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Difficulty: EasyPercentages, Percent Change, and Interest

The price of an item was increased by 25%25\% and then the new price was decreased by 20%20\%. Which of the following statements must be true regarding the final price of the item relative to its initial price? Select all such statements.

  1. The final price of the item is equal to its initial price.Answer
  2. The overall net percent change in the price of the item is 0%0\%.Answer
  3. The ratio of the final price to the initial price is 11 to 11.Answer
  4. D
    The final price of the item is 5%5\% greater than its initial price.
  5. E
    The final price of the item is 5%5\% less than its initial price.

Answer

The statements confirming that the final price equals the initial price, that the overall net percent change is 0%, and that the ratio of the final price to initial price is 1 to 1 are all correct.
Let PP be the original price. After a 25%25\% increase, the price becomes 1.25P1.25P. Decreasing this intermediate price by 20%20\% yields 1.25P×0.80=1.00P1.25P \times 0.80 = 1.00P. Because the final price is identical to the initial price, the final price is equal to the initial price, the net percentage change is 0%0\%, and the ratio of final to initial price is 1:11:1.

Step-by-Step Solution

1
Represent the initial price with a variable
Let the initial price of the item be PP.
Using a variable makes tracking successive percent changes straightforward.
2
Apply the 25%25\% increase
Increased price =P×(1+0.25)=1.25P= P \times (1 + 0.25) = 1.25P.
A 25%25\% increase corresponds to multiplying the base by 1.251.25.
3
Apply the 20%20\% decrease to the new base
Final price =1.25P×(1−0.20)=1.25P×0.80=1.00P= 1.25P \times (1 - 0.20) = 1.25P \times 0.80 = 1.00P.
The 20%20\% decrease applies to the updated base value of 1.25P1.25P.
4
Evaluate each statement based on the final price 1.00P1.00P
The final price equals the initial price (PP), the net percent change is 0%0\%, and the ratio of final to initial price is 1:11:1.
All three true statements directly reflect that the final price is equal to the initial price.

Key Concept

Successive Percentage Changes and Base Shift
Estimated Time:45s
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Topics

Percentages, Percent Change, and Interest

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