Problem 55 - College Algebra with Corequisite Support

Question

Your roommate, Sarah, offered to buy groceries for you and your other roommate. The total bill was $$\$ 82$$. She forgot to save the individual receipts but remembered that your groceries were $$\$ 0.05$$ cheaper than half of her groceries, and that your other roommate's groceries were $$\$ 2.10$$ more than your groceries. How much was each of your share of the groceries?

Step-by-Step Solution

Answer

Sarah: $40.025, You: $19.9625, Other Roommate: $22.0625.

Step 1: Understanding the Problem

We need to find the cost of groceries for you, Sarah, and your other roommate. We know the total cost is $82. Additionally, your groceries were $0.05 cheaper than half of Sarah's groceries, and your other roommate's groceries were $2.10 more expensive than yours.

Step 2: Setting Up Variables

Let Sarah's groceries be represented as \( S \), your groceries as \( Y \), and your other roommate's groceries as \( R \). We know \( S + Y + R = 82 \).

Step 3: Creating Equations from Descriptions

From the problem statement, the following relationships are given: 1. \( Y = \frac{S}{2} - 0.05 \).2. \( R = Y + 2.10 \).

Step 4: Substitute for Your Groceries

Substitute the expression for \( Y \) into the equation for \( R \): \( R = (\frac{S}{2} - 0.05) + 2.10 \), which simplifies to \( R = \frac{S}{2} + 2.05 \).

Step 5: Form a Single Equation

Substitute \( Y = \frac{S}{2} - 0.05 \) and \( R = \frac{S}{2} + 2.05 \) into the total cost equation: \[ S + \left(\frac{S}{2} - 0.05\right) + \left(\frac{S}{2} + 2.05\right) = 82 \].

Step 6: Simplify and Solve for Sarah's Share

Simplify the equation: \[ S + \frac{S}{2} - 0.05 + \frac{S}{2} + 2.05 = 82 \] becomes \[ 2S + 2 - 0.05 = 82 \]\[ 2S + 1.95 = 82 \]Subtract 1.95 from both sides:\[ 2S = 80.05 \]Solve for \( S \):\[ S = 40.025 \].

Step 7: Find Your Share

Substitute \( S = 40.025 \) into the equation for your groceries:\[ Y = \frac{40.025}{2} - 0.05 \]\[ Y = 20.0125 - 0.05 \]\[ Y = 19.9625 \].

Step 8: Determine Other Roommate's Share

Use the value of \( Y \) to find \( R \):\[ R = Y + 2.10 \]\[ R = 19.9625 + 2.10 \]\[ R = 22.0625 \].

Step 9: Verification of Total

Sum up all the shares to verify:\[ S + Y + R = 40.025 + 19.9625 + 22.0625 = 82 \]. The equation holds, so the solution is verified.

Key Concepts

Understanding Variables

In mathematics, a variable acts as a placeholder or symbol that represents an unknown value. Rather than working with unknown constant numbers, a variable lets us manipulate algebraic expressions and equations to find solutions.

In the context of our problem, variables are used to represent each person's share of the grocery bill:

• Let Sarah's groceries be represented as \( S \).
• Your groceries are represented as \( Y \).
• Your other roommate's groceries are represented as \( R \).

By doing so, we create a way to express complex relationships and derive manageable equations that describe the problem. The goal is to use these equations to ultimately solve for the values of these variables. Each variable represents a specific quantity we want to know. Defining variables clearly is a critical first step in solving algebraic equations.

Structure of Linear Equations

A linear equation is an algebraic equation where each term is either a constant or a product of a constant and a single variable. Linear equations are straight lines when graphed. They help describe relationships where change is consistent.

The problem gives us equations based on the relationships:

• Your groceries are \( Y = \frac{S}{2} - 0.05 \).
• Your other roommate's groceries are \( R = Y + 2.10 \).
• The total cost relationship is \( S + Y + R = 82 \).

These are all linear equations because they model a direct proportionality. They are written in a form that can easily be solved using algebraic techniques. By substituting one linear equation into another, we can simplify complex expressions and make solving for our desired values more straightforward.

Approach to Problem-Solving

Successful problem-solving with algebraic equations often involves several strategic steps. Here’s how we approached solving this particular problem:

• Understand the problem: Identify what you are asked to find and take note of the information given.
• Define variables: Use variables to represent unknown quantities, establishing clear definitions like we did with \( S \), \( Y \), and \( R \).
• Formulate equations: Use the relationships described in the problem to create equations that connect your variables.
• Solve the equations: Substitute and manipulate these equations to isolate each variable and find its value. This often includes eliminating variables and simplifying expressions.
• Verify the solution: Always double-check that your solution makes sense in the context of the problem. For instance, confirm that the sum of shares equals \( 82 \).

By carefully following these problem-solving steps, complex problems become easier to manage, leading to accurate and meaningful solutions.