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Difficulty: Very hardFractions, Decimals, and Percentages

An investment fund allocates its capital into three portfolios: Real Estate, Technology, and Green Energy. Initially, 14\frac{1}{4} of the total capital is allocated to Real Estate, and 25\frac{2}{5} of the total capital is allocated to Technology. The remaining capital is allocated to Green Energy. During the first year, the Real Estate portfolio decreases in value by 16%16\%, the Technology portfolio increases in value by 15%15\%, and the Green Energy portfolio increases in value by p%p\%. If the total value of the entire investment fund at the end of the first year has increased by 9%9\% of its initial value, what is the value of pp?

  1. A
    10.010.0
  2. B
    10.510.5
  3. C
    18.918.9
  4. 20.020.0Answer
  5. E
    31.431.4

Answer

The value of pp is 20.020.0.
The correct answer is 20.020.0. By assuming a total capital of 100100 units, we find that the Real Estate portfolio starts at 2525 units, the Technology portfolio at 4040 units, and the Green Energy portfolio at 3535 units. The change in Real Estate is −4-4 units, and the change in Technology is +6+6 units. To reach a total fund increase of 99 units, the change in the Green Energy portfolio must satisfy −4+6+Change in Green Energy=9-4 + 6 + \text{Change in Green Energy} = 9, which means the Green Energy portfolio must increase by 77 units. Finding what percent 77 is of 3535 yields 735=0.20\frac{7}{35} = 0.20, or 20%20\%.

Step-by-Step Solution

1
Determine the initial allocation of capital for each portfolio by assuming a total initial capital of 100100 units.
Real Estate receives 14×100=25\frac{1}{4} \times 100 = 25 units. Technology receives 25×100=40\frac{2}{5} \times 100 = 40 units. The remaining capital allocated to Green Energy is 100−(25+40)=35100 - (25 + 40) = 35 units.
This establishes the weighted share of each portfolio relative to the total capital.
2
Calculate the changes in value for the Real Estate and Technology portfolios based on their given percentage changes.
Real Estate change: −16% of 25=−0.16×25=−4-16\% \text{ of } 25 = -0.16 \times 25 = -4 units. Technology change: +15% of 40=0.15×40=+6+15\% \text{ of } 40 = 0.15 \times 40 = +6 units.
This determines the absolute change in value contributed by these two portfolios.
3
Calculate the total change in value for the entire fund.
Total fund change: +9% of 100=+9+9\% \text{ of } 100 = +9 units.
The total change in the fund is the sum of the individual changes in the portfolios.
4
Set up an equation representing the sum of the changes in all three portfolios and solve for pp.
−4+6+(p100×35)=9  ⟹  2+0.35p=9  ⟹  0.35p=7  ⟹  p=20-4 + 6 + \left(\frac{p}{100} \times 35\right) = 9 \implies 2 + 0.35p = 9 \implies 0.35p = 7 \implies p = 20.
This solves for the unknown percentage rate of increase required for the Green Energy portfolio.

Key Concept

Weighted average of percentage changes using fractional and decimal representations.

Alternative Method

Instead of choosing an arbitrary starting capital of 100100, you can use decimals directly. Let CC be the total capital. The weighted changes are 0.25C(−0.16)+0.40C(0.15)+0.35C(p100)=0.09C0.25C(-0.16) + 0.40C(0.15) + 0.35C\left(\frac{p}{100}\right) = 0.09C. Dividing both sides by CC yields −0.04+0.06+0.0035p=0.09-0.04 + 0.06 + 0.0035p = 0.09. Simplifying gives 0.02+0.0035p=0.09  ⟹  0.0035p=0.07  ⟹  p=200.02 + 0.0035p = 0.09 \implies 0.0035p = 0.07 \implies p = 20.
Estimated Time:3m 0s
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Topics

Fractions, Decimals, and Percentages

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