Question

One used-car salesperson receives a commission of \(\$ 200\) plus 4 percent of \(\$ 1,000\) less than the car's final sale price. Another car salesperson earns a straight commission of 6 percent of the car's final sale price. What is the final sale price of a car if both salespeople would earn the same commission for selling it? A \(\$ 5,000\) B \(\$ 6,000\) C \(\$ 8,000\) D \(\$ 10,000\) E \(\$ 12,000\)

Short answer

The final sale price of the car is \( \$ 8,000 \).

Step-by-step solution

Define Variables

Let the final sale price of the car be given by \( P \).

Calculate the First Salesperson's Commission

The first salesperson receives a commission of \( \$ 200 \) plus 4 percent of \( \$ 1,000 \) less than the car's final sale price. This can be written as: \ \( \text{First Salesperson's Commission} = 200 + 0.04(P - 1,000) \)

Calculate the Second Salesperson's Commission

The second salesperson earns a straight commission of 6 percent of the car's final sale price. This can be written as: \ \( \text{Second Salesperson's Commission} = 0.06P \)

Set the Commissions Equal to Each Other

Since both salespeople would earn the same commission for selling the car, set the two expressions equal to each other: \ \( 200 + 0.04(P - 1,000) = 0.06P \)

Solve for P

Isolate \( P \) by first distributing the 0.04: \ \( 200 + 0.04P - 40 = 0.06P \) \ Combine like terms: \ \( 160 + 0.04P = 0.06P \) \ Subtract \( 0.04P \) from both sides: \ \( 160 = 0.02P \) \ Solve for \( P \): \ \( P = \frac{160}{0.02} = 8,000 \)

Key concepts

Commission Calculations

Understanding commission calculations is essential for solving this GMAT math word problem. Commission refers to the payment made to a salesperson based on the value of sales they generate. In this problem, we have two different commission structures:

The first salesperson receives a fixed amount plus a variable percentage of the sale price. Specifically, they earn a commission of \(\text{\$200} + 4\% \text{ of (sale price - \$1000)}\). This means that for every sold car, they not only get a fixed payment of \(\text{\$200}\) but also a percentage based on a slightly reduced sale price.

The second salesperson has a simpler structure; they earn a straight 6 percent of the final sale price. This means that their entire commission is directly tied to the sale price without any additional fixed component.

Recognizing these different structures is the first step in setting up the equation correctly to solve the problem.

Equation Solving

Solving equations is a key part of this problem. To find the sale price where both salespeople earn the same commission, you must set up and solve an equation involving their commissions.

First, define the variable representing the unknown quantity: the final sale price of the car \(P\). Then, write expressions for the commissions of both salespeople, as shown in the problem's solution. The next step is to set these two expressions equal because we know they earn the same commission for selling the car.

For the first salesperson, the commission equation is: \(\text{Commission} = 200 + 0.04(P - 1000)\), and for the second, it's: \(\text{Commission} = 0.06P\). Setting these equal gives us: \(200 + 0.04(P - 1000) = 0.06P\).

From here, distribute and combine like terms to isolate \(P\), eventually simplifying to solve for \(P = 8000\). This step-by-step process highlights how to transform word problems into solvable algebraic equations.

Percentage Operations

Understanding percentages is crucial for solving this type of GMAT quantitative problem. Percentages describe a number as a fraction of 100. In commission calculations, we often use percentages to determine parts of total sales.

In our problem, the first salesperson earns a percentage of \(\text{P - 1000}\), given as \(0.04(P - 1000)\), which equates to 4%. For calculations, convert percentages to decimal form by dividing by 100 (e.g., 4% becomes 0.04).

The second salesperson earns 6% of the sale price directly: \(0.06P\). Performing operations with these percentages involves multiplication and addition/subtraction of these decimal values.

These operations are essential for setting up the equations correctly. For instance, you need to understand how to calculate \(4\%\) of \(P - 1000\) and how it impacts the total commission the first salesperson earns, compared to the simple \(6\%\) calculation for the second salesperson.

GMAT Quantitative Section

This problem falls under the GMAT quantitative section, which assesses your mathematical skills through various word problems, algebra, and arithmetic.

The GMAT quantitative section has two main types of questions: problem-solving and data sufficiency. Problem-solving, like the provided exercise, requires applying math skills to find solutions.

Success in these questions depends on several factors:

• Understanding the problem.
• Defining variables.
• Writing equations.
• Performing accurate calculations.

The provided problem is an excellent example requiring no advanced mathematics but solid knowledge and application of basic arithmetic, percentages, and algebra.

Getting familiar with such problems prepares you for the range of questions you might encounter on the actual exam, helping you develop a strategy for approaching and solving them efficiently.