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Zorluk: ZorRatios, Rates, and Percentages

At the beginning of a fiscal year, an asset management firm allocated funds between two portfolios, Portfolio Alpha and Portfolio Beta, in the ratio of 3:53:5, respectively. Over the course of the year, Portfolio Alpha gained 40%40\% in value while Portfolio Beta lost 10%10\% in value. At the end of the year, a total of $150,000\$150,000 in profits was transferred from Portfolio Alpha to Portfolio Beta. Following this transfer, the final value of Portfolio Beta was exactly 25%25\% greater than the final value of Portfolio Alpha. Based on the information provided, what were the initial asset values of Portfolio Alpha and Portfolio Beta, respectively, at the start of the fiscal year?

  1. Portfolio Alpha: $1,350,000\$1,350,000; Portfolio Beta: $2,250,000\$2,250,000Cevap
  2. B
    Portfolio Alpha: $2,250,000\$2,250,000; Portfolio Beta: $1,350,000\$1,350,000
  3. C
    Portfolio Alpha: $1,890,000\$1,890,000; Portfolio Beta: $2,025,000\$2,025,000
  4. D
    Portfolio Alpha: $1,740,000\$1,740,000; Portfolio Beta: $2,175,000\$2,175,000
  5. E
    Portfolio Alpha: $1,200,000\$1,200,000; Portfolio Beta: $2,000,000\$2,000,000

Cevap

Portfolio Alpha had an initial asset value of $1,350,000\$1,350,000 and Portfolio Beta had an initial asset value of $2,250,000\$2,250,000.
The correct pair identifies Portfolio Alpha's initial value as $1,350,000\$1,350,000 and Portfolio Beta's initial value as $2,250,000\$2,250,000. This preserves the initial 3:53:5 ratio. A 40%40\% increase brings Alpha to $1,890,000\$1,890,000, and a 10%10\% decrease brings Beta to $2,025,000\$2,025,000. Transferring $150,000\$150,000 results in final values of $1,740,000\$1,740,000 for Alpha and $2,175,000\$2,175,000 for Beta, satisfying the condition that Beta's final value is 25%25\% greater than Alpha's final value ($1,740,000×1.25=$2,175,000\$1,740,000 \times 1.25 = \$2,175,000).

Adım Adım Çözüm

1
Set up algebraic expressions for initial values based on the given ratio.
Let initial Alpha = 3k3k and initial Beta = 5k5k for some multiplier kk.
The initial ratio of Portfolio Alpha to Portfolio Beta is given as 3:53:5.
2
Express values after annual percentage changes.
Alpha pre-transfer = 3k×(1+0.40)=4.2k3k \times (1 + 0.40) = 4.2k. Beta pre-transfer = 5k×(1−0.10)=4.5k5k \times (1 - 0.10) = 4.5k.
Alpha gained 40%40\% and Beta lost 10%10\% over the year.
3
Account for the $150,000\$150,000 transfer and construct the final equality equation.
Alpha final = 4.2k−150,0004.2k - 150,000 and Beta final = 4.5k+150,0004.5k + 150,000. Equation: 4.5k+150,000=1.25(4.2k−150,000)4.5k + 150,000 = 1.25(4.2k - 150,000).
Beta's final value is 25%25\% (1.251.25 times) greater than Alpha's final value after receiving $150,000\$150,000.
4
Solve for the multiplier kk and compute initial values.
4.5k+150,000=5.25k−187,500  ⟹  0.75k=337,500  ⟹  k=450,0004.5k + 150,000 = 5.25k - 187,500 \implies 0.75k = 337,500 \implies k = 450,000. Thus, Alpha initial = 3(450,000)=$1,350,0003(450,000) = \$1,350,000 and Beta initial = 5(450,000)=$2,250,0005(450,000) = \$2,250,000.
Evaluating the linear equation yields the constant multiplier kk.

Anahtar Kavram

Multi-stage ratio and percentage equation systems with transition transfers.
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Ratios, Rates, and Percentages

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