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Difficulty: MediumPercent Change and Interest

At the beginning of 2021, a technology startup had an initial valuation of VV dollars. Over the next three years, its valuation experienced the following successive annual percentage changes:
- In 2021, the valuation increased by 20%20\%.
- In 2022, the valuation decreased by 10%10\% relative to its valuation at the end of 2021.
- In 2023, the valuation increased by 50%50\% relative to its valuation at the end of 2022.

If the startup's valuation at the end of 2023 was $648,000\$648,000, what was its initial valuation VV at the beginning of 2021?

  1. A
    $360,000\$360,000
  2. $400,000\$400,000Answer
  3. C
    $405,000\$405,000
  4. D
    $432,000\$432,000
  5. E
    $450,000\$450,000

Answer

The initial valuation of the startup at the beginning of 2021 was $400,000\$400,000.
To find the initial valuation VV, we must apply successive growth multipliers rather than adding simple percentages. A 20%20\% increase corresponds to a multiplier of 1.201.20, a 10%10\% decrease corresponds to 0.900.90, and a 50%50\% increase corresponds to 1.501.50. Multiplying these factors yields 1.20×0.90×1.50=1.621.20 \times 0.90 \times 1.50 = 1.62. Setting up the equation 1.62V=$648,0001.62 V = \$648,000 yields V=$400,000V = \$400,000.

Step-by-Step Solution

1
Express each year's percentage change as a multiplier of the value at the start of that year.
Year 2021 multiplier = 1+0.20=1.201 + 0.20 = 1.20; Year 2022 multiplier = 1−0.10=0.901 - 0.10 = 0.90; Year 2023 multiplier = 1+0.50=1.501 + 0.50 = 1.50.
Percentage increases and decreases compound sequentially on the updated base value of each preceding period.
2
Calculate the combined successive percentage multiplier for the three-year period.
Combined Multiplier = 1.20×0.90×1.50=1.08×1.50=1.621.20 \times 0.90 \times 1.50 = 1.08 \times 1.50 = 1.62.
Successive changes multiply together to determine the final overall factor relative to the initial value VV.
3
Set up the linear equation relating the final valuation to the initial valuation VV and solve for VV.
1.62V=648,000  ⟹  V=648,0001.62=648,000162100=4,000×100=400,0001.62 V = 648,000 \implies V = \frac{648,000}{1.62} = \frac{648,000}{\frac{162}{100}} = 4,000 \times 100 = 400,000.
Dividing the final amount by the combined multiplier gives the initial amount VV.

Key Concept

Successive Percent Change
Estimated Time:2m 0s
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Topics

Percent Change and Interest

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