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Difficulty: MediumPercentages, Percent Change, and Interest

An electronics retailer evaluated its annual revenue over a four-year period. In Year 2, the revenue increased by 25%25\% compared to Year 1. In Year 3, the revenue decreased by 20%20\% compared to Year 2. In Year 4, the revenue increased by 15%15\% compared to Year 3.

Which of the following statements regarding the retailer's revenue must be true? Select all such statements.

  1. The revenue in Year 3 is equal to the revenue in Year 1.Answer
  2. The overall percentage change in revenue from Year 1 to Year 4 is a 15%15\% increase.Answer
  3. The revenue in Year 2 is 25%25\% greater than the revenue in Year 3.Answer
  4. D
    The overall percentage change in revenue from Year 1 to Year 3 is a 5%5\% increase.
  5. E
    The percent increase from Year 3 to Year 4 is greater than the percent decrease from Year 2 to Year 3.

Answer

The statements asserting that Year 3 revenue equals Year 1 revenue, that the overall revenue change from Year 1 to Year 4 is a 15% increase, and that Year 2 revenue is 25% greater than Year 3 revenue are all true.
Let the revenue in Year 1 be RR. Revenue in Year 2 is 1.25R1.25R. Revenue in Year 3 is 1.25R×0.80=1.00R1.25R \times 0.80 = 1.00R, which equals the revenue in Year 1. Revenue in Year 4 is 1.00R×1.15=1.15R1.00R \times 1.15 = 1.15R, representing a net 15%15\% increase from Year 1. Furthermore, Year 2 revenue (1.25R1.25R) compared to Year 3 revenue (1.00R1.00R) is calculated as 1.25R−1.00R1.00R=0.25\frac{1.25R - 1.00R}{1.00R} = 0.25, or a 25%25\% increase. Therefore, the first three statements are mathematically true.

Step-by-Step Solution

1
Define variables and calculate Year 2 revenue relative to Year 1
Let Year 1 revenue be RR. Year 2 revenue = R×(1+0.25)=1.25RR \times (1 + 0.25) = 1.25R.
A 25% increase corresponds to a multiplier of 1.25.
2
Calculate Year 3 revenue relative to Year 2 and Year 1
Year 3 revenue = 1.25R×(1−0.20)=1.25R×0.80=1.00R1.25R \times (1 - 0.20) = 1.25R \times 0.80 = 1.00R.
A 20% decrease applies to the new base value of 1.25R1.25R.
3
Calculate Year 4 revenue relative to Year 3 and Year 1
Year 4 revenue = 1.00R×(1+0.15)=1.15R1.00R \times (1 + 0.15) = 1.15R.
A 15% increase applies to Year 3 revenue, which equals RR.
4
Evaluate each statement against the calculated values
Year 3 revenue (1.00R1.00R) equals Year 1 revenue (RR). Year 4 revenue (1.15R1.15R) is 15% greater than Year 1 revenue (RR). Year 2 revenue (1.25R1.25R) is 25% greater than Year 3 revenue (1.00R1.00R).
Determines which statements hold true mathematically.

Key Concept

Successive Percent Changes and Base Value Shifting
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Topics

Percentages, Percent Change, and Interest

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