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Difficulty: MediumPercentages, Percent Change, and Interest

An employee's annual salary was increased by 20%20\% at the beginning of 2024. At the end of 2024, the salary was decreased by 20%20\% from its updated value. At the beginning of 2025, the salary was increased by 10%10\% from its end-of-2024 value. If the initial salary before any of these changes was SS, which of the following statements must be true? Select all such statements.

  1. The salary at the end of 2024 was 4%4\% less than the original salary SS.Answer
  2. The overall percentage change from the original salary SS to the salary at the beginning of 2025 is a 5.6%5.6\% increase.Answer
  3. C
    The salary at the end of 2024 was equal to the original salary SS because the 20%20\% increase and 20%20\% decrease offset each other.
  4. D
    The salary at the beginning of 2025 was 10%10\% greater than the original salary SS.
  5. E
    The net change over the entire period is equivalent to a single decrease of 10%10\% followed by an increase of 15.6%15.6\%.

Answer

The statement that the salary at the end of 2024 was 4% less than the original salary S, and the statement that the overall percentage change from the original salary S to the salary at the beginning of 2025 is a 5.6% increase, are both correct.
The initial salary SS experiences successive changes: first multiplied by 1.201.20, then by 0.800.80, and finally by 1.101.10. At the end of 2024, the value is 1.20×0.80S=0.96S1.20 \times 0.80 S = 0.96S, which is 4%4\% less than SS. At the beginning of 2025, the value is 0.96S×1.10=1.056S0.96S \times 1.10 = 1.056S, which reflects a net increase of 5.6%5.6\% over the initial amount SS.

Step-by-Step Solution

1
Calculate the salary after the first 20% increase at the beginning of 2024.
Salary = S×(1+0.20)=1.20SS \times (1 + 0.20) = 1.20S
An increase of 20% scales the initial value SS by a multiplier of 1.20.
2
Calculate the salary after the 20% decrease at the end of 2024.
Salary = 1.20S×(1−0.20)=1.20S×0.80=0.96S1.20S \times (1 - 0.20) = 1.20S \times 0.80 = 0.96S
The 20% decrease is calculated using the updated base of 1.20S1.20S, yielding 0.96S0.96S, which represents a 4% overall decrease from SS.
3
Calculate the salary after the 10% increase at the beginning of 2025.
Final Salary = 0.96S×(1+0.10)=0.96S×1.10=1.056S0.96S \times (1 + 0.10) = 0.96S \times 1.10 = 1.056S
The 10% increase is calculated using the base value of 0.96S0.96S.
4
Determine the net percent change from the original salary SS to the beginning of 2025.
Net Change = 1.056S−SS×100%=0.056×100%=5.6%\frac{1.056S - S}{S} \times 100\% = 0.056 \times 100\% = 5.6\% increase.
Comparing the final multiplier 1.0561.056 to the original base multiplier 1.0001.000 shows a 5.6% overall growth.

Key Concept

Successive Percentage Changes and Base Value Shifts
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Topics

Percentages, Percent Change, and Interest

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