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Zorluk: ZorPercent Change and Interest

An investor allocated a total initial capital of $60,000\$60,000 between two funds, Account X and Account Y. During the first year, Account X increased in value by 40%40\%, while Account Y increased in value by 20%20\%. During the second year, Account X decreased in value by 15%15\% relative to its value at the end of the first year, while Account Y increased in value by 10%10\% relative to its value at the end of the first year. If the total combined value of both accounts at the end of the second year was $74,000\$74,000, what was the initial amount invested in Account X?

  1. A
    $20,000\$20,000
  2. B
    $30,000\$30,000
  3. $40,000\$40,000Cevap
  4. D
    $45,000\$45,000
  5. E
    $50,000\$50,000

Cevap

The initial amount invested in Account X was $40,000\$40,000.
The correct answer is $40,000\$40,000. Account X increases by 40%40\% in Year 1 to 1.40X1.40X and decreases by 15%15\% in Year 2, yielding 1.40X×0.85=1.19X1.40X \times 0.85 = 1.19X. Account Y increases by 20%20\% in Year 1 and 10%10\% in Year 2, yielding (60,000−X)×1.20×1.10=1.32(60,000−X)(60,000 - X) \times 1.20 \times 1.10 = 1.32(60,000 - X). Setting 1.19X+1.32(60,000−X)=74,0001.19X + 1.32(60,000 - X) = 74,000 gives −0.13X=−5,200-0.13X = -5,200, so X=40,000X = 40,000.

Adım Adım Çözüm

1
Calculate the net multiplier for Account X over the two-year period.
Multiplier for Account X = (1+0.40)×(1−0.15)=1.40×0.85=1.19(1 + 0.40) \times (1 - 0.15) = 1.40 \times 0.85 = 1.19
Successive percentage changes must be applied sequentially to updated base values.
2
Calculate the net multiplier for Account Y over the two-year period.
Multiplier for Account Y = (1+0.20)×(1+0.10)=1.20×1.10=1.32(1 + 0.20) \times (1 + 0.10) = 1.20 \times 1.10 = 1.32
Successive percentage increases compound on the value at the end of the first year.
3
Set up an algebraic equation for the total combined final value.
Let XX be the initial capital in Account X. Then 60,000−X60,000 - X is the initial capital in Account Y. 1.19X+1.32(60,000−X)=74,0001.19X + 1.32(60,000 - X) = 74,000
The sum of the final values of both accounts equals the total combined value of $74,000\$74,000.
4
Solve the linear equation for XX.
1.19X+79,200−1.32X=74,000  ⟹  −0.13X=−5,200  ⟹  X=40,0001.19X + 79,200 - 1.32X = 74,000 \implies -0.13X = -5,200 \implies X = 40,000
Dividing −5,200-5,200 by −0.13-0.13 yields the initial amount invested in Account X.

Anahtar Kavram

Successive Percent Change and Base Value Determination
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Konular

Percent Change and Interest

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