Question

In a factory producing parts of an automobile, the parts manufactured on the shop floor are required to go through three quality checks, each conducted after a specific part of the processing on the raw material is completed. Only parts that are not rejected at one stage are put through the subsequent stages of production and testing. If average rejection rates at these three testing machines during a month are 10%, 5% and 2% respectively, then what is the effective rejection rate for the whole plant?

Answer 1

Correct Answer: 3

Step 1: Understanding the Question:
A manufacturing process has three sequential quality checks. Only parts that pass a stage move to the next. We are given the rejection rates at each specific stage and need to find the overall (effective) rejection rate for the entire process from start to finish.
Step 2: Key Formula or Approach:
Since the process is sequential, calculate the percentage of parts that successfully pass all three stages to find the final acceptance rate. The effective rejection rate is then $100% - \text{Final Acceptance Rate}$.
Step 3: Detailed Explanation:

Let the initial number of raw parts entering the first testing machine be 100.

Stage 1: The rejection rate is 10%.
Number of parts rejected here = 10% of 100 = 10.
Number of parts that pass and move to Stage 2 = $100 - 10 = 90$ parts.

Stage 2: The rejection rate is 5% \textit{on the remaining parts}.
Number of parts rejected here = 5% of 90 = $0.05 \times 90 = 4.5$ parts.
Number of parts that pass and move to Stage 3 = $90 - 4.5 = 85.5$ parts.

Stage 3: The rejection rate is 2% \textit{on the remaining parts}.
Number of parts rejected here = 2% of 85.5 = $0.02 \times 85.5 = 1.71$ parts.
Number of parts that pass the final stage = $85.5 - 1.71 = 83.79$ parts.

After all three stages, exactly 83.79 parts out of the initial 100 are successfully manufactured and accepted.

The effective rejection rate is the total percentage of parts that were rejected throughout the whole process.

Effective Rejection Rate = $100% - \text{Final Accepted Percentage}$.

Effective Rejection Rate = $100 - 83.79 = 16.21%$.

Step 4: Final Answer:
The effective rejection rate for the whole plant is 16.21%.