Problem 9 - Algebra 2

Question

SHOPPING For Exercises \(7-9,\) use the following information. Gwen wants to buy some used CDs that cost \(\$ 10\) each and some used DVDs that cost \(\$ 13\) each. She has \(\$ 40\) to spend. Can she buy 2 \(\mathrm{CDs}\) and 3 \(\mathrm{DVDs} ?\) Explain.

Step-by-Step Solution

Answer

No, she cannot buy them because they cost \$59, but she only has \$40.

Step 1: Understand the Total Cost of CDs

First, calculate the total cost for the CDs Gwen wants to buy. She plans to buy 2 CDs, each costing \\(10. Multiply the number of CDs by the price per CD: \(2 \times 10 = 20\). So, the total cost for the CDs is \\)20.

Step 2: Understand the Total Cost of DVDs

Next, calculate the total cost for the DVDs Gwen wants to buy. She plans to buy 3 DVDs, each costing \\(13. Multiply the number of DVDs by the price per DVD: \(3 \times 13 = 39\). So, the total cost for the DVDs is \\)39.

Step 3: Calculate the Total Expense

Add the total costs of the CDs and DVDs to find the total expense. That gives us: \(20 + 39 = 59\). Therefore, the total amount Gwen needs to spend is \$59.

Step 4: Compare Total Expense with Available Budget

Gwen has \\(40 to spend. Compare this amount with the total expense calculated in Step 3. \\)59 (total expense) > \$40 (available budget), so Gwen does not have enough money to buy 2 CDs and 3 DVDs.

Key Concepts

Budget Constraint

A budget constraint represents the limit on the spending power when one wants to purchase goods or services, like Gwen in her shopping exercise. It is the total amount of money available to spend, which in this situation, is \\(40. Think of it as a boundary, or a fence, that you cannot cross without breaking your budget.
When assessing a budget constraint, it's essential to calculate whether the total cost of the items fits within that boundary. For Gwen, her budget constraint is clear: she cannot spend more than her \\)40 allowance. This constraint ensures resource allocation is optimized and helps prevent overspending.
A practical tip when working with budget constraints is to list the prices of all items and consider all possibilities, like opting for different quantities or switching items if necessary. Always keep checking to see if various combinations still let you stay within your defined financial boundary.

Cost Calculation

Cost calculation is a step-by-step method used to determine the total expenditure involved in purchasing goods or services. In Gwen's case, determining her total costs involved calculating the price of her desired CDs and DVDs.
Here's how you do it:

• Multiply the number of each product by its unit price.
• Add up the results to find the total spending.

Gwen wants to buy 2 CDs, each costing \\(10. For the CDs, the calculation is: \(2 \times 10 = \\)20\). Next, with DVDs priced at \\(13 each, she plans to buy 3, leading to: \(3 \times 13 = \\)39\). Finally, sum these two results to find the total cost: \(20 + 39 = \$59\).
Remember, accurate calculations are essential to prevent misunderstandings about how much you can spend, especially when dealing with a limited budget.

Comparison of Values

Once you have calculated costs, you'll need to compare the total expenditure with the budget to ensure everything aligns within your limit. This step is crucial to making informed financial decisions.
For Gwen’s example, you compare her total expense of \\(59 with her budget of \\)40. The comparison is expressed as an inequality:

\(59 > 40\), which clearly shows the total spending exceeds the available budget. Such a mismatch highlights that she cannot proceed with buying 2 CDs and 3 DVDs.
Comparing values helps to see potential financial shortfalls early, allowing for re-adjustments in purchasing plans to fit within budget constraints without overspending. It’s important to practice this, making it a habit to ensure financial health is maintained effectively.