Question

Translate to an equation and solve. Rosita sells real estate and receives a commission of \(\$ 750 .\) If her commission rate is \(6 \%\) of the seller's net, what is the seller's net?

Short answer

Answer: The seller's net is $12,500.

Step-by-step solution

Declare the variables

Let's use the symbol S to represent the seller's net.

Write down the equation

Since Rosita's commission is 6% of the seller's net, we can write this as an equation: \(Commission = 6\% \times Seller's \; net\) So, we have the equation: \(750 = 0.06 \times S\)

Solve the equation

To find the seller's net (S), we'll divide both sides of the equation by 0.06: \(S = \frac{750}{0.06}\)

Calculate the result

Now, we can calculate the seller's net: \(S = \frac{750}{0.06} = 12,500\) So, the seller's net is \(\$12,500\).

Key concepts

Equation Translation

Understanding how to translate word problems into mathematical equations is a fundamental skill in prealgebra. In the given exercise, the problem involves real-life scenarios, which is common in percent problems where establishing the relevant equation is crucial. The first step is identifying the key quantities and their relationships. Here, Rosita’s commission and the seller's net are the two primary quantities. The commission is described as a percentage (6%) of the seller's net. This percentage is then converted into a decimal (0.06) to create a mathematical representation of the situation. By translating '6% of the seller's net' into an equation, we formulate the relation as
\[ 750 = 0.06 \times S \]
where S represents the seller's net. This simple equation serves as the backbone for solving the entire problem.

Commission Calculation

Commission calculation is a practical application of percentage mathematics, often involving sales or brokerage jobs. In the commission problem at hand, we are dealing with a direct proportionality where the commission is a fixed percentage of an amount. To calculate the commission, we first establish the rate (in this case, 6%) and translate it into a decimal form for easier multiplication. Here’s the key:
\[ Commission = Rate \times Base \]
The 'rate' is the commission rate, while the 'base' is the amount the commission is calculated on—in this exercise, the seller's net. Understanding this relationship, we can solve for the unknown when provided with the other two variables, as demonstrated in the solution steps.

Solving Linear Equations

Solving linear equations is a vital skill in algebra. Linear equations involve variables that are raised to the power of one, like we see in the equation for Rosita’s commission. The solution process typically involves isolating the variable on one side of the equation. Here, we want to solve for S (the seller's net), so we must get S alone on one side. This involves dividing both sides of the equation by the commission rate in decimal form (0.06). Here is the performed operation:
\[ S = \frac{750}{0.06} \]
By performing this operation correctly, you obtain the value of the seller's net. Clear and logical steps to isolate the variable will lead to the accurate solution of linear equations like this one.

Percentage Mathematics

Percentage mathematics is all about understanding parts per hundred, an essential concept for dealing with various real-world situations, including discounts, interest rates, and commission problems like the one at hand. A percentage is a dimensionless ratio, which means that it describes the relationship between two quantities without the need for specific units. In this case, '6%' needs to be converted into a decimal for calculations, which is done by dividing by 100, yielding 0.06. This conversion is critical because it allows us to perform multiplications and divisions in equations. Remember:
\[ Percent \; in \; Decimal = \frac{Percent}{100} \]
Once a percent is converted into a decimal, it can be used to find parts of the whole in problems involving percentage calculations, which is exactly what we do when finding Rosita’s commission from the seller's net.