Problem 18 - Prealgebra

Question

Solve each of the following problems by first restating it as one of the three basic percent problems of Section 7.2 . In each case, be sure to show the equation. In a shipment of airplane parts, \(3 \%\) are known to be defective. If 15 parts are found to be defective, how many parts are in the shipment?

Step-by-Step Solution

Answer

The shipment contains 500 airplane parts.

Step 1: Understand the Problem

We need to find out how many total airplane parts are in the shipment given that 3% of them are defective and 15 parts are known to be defective.

Step 2: Set Up the Equation

Restate the problem as: 3% of the total number of airplane parts equals 15 defective parts. The equation representing this is: \( 0.03 \times x = 15 \), where \( x \) is the total number of airplane parts.

Step 3: Solve for x

Solve the equation \( 0.03 \times x = 15 \) for \( x \) by dividing both sides by 0.03: \( x = \frac{15}{0.03} \).

Step 4: Calculate the Solution

Perform the division: \( x = 15 \div 0.03 = 500 \). This means there are 500 airplane parts in the shipment.

Key Concepts

Basic Percent Problems

Understanding basic percent problems is crucial in solving many real-life scenarios. Percent problems revolve around finding three key elements: the part, the whole, and the percent itself. When dealing with these problems, one of the first steps is to identify which of these components you are given and what you need to find. For instance, in the problem about airplane parts, we are given:

• Percent: 3% are defective
• Part: 15 defective parts
• Whole: Total airplane parts, which is what we need to find.

The goal is to use these parts to form an equation where the percentage relates one part of the quantity to the entire amount. Familiarity with this process helps in rearranging the problem into a mathematical expression and gives insight into how percentages interact with real-world quantities.
By continuously practicing these setups, students can flexibly adapt to varying percent problems with ease.

Problem-Solving Steps

To effectively solve percent problems, having a structured approach can be immensely beneficial. Let's break it down into specific steps:

• Understand the Problem: Clearly read the problem and determine what is given and what needs to be found. In our case, 3% represents a part of the total airplane parts, and we need to find the total parts.
• Set Up the Equation: Translate the words into a mathematical equation. For example, 3% becomes the decimal 0.03, and we can establish that 0.03 of the total number of parts equals 15, leading to the equation: \( 0.03 \times x = 15 \).
• Solve the Equation: Isolate the unknown by appropriate operations. Divide both sides by 0.03 to find the total number of parts, resulting in \( x = \frac{15}{0.03} \).
• Calculate the Solution: Perform the actual division to get the answer. In the exercise, \( 15 \div 0.03 = 500 \).

Following these steps methodically ensures that no detail is overlooked and the solution is clear.

Prealgebra Concepts

Prealgebra forms the foundation for understanding basic percent problems and encompasses various mathematical principles. Here are some key concepts:

• Decimal Representation: Converting percentages into decimals is often required in equations, such as converting 3% to 0.03 in our problem.
• Equation Solving: Understanding how to manipulate equations to isolate variables is a pivotal prealgebra skill. Solving \( 0.03 \times x = 15 \) involves dividing by the coefficient of \( x \).
• Division: Fundamental division skills are necessary, such as performing \( 15 \div 0.03 \) to find the solution.
• Logical Reasoning: Developing the ability to follow logical steps from problem to solution helps in organizing thoughts and ensuring accuracy.

These fundamental skills are essential not just in solving the exercise at hand but also as building blocks for more complex mathematical tasks.