Question

To rent a community center for a school prom costs a base fee of $$\$ 550$$ and an additional fee of $$\$ 6$$ per person in attendance. The organizers expected 350 attendees, but in fact a total of 377 people attended. How much greater was the actual fee than the fee the organizers had expected to pay? A. $$\$ 162$$ B. $$\$ 712$$ C. $$\$ 2,262$$ D. $$\$ 2,812$$

Short answer

The short answer is: 1. Expected fee = \( \$550 + (\$6 \times 350) = \$2,650 \) 2. Actual fee = \( \$550 + (\$6 \times 377) = \$2,812 \) 3. Difference = Actual fee - Expected fee = \( \$2,812 - \$2,650 = \$162 \) So, the correct answer is (A) \( \$162 \).

Step-by-step solution

Calculate the expected fee.

First, we need to calculate the fee that the organizers had expected to pay. To do this, we use the formula: Expected fee = Base fee + (Additional fee per person × Expected attendees) Here, the base fee is $$\$550$$, the additional fee per person is $$\$6$$, and the expected attendees were 350. So, Expected fee = $$\$550 + (\$6 × 350)$$

Calculate the actual fee.

Now, we need to calculate the actual fee, which the organizers paid. To do this, we again use the formula: Actual fee = Base fee + (Additional fee per person × Actual attendees) Here, the base fee is $$\$550$$, the additional fee per person is $$\$6$$, and the actual attendees were 377. So, Actual fee = $$\$550 + (\$6 × 377)$$

Find the difference between the expected fee and the actual fee.

Finally, we need to find the difference between the expected fee and the actual fee, using the formula: Difference = Actual fee - Expected fee Calculate based on the expected fee and actual fee found in steps 1 and 2.

Compare the calculated difference to the given answer choices.

After finding the difference, we'll compare our result to the answer choices (A) $$\$162$$, (B) $$\$712$$, (C) $$\$2,262$$, and (D) $$\$2,812$$ to find the correct answer.

Key concepts

Mathematical Reasoning

Understanding mathematical reasoning is pivotal in solving not just GED math problems but any mathematical problem effectively. It involves the ability to think logically about the relationships between numbers and how they apply to real-world scenarios. In our rental community center problem, mathematical reasoning is applied by interpreting the given information - the base fee, additional fee per person, and the number of attendees - to establish relevant equations.|n|nFor example, recognizing that the total cost is a combination of a fixed base fee and a variable component that depends on the number of people ultimately leads to a linear equation. Additionally, the reasoning extends to predicting outcomes, as seen when calculating the difference in costs between expected and actual attendees. This problem-solving skill is essential for breaking down complex word problems into manageable steps that lead to the correct solution.|n|nImproving mathematical reasoning can require practice in identifying patterns, understanding the significance of numbers, and seeing the structure of problems. Employing strategies such as drawing diagrams, making lists, or acting out the problem could further enhance reasoning skills.

Word Problem Solving

Word problem solving is a crucial skill that combines reading comprehension with mathematical operations. It demands that students first understand the scenario presented and then translate it into mathematic expressions or equations that can be solved. In the GED problem we face, the student must decipher the costs associated with renting a venue and then relate them to the number of attendees, a common real-life task.|n|nStrategies for Solving Word Problems:

• Identify the key information given and what is being asked for.
• Determine what operations will yield the correct relationship and outcome. In our example, multiplication is used to calculate the additional fees, and subtraction finds the difference between the expected and actual fees.
• Translate the words into mathematical symbols and numbers, ensuring that you maintain the correct order of operations.
• Check the answer back in the context of the problem to ensure it makes sense.

|n|nBy becoming proficient in word problem solving, students bolster their ability to tackle a wide range of questions, linking the language of the problem to the math needed for its solution. Repeated practice and exposure to various types of word problems are highly beneficial in honing this competency.

Algebraic Expressions

Algebraic expressions are mathematical phrases that represent numbers and operations without showing the exact values of those numbers. In GED math problems, understanding how to create and manipulate these expressions is key. Referring to our prom rental scenario, two algebraic expressions were constructed to represent the expected and actual fees.|n|nHere's a breakdown of how to approach these algebraic expressions:

• The base fee, which does not change with the number of attendees, is represented as a constant, in this case, $550.
• The variable component—a fee per person—is an unknown that can be represented by a variable. However, here we have specific quantities (expected and actual attendees), so we directly include these known values within our expression to calculate the total cost.
• The total cost expression combines the fixed and variable components with addition: Total cost = Base fee + (Additional fee per person × Number of attendees).

|n|nDeveloping skills in algebraic expressions involves recognizing and translating real-world scenarios into mathematical form, choosing the correct operations, and simplifying expressions. While the exercise provided uses concrete numbers, being comfortable with variables is also essential as the complexity of algebra problems increases. Regular practice in simplifying and evaluating expressions is crucial to mathematical competence.