Question

A company manufactures light bulbs. If 1 out of 25 light bulbs produced by the company is found to be defective, how many defective light bulbs will there be in a batch of 3,075 light bulbs? (A) 25 (B) 123 (C) 500 (D) 76,875

Short answer

There will be 123 defective light bulbs in a batch of 3,075 light bulbs. Answer: (B) 123.

Step-by-step solution

Write the proportions

Begin by writing the proportion: $\frac{numberofdefectivebulbs}{totalnumberofbulbs}$ in both situations (1 out of 25 and the 3,075 light bulbs).

Set up an equation

Now, set up an equation with the proportions, where $x$ represents the number of defective bulbs in the batch of 3,075 light bulbs: $$\frac{1}{25}=\frac{x}{3075}$$

Solve for x

To solve for $x$, cross-multiply and divide: $$x=\frac{1\times 3075}{25}$$

Calculate x

Calculate the value of $x$: $$x=\frac{3075}{25}=123$$

Choose the correct answer

Since we found that there will be 123 defective light bulbs in a batch of 3,075 light bulbs, the correct answer is: (B) 123

Key concepts

Mathematics Problem Solving

Mathematics problem solving is a fundamental skill that involves understanding, analyzing, and solving problems. It goes beyond simple arithmetic and requires logical thinking.
Problem-solving helps us apply math concepts to real-life scenarios. To tackle a problem effectively, it's essential to follow a structured approach. Here’s a brief breakdown of the steps involved in problem solving for numerical questions like our original exercise:

• **Understand the Problem:** Read through the problem carefully and identify what is being asked. Look for keywords and numbers that will guide you.
• **Develop a Plan:** Determine the best strategy or formula to use. This may involve setting up equations, writing proportions, or drawing diagrams.
• **Execute the Plan:** Carry out your strategy with accurate calculations. Cross-multiply, add or subtract, whatever is necessary!
• **Check the Results:** After solving the problem, it’s important to verify your solution. Double-check your work to ensure that your answer makes sense.

The more you practice, the better you'll become at recognizing patterns and choosing effective strategies, improving your problem-solving skills.

Proportions and Ratios

Proportions and ratios are mathematical methods used to compare quantities and find relationships between them. These concepts are crucial when dealing with scenarios like the one in our exercise regarding defective items in manufacturing.
A ratio is a way to express the relative sizes of two or more values. For instance, the problem statement gives us a ratio of 1 defective light bulb to every 25 bulbs produced. It can be written as a fraction: $\frac{1}{25}$.
On the other hand, a proportion is an equation that says two ratios are equal. To find out how many defective bulbs exist in a larger batch, we set up a proportion using the known ratio and the total number of bulbs. The equation looks like this:

• $$\frac{1}{25}=\frac{x}{3075}$$

This equation tells us that the ratio of defective bulbs in the larger batch is the same as the initial ratio. Cross-multiplication is then used to solve for $x$, determining how many defective bulbs there are relative to the larger quantity.
Understanding how proportions work allows us to solve for unknowns in similar contexts, providing a valuable skill in quality control and other fields.

Quality Control in Manufacturing

Quality control is a crucial aspect of manufacturing, focusing on ensuring products meet certain standards and are free from defects. In the context of our exercise, we've looked at how ratios help assess product quality.
Defective items in a batch can impact profitability and customer satisfaction. Here’s why quality control matters:

• **Maintains Standards:** A consistent level of quality is crucial for maintaining product reliability and safety.
• **Detects Issues Early:** Identifying defective products quickly helps prevent larger problems and losses.
• **Improves Efficiency:** Effective quality control minimizes waste and ensures resources are used optimally.
• **Enhances Consumer Trust:** Reliable quality control processes boost customer satisfaction and trust in the brand.

Quality control often involves sampling methods where a small portion of the batch is tested to predict the quality of the whole. Ratios like the one used in the exercise can guide these assessments and decide the acceptability of a production run.
By applying mathematical concepts, quality control not only safeguards the product quality but also supports the overall efficiency of manufacturing operations.