percentage of his gross salary. To understand this better, imagine percentages as parts of a whole:

• 100% means the whole thing. In this case, it's Luther's total (or gross) salary.
• 28% means 28 out of every 100 units of his salary go to deductions.
• The remaining percentage represents what he takes home after deductions.

When you see a percentage like 28%, you can convert it to a decimal by dividing by 100, so it becomes 0.28. This decimal helps calculate actual deductions based on a salary or any amount in mathematical equations.
Understanding how to work with percentages is key in solving problems where you need to find out parts deducted from totals, as in Luther's case.

Equations

Equations are mathematical statements that assert the equality of two expressions. Think of an equation as a balance scale—it helps you find the unknown value, keeping everything even on both sides. In our exercise, the equation balances Luther's gross salary against his net pay after deductions.
Here is how it works:

• The left side of the equation, \( x - 0.28x \), represents the gross salary minus deductions.
• The right side, \( 1850.40 \), is his net salary after those deductions.

By solving the equation, you essentially "balance" it to find the value of \( x \). This involves combining like terms and performing inverse operations to isolate \( x \):

• Combine \( x \) terms: \( 1x - 0.28x = 0.72x \).
• Solve for \( x \) by dividing both sides by 0.72.

Once you perform these steps, you determine the gross salary amount. Understanding equations is crucial for unraveling similar prealgebra problems.

Gross and Net Salary Calculations

Understanding the difference between gross and net salary is a practical skill in managing personal finances. Gross salary is the amount earned before any deductions, such as taxes or benefits. In Luther's scenario, he starts with a gross salary which gets reduced by deductions.
Here's a breakdown of the process:

• Gross Salary: The total earnings before deductions (unknown value \( x \) in the equation).
• Deductions: Specific deductions (28% of the gross salary in this case), calculated as \( 0.28x \).
• Net Salary: The remaining salary after deductions, known here as \( 1850.40 \).

The relationship between these values can help you determine either the gross salary or the deductions if you know the other values like the net salary and the deduction percentage. By performing calculations as shown in the exercise, you unveil the value of the earnings before any deductions, providing clear insight into financial exposition.