Question

Reliability Engineering - Quality control. (a) Quality control checks find six faulty items from 200 that were tested. Express the number of faulty items as a fraction, in its simplest form, of the total number tested. (b) Improvements to the production process mean that the number of faulty items is halved. Express the number of faulty items now as a fraction of the total number tested.

Short answer

Answer: The initial fraction of faulty items was 3/100, and the fraction of faulty items after improvements is 3/400.

Step-by-step solution

Find the fraction of faulty items initially

To find the fraction of faulty items initially, divide the number of faulty items by the total number of items tested. The fraction will be: 6 (faulty items) / 200 (total items tested)

Simplify the initial fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of 6 and 200 is 2. So, dividing both by 2 gives: (6/2) / (200/2) = 3/100 So, the initial fraction of faulty items is 3/100.

Calculate the fraction after improvements

After improvements, the number of faulty items is halved. To find the new fraction, divide the number of faulty items after improvements by the total number of items tested. The new fraction of faulty items is: (3 (initial faulty items)/2) / 200 (total items tested)

Simplify the fraction after improvements

Simplify the fraction as follows: (3/2) / 200 = (3/2) * (1/200) = (3/400) So, the fraction of faulty items after improvements is 3/400. The answers to the questions are: (a) The initial fraction of faulty items is 3/100. (b) The fraction of faulty items after improvements is 3/400.

Key concepts

Quality Control

Quality control is a crucial aspect of reliability engineering, ensuring that products meet specific standards of quality before reaching customers. It involves testing and inspecting products to identify any defects or issues. Implementing effective quality control processes can prevent faulty items from reaching the consumer, thereby enhancing customer satisfaction and reducing costs.
In the given exercise, quality control helps detect the number of items with faults from a batch. It plays a key role in maintaining product standards, ensuring that any defective items are found and dealt with appropriately. By identifying the six faulty items out of 200 tested, the quality control assessment could highlight issues in the production process that need fixing.

• Quality control helps to maintain product standards.
• Ensures faulty items are detected before reaching consumers.
• Aids in identifying production process issues.

This assurance guarantees that the majority of produced items meet the established criteria and are reliable for consumer use.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common divisors other than 1. This not only makes fractions easier to work with but also provides a clearer interpretation of proportions.
In the exercise, simplifying the fraction involving faulty items is essential for clear communication of the data. Initially, the fraction of faulty items was \(\frac{6}{200}\) which simplifies to \(\frac{3}{100}\).
To simplify a fraction, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.
• Write down the simplified fraction.

The greatest common divisor of 6 and 200 in this case is 2, allowing the fraction to be simplified accurately and effectively.

Faulty Items Calculation

Faulty items calculation forms the groundwork of evaluating product reliability and improving processes. By determining the fraction of faulty items, engineers and managers can assess the current quality levels.
In the exercise, after halving the number of faulty items due to improvements, we calculate the fraction of faulty items as follows: originally, 3 initialized faulty items become 1.5 after improvement.
To find the new fraction:

• Divide the number of faulty items after improvements by the total number tested: \(\frac{3/2}{200}\).
• Multiply this by the reciprocal of the total number of items to find a single figure: \(\frac{3}{400}\).

This calculation illustrates the impact improvements have on product reliability, cutting the number of faulty items significantly, hence enhancing the overall quality.