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Zorluk: Çok zorRatio and Proportion Word Problems

An investment consultancy allocates capital across three asset classes: Equities, Fixed Income, and Real Estate. Initially, the ratio of Equities to Fixed Income is 3:43 : 4, and the ratio of Fixed Income to Real Estate is 2:32 : 3. Following a market expansion, the total value of the portfolio increases by 50%50\%. To comply with updated risk guidelines, the firm reallocates the expanded capital such that the amount in Fixed Income decreases by 25%25\%, and the remaining portfolio value is divided between Equities and Real Estate in the ratio 3:23 : 2, respectively. If the new allocation in Real Estate is $18,000\$18,000 greater than its initial allocation, what was the initial total value of the investment portfolio?

  1. A
    $130,000\$130,000
  2. B
    $300,000\$300,000
  3. $390,000\$390,000Cevap
  4. D
    $520,000\$520,000
  5. E
    $585,000\$585,000

Cevap

The initial total value of the investment portfolio was $\$ 390,000.
Combining the initial ratios gives an Equities to Fixed Income to Real Estate ratio of 3:4:63 : 4 : 6, making the initial total 13x13x. After a 50%50\% increase in total value (19.5x19.5x) and a 25%25\% decrease in Fixed Income (3x3x), the remaining 16.5x16.5x yields a new Real Estate value of 25×16.5x=6.6x\frac{2}{5} \times 16.5x = 6.6x. The difference 6.6x−6x=0.6x=18,0006.6x - 6x = 0.6x = 18,000 gives x=30,000x = 30,000, resulting in an initial total portfolio of 13×30,000=$390,00013 \times 30,000 = \$390,000.

Adım Adım Çözüm

1
Combine the initial two-part ratios into a single three-part ratio for Equities (E1E_1), Fixed Income (F1F_1), and Real Estate (R1R_1).
E1:F1=3:4E_1 : F_1 = 3 : 4 and F1:R1=2:3=4:6F_1 : R_1 = 2 : 3 = 4 : 6, giving E1:F1:R1=3:4:6E_1 : F_1 : R_1 = 3 : 4 : 6.
A common term for Fixed Income (44) is needed to express all three asset classes in terms of a single variable xx.
2
Express initial amounts and initial total portfolio value in terms of xx.
E1=3xE_1 = 3x, F1=4xF_1 = 4x, R1=6xR_1 = 6x, and Total Initial Portfolio T1=3x+4x+6x=13xT_1 = 3x + 4x + 6x = 13x.
Summing the ratio units gives the total initial quantity representation.
3
Calculate the updated total portfolio value (T2T_2) and updated Fixed Income amount (F2F_2).
T2=1.50×13x=19.5xT_2 = 1.50 \times 13x = 19.5x, and F2=(1−0.25)×4x=3xF_2 = (1 - 0.25) \times 4x = 3x.
The overall portfolio increases by 50%50\% while Fixed Income decreases by 25%25\%.
4
Determine the remaining portfolio value for Equities and Real Estate, and calculate the new Real Estate value (R2R_2).
Remaining Value =19.5x−3x=16.5x= 19.5x - 3x = 16.5x. R2=23+2×16.5x=25×16.5x=6.6xR_2 = \frac{2}{3 + 2} \times 16.5x = \frac{2}{5} \times 16.5x = 6.6x.
The remaining capital is divided between Equities and Real Estate in a 3:23 : 2 ratio.
5
Set up the change equation for Real Estate to solve for xx and determine T1T_1.
R2−R1=6.6x−6x=0.6x=18,000  ⟹  x=30,000R_2 - R_1 = 6.6x - 6x = 0.6x = 18,000 \implies x = 30,000. Therefore, T1=13×30,000=390,000T_1 = 13 \times 30,000 = 390,000.
The problem states the final Real Estate amount is $\$ 18,000 greater than its initial value.

Anahtar Kavram

Combining compound ratios and applying sequential percentage modifications to part-to-part and part-to-whole relationships.
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Ratio and Proportion Word Problems

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