Problem 55 - College Algebra

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, You: $19.95, Other roommate: $22.05.

Step 1: Define Variables

Let Sarah's groceries cost \( S \), your groceries cost \( Y \), and your other roommate's groceries cost \( O \).

Step 2: Set Up Equations

Using the information given, we can set up the equations: - Total bill: \( S + Y + O = 82 \)- Your groceries: \( Y = \frac{1}{2}S - 0.05 \)- Other roommate's groceries: \( O = Y + 2.10 \).

Step 3: Substitute for Y and O

Substitute the expressions for \( Y \) and \( O \) in terms of \( S \) from the second step into the first equation: \( S + \frac{1}{2}S - 0.05 + \left( \frac{1}{2}S - 0.05 + 2.10 \right) = 82 \).

Step 4: Simplify the Equation

Simplify the equation: \( S + \frac{1}{2}S - 0.05 + \frac{1}{2}S - 0.05 + 2.10 = 82 \). Combine like terms: \( 2S + 2.00 = 82 \).

Step 5: Solve for S

Rearrange the equation to solve for \( S \):\( 2S = 80 \) Thus, \( S = 40 \).

Step 6: Find Your Groceries (Y)

Using \( S = 40 \) in the equation for \( Y \):\( Y = \frac{1}{2}(40) - 0.05 = 20 - 0.05 = 19.95 \).

Step 7: Find Other Roommate's Groceries (O)

Using \( Y = 19.95 \) in the equation for \( O \):\( O = 19.95 + 2.10 = 22.05 \).

Key Concepts

Defining Variables

Whenever you encounter a problem that involves unknown quantities, it's crucial to clearly define your variables.This helps simplify and organize the information in the problem. In our example, the task is to find the cost of groceries for each roommate.
You will use variables to represent these costs:

• Let \( S \) represent the cost of Sarah's groceries.
• Let \( Y \) represent the cost of your groceries.
• Let \( O \) represent the cost of your other roommate's groceries.

By assigning these variables, we make it easier to see the relationships between different parts of the problem. Being consistent with variable names is key to minimizing confusion as you proceed with solving the exercise.

Setting Up Equations

After defining the variables, the next step is to translate the word problem into mathematical equations.This involves using the relationships described in the problem.

• The total cost of groceries was known: \( S + Y + O = 82 \).
• Your groceries were \( \\(0.05 \) cheaper than half of Sarah’s groceries: \( Y = \frac{1}{2}S - 0.05 \).
• The other roommate’s groceries cost \\)2.10 more than yours: \( O = Y + 2.10 \).

Turning relationships into equations is like creating a blueprint for solving the problem.Everything you need to find out is now mathematically represented.

Substitution Method

The substitution method is a technique used to solve systems of equations.After setting up your equations, you can use substitution to solve for one of the variables.Start by using one equation to express one variable in terms of another.In this problem, use the equations for \( Y \) and \( O \) to substitute into the total cost equation.Replace \( Y \) and \( O \) in the total cost equation:\[S + \frac{1}{2}S - 0.05 + \left(\frac{1}{2}S - 0.05 + 2.10\right) = 82\]This allows you to reduce the number of variables in one equation, making it easier to solve.

Simplification

Simplification in algebra involves combining like terms and clearing out any unnecessary elements from an equation.It's an important step to make solving the equations easier and more straightforward.In this problem, simplify the substitution equation from:\[S + \frac{1}{2}S - 0.05 + \frac{1}{2}S - 0.05 + 2.10 = 82\]Combine like terms:\[2S + 2.00 = 82\]Simplification helps to see the direct relationships and to better manipulate the equation to find the solution.

Solving Linear Equations

The final step is to solve the simplified linear equation for the variable of interest. Once simplified, our equation is: \[2S + 2.00 = 82\]The next move is to isolate \( S \).Subtract \( 2.00 \) from both sides to get:\[2S = 80\]Divide both sides by \( 2 \) to solve for \( S \):\[S = 40\]Once \( S \) is found, you can use it to find other variable values:

• For \( Y \), substitute \( S = 40 \) into \( Y = \frac{1}{2}(40) - 0.05 = 19.95 \).
• For \( O \), use \( Y = 19.95 \) in \( O = 19.95 + 2.10 = 22.05 \).

By following these steps, each variable's value is revealed, solving the whole problem.