In a factory, producing parts for 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 testing machines during a month are 10%, 5% and 2% respectively, then what is the effective rejection rate for the whole plant?

Concept & Logic The effective rejection rate cannot be found by simply adding the individual rejection rates because the subsequent checks are only performed on the parts that *survived* the previous checks. We must calculate the successive acceptance rates first to find the final percentage of parts that pass all tests. $$Final Acceptance Rate = Initial \times \left(1 - \frac{R_1}{100}\right) \times \left(1 - \frac{R_2}{100}\right) \times \left(1 - \frac{R_3}{100}\right)$$ Step-by-Step Solution * Let the initial number of parts manufactured be 100. * **Stage 1 (10% rejection):** Rejected = $10\%$ of $100 = 10$. Passed = $100 - 10 = 90$ parts. * **Stage 2 (5% rejection):** Only the 90 passed parts are tested. Rejected = $5\%$ of $90 = 4.5$. Passed = $90 - 4.5 = 85.5$ parts. * **Stage 3 (2% rejection):** Only the 85.5 passed parts are tested. Rejected = $2\%$ of $85.5 = 1.71$. Passed = $85.5 - 1.71 = 83.79$ parts. * **Final Calculation:** Out of 100 initial parts, 83.79 parts passed all quality checks. Total rejected parts = Initial Parts - Final Passed Parts $$Total Rejected = 100 - 83.79 = 16.21$$ Since we started with 100, this is exactly $16.21\%$. Exam Strategy & Shortcut Use decimal multipliers for speed. The acceptance rates are $90\%$ (0.9), $95\%$ (0.95), and $98\%$ (0.98). Total acceptance = $0.9 \times 0.95 \times 0.98$ $= 0.855 \times 0.98$ $= 0.8379 = 83.79\%$ Effective Rejection = $100\% - 83.79\% = 16.21\%$. Common Pitfall The most frequent error is simply adding the percentages together ($10\% + 5\% + 2\% = 17\%$, Option D). This is incorrect because the 5% and 2% rejection rates apply to a shrinking pool of parts, not the original total batch. Final Answer **Therefore, the correct answer is 16.21%.**