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Zorluk: OrtaPercentages, Percent Change, and Interest

At the beginning of 2024, a university endowment fund was reallocated across various asset classes. Over the course of 2024, the total value of the endowment fund increased by 30%30\%. In 2025, market downturns caused the total value of the fund to decrease by 20%20\% relative to its value at the end of 2024. If the total value of the endowment fund at the end of 2025 was $520,000\$520,000, what was the total value of the fund at the beginning of 2024?

  1. A
    $455,000\$455,000
  2. B
    $468,000\$468,000
  3. C
    $480,000\$480,000
  4. D
    $499,200\$499,200
  5. $500,000\$500,000Cevap

Cevap

$500,000\$500,000
The option stating $500,000\$500,000 is correct because a 30%30\% increase followed by a 20%20\% decrease corresponds to multiplying the initial value VV by 1.30×0.80=1.041.30 \times 0.80 = 1.04. Solving 1.04V=520,0001.04V = 520,000 yields V=500,000V = 500,000.

Adım Adım Çözüm

1
Define the initial value variable and express successive percent changes as growth/decay multipliers.
Let VV be the initial value at the start of 2024. A 30%30\% increase corresponds to a multiplier of 1+0.30=1.301 + 0.30 = 1.30. A 20%20\% decrease corresponds to a multiplier of 1−0.20=0.801 - 0.20 = 0.80.
Successive percentage changes compound multiplicatively on the original base value.
2
Formulate the net equation for the final value.
V×1.30×0.80=520,000  ⟹  V×1.04=520,000V \times 1.30 \times 0.80 = 520,000 \implies V \times 1.04 = 520,000
Multiplying 1.301.30 by 0.800.80 yields a net factor of 1.041.04, representing a net 4%4\% increase over the two-year period.
3
Solve for the initial value VV.
V=520,0001.04=500,000V = \frac{520,000}{1.04} = 500,000
Dividing the final amount by the combined multiplier isolates the original starting value.

Anahtar Kavram

Successive Percent Changes and Base Value Calculation
Tahmini Süre:1m 30s
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Percentages, Percent Change, and Interest

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