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Difficulty: MediumPercentages, Percent Change, and Interest

During an operational quality review, a manufacturing plant evaluated its annual defect rate over a three-year period. In Year 1, the defect rate was d%d\%, where d>0d > 0. In Year 2, the defect rate decreased by 20%20\% from Year 1. In Year 3, the defect rate increased by 25%25\% from Year 2. Which of the following statements must be true? Select all that apply.

  1. The defect rate in Year 3 is equal to the defect rate in Year 1.Answer
  2. The defect rate in Year 2 is equal to 80%80\% of the defect rate in Year 1.Answer
  3. C
    The net percentage change in the defect rate from Year 1 to Year 3 is an increase of 5%5\%.
  4. D
    The defect rate in Year 1 is 20%20\% greater than the defect rate in Year 2.
  5. E
    The ratio of the defect rate in Year 3 to the defect rate in Year 2 is 4:54 : 5.

Answer

The statements confirming that the defect rate in Year 3 equals the defect rate in Year 1, and that the defect rate in Year 2 is 80% of Year 1, are correct.
The statement asserting that Year 3 defect rate equals Year 1 defect rate is true because 0.80×1.25=1.000.80 \times 1.25 = 1.00. The statement asserting that Year 2 defect rate is 80% of Year 1 is true because a 20% reduction leaves 80% of the initial value.

Step-by-Step Solution

1
Express the Year 2 defect rate in terms of Year 1.
d2=d1×(1−0.20)=0.80d1d_2 = d_1 \times (1 - 0.20) = 0.80 d_1
A 20%20\% decrease from Year 1 corresponds to multiplying the base value d1d_1 by 0.800.80.
2
Express the Year 3 defect rate in terms of Year 2 and Year 1.
d3=d2×(1+0.25)=(0.80d1)×1.25=1.00d1d_3 = d_2 \times (1 + 0.25) = (0.80 d_1) \times 1.25 = 1.00 d_1
A 25%25\% increase over Year 2 multiplies d2d_2 by 1.251.25. Substituting d2=0.80d1d_2 = 0.80 d_1 shows d3=d1d_3 = d_1.
3
Evaluate each given choice against the derived relationships.
Year 3 equals Year 1 (True). Year 2 is 80%80\% of Year 1 (True). Net change is 0%0\% not 5%5\% (False). Year 1 is 25%25\% greater than Year 2, not 20%20\% (False). Ratio of Year 3 to Year 2 is 5:45:4, not 4:54:5 (False).
Only the first two statements hold true under rigorous percentage calculations.

Key Concept

Successive Percent Change and Base Value Shifts
Estimated Time:1m 30s
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Topics

Percentages, Percent Change, and Interest

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