Question

Tickets for a community theater production cost \(\$ 6\) each when bought in advance and \(\$ 8\) each when bought at the door. The theater group's goal is at least \(\$ 2,000\) in ticket sales for opening night. The theater group sold 142 opening-night tickets in advance. What is the minimum number of tickets they need to sell at the door on opening night to make their goal? A. 143 B. 144 C. 192 D. 250 E. 357

Short answer

Answer: B. 144.

Step-by-step solution

Calculate the revenue from advance sales

In order to calculate how much money is generated from the 142 advance ticket sale, we can multiply the number of tickets (142) by the advance price ($6): Advance revenue = 142 * \(6 = \)852

Calculate the amount needed to reach the goal

Now we need to find out how much money they still need to earn in order to reach the goal of $2,000. We will do this by subtracting the advance revenue from the goal: Amount needed = Goal - Advance revenue = \(2,000 - \)852 = $1,148

Calculate the minimum number of tickets needed to be sold at the door

In order to calculate the minimum number of tickets needed to be sold at the door, we will divide the amount needed to reach the goal (\(1,148) by the price of a ticket at the door (\)8): Minimum door tickets = Amount needed / Door ticket price = \(1,148 / \)8 = 143.5 Since they cannot sell a fractional ticket, they need to sell at least 144 tickets at the door. So, the correct answer is B. 144.

Key concepts

Arithmetic problem-solving

Arithmetic problem-solving forms the backbone of many standardized tests like the ACT Math section. It involves the ability to use basic mathematical operations such as addition, subtraction, multiplication, and division to solve problems. In the context of the ticket sales problem we’re considering, we first multiply the number of advance tickets by the price per ticket to determine the advance revenue.

Understanding how to manipulate numbers accurately is crucial. For instance, multiplying the 142 advance tickets by \(6 each yields an advance revenue of \)852. This straightforward arithmetic is the first step in problem-solving. It's the process of breaking down a problem into smaller, more manageable steps, which is a vital skill not just for tests, but for real-world scenarios as well.

Algebraic equations

Algebraic equations can be thought of as puzzles. You are given a set of pieces (the numbers and variables), and your job is to put them together correctly to find the solution. In the example of the theater tickets, we used an equation to represent the situation: the goal amount minus the advance revenue equals the amount needed from sales at the door.

This equation, in its simplest form, is \( 2,000 - 852 = 1,148 \), where 1,148 represents the amount needed from door sales. To find out how many tickets necessitate sale at the door, we set up another equation \(1,148 = 8 \times \text{{number of tickets}}}\), and solve for the number of tickets. This equation reinforces the concept of isolating the variable to determine its value, an essential strategy in algebra.

Word problem strategies

Word problems pose a unique challenge, combining language comprehension with mathematical reasoning. To effectively tackle word problems on the ACT Math test, a systematic strategy is advisable. Read the problem thoroughly to understand what's being asked. Identify the quantities involved and the question.

In our ticket sale example, knowing that the theater group needs a minimum of $2,000 and that the tickets are priced differently allows us to frame the problem. Translate the situation into mathematical expressions or equations, and pay attention to the details like 'at least' which suggests a goal to be met or exceeded.

Finally, don't forget to check whether the answer makes sense within the context. In our case, we needed at least one more ticket than 143.5—which isn't possible to sell—so we round up to the next whole number, leading to the minimum of 144 tickets. This logical conclusion is indicative of successful word problem-solving.