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Question

Difficulty: EasyPercentages, Percent Change, and Interest

A boutique owner increases the original price of a handbag by 25%25\%. One month later, the owner reduces the updated price by 20%20\%. Which of the following statements about the price of the handbag must be true? Select all such statements.

  1. The final price of the handbag is equal to its original price.Answer
  2. The overall net percent change in the price of the handbag is 0%0\%.Answer
  3. The dollar amount of the 25%25\% increase is equal to the dollar amount of the 20%20\% discount.Answer
  4. D
    The final price of the handbag is 5%5\% greater than its original price.
  5. E
    The dollar amount of the 20%20\% discount is less than the dollar amount of the 25%25\% increase.

Answer

The correct statements are: the final price is equal to the original price, the overall net percent change is 0%, and the dollar amount of the 25% increase equals the dollar amount of the 20% discount.
Applying a 25% increase to an initial amount PP yields 1.25P1.25P, representing a 0.25P0.25P increase. Taking 20% off of 1.25P1.25P calculates to 0.20×1.25P=0.25P0.20 \times 1.25P = 0.25P. Subtracting 0.25P0.25P from 1.25P1.25P returns the price to 1.00P1.00P. Therefore, the final price is equal to the original price, the overall percent change is 0%, and the two dollar changes are equal.

Step-by-Step Solution

1
Define a variable for the original price and calculate the price after the 25% increase.
Let the original price be PP. The price after a 25%25\% increase is P+0.25P=1.25PP + 0.25P = 1.25P. The dollar increase is 0.25P0.25P.
Establishing the intermediate price is necessary to apply the subsequent discount correctly.
2
Calculate the price after the 20% discount applied to the updated price.
The discount amount is 20%20\% of 1.25P1.25P, which is 0.20×1.25P=0.25P0.20 \times 1.25P = 0.25P. The final price is 1.25P−0.25P=1.00P1.25P - 0.25P = 1.00P.
Percentage discounts must be evaluated using the price immediately preceding the discount as the base value.
3
Evaluate the net change and compare dollar amounts.
Final price = Original price (PP). Net percent change = 1.00P−PP×100%=0%\frac{1.00P - P}{P} \times 100\% = 0\%. Dollar increase (0.25P0.25P) = Dollar decrease (0.25P0.25P).
Comparing final values to initial values confirms which statements are true.

Key Concept

Successive Percentage Changes and Base Value Shifts
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Topics

Percentages, Percent Change, and Interest

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