Problem 13 - Prealgebra

Question

Commission and Purchase Price Suppose a salesperson gets a commission rate of \(12 \%\) on the lawnmowers she sells. If the commission on one of the mowers is $$ 24$ what is the purchase price of the lawnmower?

Step-by-Step Solution

Answer

The purchase price of the lawnmower is $200.

Step 1: Understand the Problem

We are given a commission of $24 earned by a salesperson who has a commission rate of 12%. We need to find the purchase price of the lawnmower which results in this commission.

Step 2: Identify the Relationship

The commission earned is a percentage of the purchase price. The relationship can be expressed as: \( \text{Commission} = \text{Commission Rate} \times \text{Purchase Price} \).

Step 3: Set Up the Equation

Using the given information, set up the equation: \( 24 = 0.12 \times \text{Purchase Price} \).

Step 4: Solve for Purchase Price

To find the purchase price, we need to isolate it in the equation. Divide both sides by 0.12: \( \text{Purchase Price} = \frac{24}{0.12} \).

Step 5: Calculate the Purchase Price

Perform the division: \( \text{Purchase Price} = 200 \). Thus, the purchase price of the lawnmower is $200.

Key Concepts

Understanding Percentage Problems

Percentage problems are a common occurrence in everyday math, especially in financial contexts like commission calculations. They help in determining how much a certain percentage of a number is. In our exercise, we are given that a salesperson earns a commission of 12% on the sale of lawnmowers. This means that 12% of the sale price of the lawnmower is given to the salesperson as their earnings.

To handle percentage calculations, remember that "percent" means per hundred. So, 12% can be expressed as the fraction \( \frac{12}{100} \) or the decimal 0.12. This conversion is vital for calculating the commission when you have the purchase price, or vice versa.

In the context of our problem, understanding that 12% of the purchase price equals the commission ($24 in this case) sets the stage for solving the equation. This approach involves expressing percentages as decimals to perform multiplication or division as needed.

Solving Equations for Commission Calculations

Solving equations is a systematic way to find an unknown value using given data. In our exercise, the unknown value is the purchase price of the lawnmower, and we are given the commission earned and the commission rate. The relationship between these is expressed by a simple equation: \( \text{Commission} = \text{Commission Rate} \times \text{Purchase Price} \).

Given: Commission = \(24 and Commission Rate = 12%
To find the Purchase Price, you set up the equation as:

• \( 24 = 0.12 \times \text{Purchase Price} \)

This equation implies that 0.12 times the purchase price equals 24. To isolate the purchase price, divide both sides of the equation by 0.12, ending up with:

• \( \text{Purchase Price} = \frac{24}{0.12} \)

Through division, you'll find that the purchase price is \)200. Breaking down these equations step-by-step helps ensure you find the correct solution without mistakes.

Determining the Purchase Price

The purchase price is an important figure in calculating commissions, as it is the sale amount before any deduction or addition. To determine the purchase price in our scenario, we used the relationship between the commission rate and the amount of commission earned.

Knowing that the commission is a percentage of the purchase price is crucial. It allows us to rearrange our equation to solve for the purchase price. By dividing the given commission (\(24) by the commission rate expressed as a decimal (0.12), we came to a solution:

• \( \text{Purchase Price} = \frac{24}{0.12} = 200 \)

This final division tells us that the lawnmower costs \)200. Thus, understanding how to manipulate equations and percentages allows us to find key information, like the original purchase price, that affects overall financial calculations.