Problem 10 - Prealgebra

Question

Solve each of the following word problems by translating the statement into a proportion. Be sure to show the proportion used in each case. [Examples \(1-4]\) Manufacturing Of every 17 parts manufactured by a certain machine, only 1 will be defective. How many parts were manufactured by the machine if 8 defective parts were found?

Step-by-Step Solution

Answer

136 parts were manufactured.

Step 1: Understand the Ratio

The problem states that for every 17 parts manufactured, 1 part is defective. This gives us a ratio of defective parts to total parts as 1:17.

Step 2: Set Up the Proportion

We need to find out how many total parts correspond to 8 defective parts. Let the total number of parts be represented by \( x \). The proportion based on the given information is \( \frac{1}{17} = \frac{8}{x} \).

Step 3: Cross Multiply to Solve for x

Cross multiply the proportions to form an equation: \( 1 \cdot x = 17 \cdot 8 \). This simplifies to \( x = 136 \).

Step 4: Verify the Solution

The solution \( x = 136 \) means that 136 total parts correspond to 8 defective parts. Check if the ratio \( \frac{8}{136} \) matches \( \frac{1}{17} \). Simplifying \( \frac{8}{136} \), we indeed get \( \frac{1}{17} \), confirming our solution is correct.

Key Concepts

Ratios in Mathematics

Ratios are everywhere in mathematics and in daily life situations. A ratio describes how one quantity relates to another. In the example problem, the ratio is between defective parts and total parts produced by a machine.
For instance, if a machine produces a ratio of 1 defective part for every 17 produced, it means there are many parts produced collectively where only one would typically be defective.
Understanding ratios involves acknowledging that they can be expressed in different forms, such as fractions or with a colon. In this case, the ratio of defective to total parts can be written as \( \frac{1}{17} \) or 1:17.

• Ratios help in comparing quantities.
• They provide insights into the relative size of two numbers.
• They make solving related equations simpler.

Overall, mastering the concept of ratios is crucial for analyzing situations like the one introduced in the problem.

Cross Multiplication

Cross multiplication is a powerful tool used to solve proportions, which involves equations where two ratios are set equal. When dealing with proportions, cross multiplication can help eliminate fractions, making calculations easier.
To illustrate, consider our problem where the proportion is set as \( \frac{1}{17} = \frac{8}{x} \). Using cross multiplication, you multiply the numerator of one side by the denominator of the other and vice versa. This leads us to the equation:

• \( 1 \times x = 17 \times 8 \)
• Solving this, we find \( x = 136 \).

By completing this process, you bypass the need to directly solve with fractions. By understanding and applying cross multiplication, problem solvers can effectively tackle proportion-based problems in a straightforward manner.

Problem-Solving Strategies

Effective problem-solving often requires analyzing the problem, planning a solution, executing the plan, and reviewing the results. In the context of our exercise, you can follow this structured approach as a strategy:

• Understand the Problem: Identify what is given versus what needs to be found. Here, we had the rate of defects and needed the total output.
• Create a Plan: Set up the proportion to model the problem mathematically.
• Execute: Solve the equation using cross multiplication for precise results.
• Review: Verify by ensuring the solution satisfies the initial conditions (i.e., the ratios match).

This strategy ensures accuracy and helps build confidence in tackling similar problems in mathematics and real-world scenarios. Being methodical allows problem solvers to detect errors early and confirm accuracy reliably.