Problem 66 - Algebra 2

Question

Desiree works in a photography studio and makes a commission of \(\$ 8\) per photo package she sels. On Tuesday, she sold 3 more packages than she sold on Monday. For the two days, Victoria earned \(\$ 264 .\) How many photo packages did she sell on these two days?

Step-by-Step Solution

Answer

Desiree sold 33 photo packages in total.

Step 1: Define Variables

Let's define variables for the photo packages sold. Let \( p \) be the number of photo packages Desiree sold on Monday. Since she sold 3 more on Tuesday, she sold \( p + 3 \) packages on Tuesday.

Step 2: Write the Equation for Total Earnings

Desiree earns an \( \$8 \) commission per package. Her earnings from Monday are \( 8p \), and from Tuesday, they are \( 8(p + 3) \). The total earnings for the two days are \( 264 \). Thus, the equation is:\[ 8p + 8(p + 3) = 264 \]

Step 3: Simplify and Solve the Equation

Simplify the equation:\[ 8p + 8p + 24 = 264 \]Combine like terms:\[ 16p + 24 = 264 \]Subtract 24 from both sides:\[ 16p = 240 \]Divide by 16:\[ p = 15 \]

Step 4: Calculate the Total Packages Sold

Desiree sold \( 15 \) packages on Monday. She sold \( 15 + 3 = 18 \) packages on Tuesday. Therefore, the total number of packages sold is:\[ 15 + 18 = 33 \]

Key Concepts

Understanding Variables in Math Problems

Variables are essential in algebra as they allow us to represent unknown quantities. In this problem, the variable \( p \) is used to symbolize the number of photo packages Desiree sold on Monday. Variables are like placeholders and can stand in for numbers we want to find. Here's the cool part: once you define a variable, you can use it to write equations that model real-world situations.

• Variables help simplify complex problems by turning them into a mathematical expression.
• They provide a way to compute solutions to problems without having all values initially known.

Understanding how variables work enables you to solve equations, just like the number of packages sold here.

Commission Calculation Made Simple

Commission is a payment based on the sales an employee makes. Desiree earns a commission of \( \$8 \) per photo package. Calculating commissions involves multiplying the number of items sold by the commission rate.

• For example, if she sells 15 packages, her earnings would be \( 8 \times 15 = 120 \) dollars.
• The formula is: Total commission = Number of Items Sold \( \times \) Rate per Item.

This straightforward calculation helps businesses determine how much to pay their sales staff based on their performance. Knowing how to calculate commission allows you to manage earnings and understand income better.

Simplifying Equations for Clarity

Simplifying an equation involves combining like terms and making the equation easier to solve. When you simplify, you're essentially making the math cleaner and more approachable.

• In Desiree’s case, the initial equation was: \( 8p + 8(p + 3) = 264 \).
• After expanding and combining like terms, we arrived at: \( 16p + 24 = 264 \).

Simplifying is vital because it reduces potential errors in calculations and makes solving the equation easier. Think of it like tidying up your workspace before starting a project—it just makes everything smoother!

Efficiently Solving Equations

Solving equations means finding the value of the variable that makes the equation true. Here, we found \( p \) by isolating it.

• First, subtract any constants from both sides of the equation. For our example: \( 16p = 240 \).
• Then, divide both sides by the coefficient of the variable to solve for the variable: \( p = 15 \).

This two-step process—subtract and divide—helps solve linear equations effectively. Being comfortable with solving equations allows you to tackle a wide variety of math problems, from simple commissions to complex algebraic expressions.