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Zorluk: ZorRatio and Proportion Word Problems

An investment fund initially divides its total capital among three portfolios—Real Estate, Stocks, and Bonds—in the ratio 3:4:53 : 4 : 5. During a financial restructuring, a portion of capital is transferred directly from the Bonds portfolio to the Real Estate portfolio while the capital in Stocks remains unchanged, resulting in a new ratio of Real Estate to Stocks of 5:45 : 4. Subsequently, an additional $60,000\$60,000 in external capital is deposited into the Bonds portfolio, making the ratio of Stocks to Bonds 2:32 : 3. What was the total initial capital in the fund across all three portfolios?

  1. A
    $180,000
  2. B
    $200,000
  3. $240,000Cevap
  4. D
    $300,000
  5. E
    $720,000

Cevap

The total initial capital in the fund was $240,000.
The correct answer of 240,000isobtainedbytrackingeachportfolio′sbalanceacrossthetwosequentialchanges.Initially,RealEstate,Stocks,andBondsare240,000 is obtained by tracking each portfolio's balance across the two sequential changes. Initially, Real Estate, Stocks, and Bonds are 3x ,, 4x ,and, and 5x respectively(total respectively (total 12x ).WhenRealEstateincreasesto). When Real Estate increases to 5x relativetoStocks( relative to Stocks ( 4x ),therequiredcapitalcomesentirelyfromBonds,reducingBondsfrom), the required capital comes entirely from Bonds, reducing Bonds from 5x to to 3x .Adding. Adding \60,00060,000 to Bonds gives a new balance of 3x+60,0003x + 60,000. Establishing the new Stocks to Bonds ratio 4x3x+60,000=23\frac{4x}{3x + 60,000} = \frac{2}{3} yields x=20,000x = 20,000. Multiplying by the initial total of 12 parts yields 12×20,000=12 \times 20,000 = 240,000$.

Adım Adım Çözüm

1
Express initial portfolio amounts in terms of a multiplier xx
Real Estate = 3x3x, Stocks = 4x4x, Bonds = 5x5x. Total initial capital = 3x+4x+5x=12x3x + 4x + 5x = 12x.
The given initial ratio of Real Estate : Stocks : Bonds is 3:4:53 : 4 : 5.
2
Account for the internal capital transfer between Bonds and Real Estate
Real Estate becomes 5x5x, Stocks remains 4x4x, and Bonds becomes 3x3x.
Stocks remains 4x4x. Since the ratio of Real Estate to Stocks becomes 5:45 : 4, Real Estate must equal 5x5x. Because capital was only moved internally from Bonds to Real Estate, the sum of Real Estate and Bonds remains 3x+5x=8x3x + 5x = 8x, leaving Bonds with 8x−5x=3x8x - 5x = 3x.
3
Set up an algebraic proportion after adding $60,000\$60,000 to Bonds
4x3x+60,000=23\frac{4x}{3x + 60,000} = \frac{2}{3}
The new ratio of Stocks to Bonds is given as 2:32 : 3.
4
Solve for the multiplier xx and determine the total initial capital
3(4x)=2(3x+60,000)⇒12x=6x+120,000⇒6x=120,000⇒x=20,0003(4x) = 2(3x + 60,000) \Rightarrow 12x = 6x + 120,000 \Rightarrow 6x = 120,000 \Rightarrow x = 20,000. Total initial capital =12x=12×20,000== 12x = 12 \times 20,000 = 240,000$.
Cross-multiplying yields the value of xx, which is then scaled by the total initial ratio sum of 12 parts.

Anahtar Kavram

Multi-Step Ratio Adjustments and Internal vs. External Capital Changes
Tahmini Süre:2m 0s
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Ratio and Proportion Word Problems

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