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Zorluk: ZorPercentages, Percent Change, and Interest

A retail store tracks the price of a commercial oven across three consecutive quarters.

- In Quarter 1, the original price PP was increased by x%x\%.
- In Quarter 2, the price was decreased by x%x\% from its Quarter 1 price.
- In Quarter 3, the price was increased by 25%25\% from its Quarter 2 price.

If the price at the end of Quarter 3 is 20%20\% greater than the original price PP, which of the following statements must be true? Select all such statements.

  1. The value of xx is 2020.Cevap
  2. The price at the end of Quarter 2 was 4%4\% less than the original price PP.Cevap
  3. C
    The dollar amount of the price decrease in Quarter 2 was equal to the dollar amount of the price increase in Quarter 1.
  4. D
    The overall net percent change from the original price PP to the price at the end of Quarter 2 was 0%0\%.
  5. E
    The price at the end of Quarter 1 was 20%20\% greater than the price at the end of Quarter 2.

Cevap

The statements holding that the value of xx is 2020 and that the price at the end of Quarter 2 was 4%4\% less than the original price PP are both true.
The statement showing x=20x = 20 is correct because solving the multi-stage percent equation 1.25P(1−x210,000)=1.20P1.25 P (1 - \frac{x^2}{10,000}) = 1.20P yields x=20x = 20. The statement showing that the Quarter 2 price was 4%4\% less than PP is correct because P2=P(1−40010,000)=0.96PP_2 = P(1 - \frac{400}{10,000}) = 0.96P, which represents a 4%4\% reduction relative to PP.

Adım Adım Çözüm

1
Express the price after each quarter algebraically in terms of initial price PP and rate xx.
Quarter 1 price: P1=P(1+x100)P_1 = P\left(1 + \frac{x}{100}\right). Quarter 2 price: P2=P1(1−x100)=P(1−x210,000)P_2 = P_1\left(1 - \frac{x}{100}\right) = P\left(1 - \frac{x^2}{10,000}\right). Quarter 3 price: P3=1.25P2=1.25P(1−x210,000)P_3 = 1.25 P_2 = 1.25 P\left(1 - \frac{x^2}{10,000}\right).
Successive percentage changes compound sequentially on updated base values.
2
Set the Quarter 3 price expression equal to 1.20P1.20P and solve for xx.
1.25P(1−x210,000)=1.20P  ⟹  1−x210,000=1.201.25=0.96  ⟹  x210,000=0.04  ⟹  x2=400  ⟹  x=201.25 P\left(1 - \frac{x^2}{10,000}\right) = 1.20 P \implies 1 - \frac{x^2}{10,000} = \frac{1.20}{1.25} = 0.96 \implies \frac{x^2}{10,000} = 0.04 \implies x^2 = 400 \implies x = 20.
A price 20%20\% greater than PP equals 1.20P1.20P.
3
Evaluate intermediate values to test each statement.
P1=1.20PP_1 = 1.20P, P2=0.96PP_2 = 0.96P. The Quarter 1 increase is 0.20P0.20P, while the Quarter 2 decrease is 0.20×1.20P=0.24P0.20 \times 1.20P = 0.24P.
Comparing exact calculated dollar and percentage values allows for direct validation of each option.

Anahtar Kavram

Successive Percentage Changes and Base Shift Dependencies
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Konular

Percentages, Percent Change, and Interest

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