Question

A museum offers a 2 -hour guided group tour. For groups with fewer than 25 people the cost is $\mathrm{\$}9.25$ per person; for groups with 25 people the cost is $\mathrm{\$}9.25$ per person; for groups with 25 people or more the cost is $\mathrm{\$}8.50$ per person. The 27 people in the $9:00$ a.m. tour group each paid $\mathrm{\$}9.25$ in advance. What is the total refund that the museum owes the $9:00$ a.m. group? F. $\mathrm{\$}12.50$ G. $\mathrm{\$}13.00$ H. $\mathrm{\$}18.75$ J. $\mathrm{\$}20.25$ K. $\mathrm{\$}25.00$

Short answer

Answer: $20.25

Step-by-step solution

Find the price difference per person

Based on the given prices, we can find that the price difference per person is $9.25-8.50=\mathrm{\$}0.75$.

Calculate the total refund

Now, we multiply this difference by the number of people in the group (27) to find the total refund: $0.75\ast 27=\mathrm{\$}20.25$. Therefore, the total refund owed to the 9:00 a.m. group is $\boxed{{\displaystyle \mathrm{\$}20.25}}$ (option J).

Key concepts

Group Discount Calculation

Understanding group discount calculation is a fundamental skill, particularly when dealing with budgeting and finance-related matters. In the context of an ACT Math problem, such as the one concerning a museum group tour, the principles of group discounts can be illustrated by differences in per-person costs based on group size. Essentially, the larger the group, the lower the cost per individual.In our example, the tour price decreases from $\text{(}9.25$ to $$8.50 \) per person when the group has 25 or more members. The incentive for a group discount comes from the idea that servicing a larger number of individuals at once can reduce overhead costs for the service provider, which can then be passed on as savings to the group.One common pitfall for students is not recognizing that a threshold needs to be met before the discount is applied. It's also essential to note that the discount applies to all members of the group once the threshold is met, not just the members beyond the threshold. This factor was key in calculating the total refund for the 27 people who paid the individual price in advance but qualified for the group discount rate.

Basic Arithmetic Operations

At the heart of many ACT Math problems are the basic arithmetic operations: addition, subtraction, multiplication, and division. These are the building blocks for solving a wide range of problems, including finance calculations such as determining discounts and refunds. In the museum group tour problem, subtraction is used to find the difference in cost per person before and after applying the group discount ($\text{(}9.25-$8.50 = $0.75$). Next, multiplication is applied to calculate the total refund amount owed to the group ($$0.75 \times 27 = $20.25 \)).Familiarity with these operations is crucial since they are routinely used in calculations of percentages, proportions, and other mathematical concepts. It's beneficial for students to practice these basic operations until they become second nature, allowing more mental resources to be focused on understanding the problem's context rather than the calculations themselves.

Word Problem Analysis

Word problems combine mathematical computation with reading comprehension. To successfully navigate word problems, such as calculating refunds for a group tour, it's essential to thoroughly understand the scenario presented and identify the relevant information. First, students must differentiate between significant and extraneous details—the cost per person for groups below 25 members is not required for the actual calculation but serves as context. Then, students must translate the verbal description into a mathematical equation that can be solved with appropriate operations. In the provided example, recognizing that the price difference per person is the key to calculating the total refund is paramount.Word problems often contain 'triggers' or keywords that hint at the operations needed. In our case, 'refund' and 'paid in advance' indicate a subtraction and a possible multiplication to find the total amount. Teaching students to identify and interpret these triggers helps develop their analytical skills, making them more adept at breaking down complex word problems into manageable parts.