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Zorluk: ZorPercent Change and Interest

A venture capital firm allocated its initial investment fund between two startups, Enterprise X and Enterprise Y. Over a two-year period, the investment in Enterprise X earned simple interest at an annual rate of 15%15\%. During the same period, the value of the investment in Enterprise Y increased by 25%25\% in the first year and then decreased by 20%20\% of its new value in the second year. If the total combined value of the two investments at the end of the two years was 12%12\% greater than the initial investment fund, what percentage of the initial fund was invested in Enterprise X?

  1. A
    28%28\%
  2. 40%40\%Cevap
  3. C
    50%50\%
  4. D
    60%60\%
  5. E
    80%80\%

Cevap

40%40\% of the initial fund was invested in Enterprise X.
The correct answer is 40%40\%. Enterprise X earns simple interest of 15%15\% per year for 2 years, yielding a growth factor of 1+2(0.15)=1.301 + 2(0.15) = 1.30. Enterprise Y increases by 25%25\% and then decreases by 20%20\%, yielding a net growth factor of 1.25×0.80=1.001.25 \times 0.80 = 1.00. Setting the overall weighted multiplier equal to 1.121.12 gives 1.30x+1.00(1−x)=1.121.30x + 1.00(1-x) = 1.12, which yields x=0.40x = 0.40, or 40%40\%.

Adım Adım Çözüm

1
Define variables and determine the growth factor for Enterprise X.
Let xx be the fraction of the fund invested in Enterprise X, so (1−x)(1-x) is the fraction invested in Enterprise Y. Over 2 years at an annual simple interest rate of 15%15\%, the growth multiplier for Enterprise X is 1+2×0.15=1.301 + 2 \times 0.15 = 1.30.
Simple interest accumulates linearly over time.
2
Calculate the net successive percentage multiplier for Enterprise Y.
A 25%25\% increase followed by a 20%20\% decrease gives a net multiplier of (1+0.25)×(1−0.20)=1.25×0.80=1.00(1 + 0.25) \times (1 - 0.20) = 1.25 \times 0.80 = 1.00.
Successive percentage changes must be multiplied sequentially based on the intermediate values.
3
Set up the weighted average equation for the total investment.
1.30x+1.00(1−x)=1.121.30x + 1.00(1-x) = 1.12.
The total combined value at the end of 2 years is 12%12\% greater than the initial fund, corresponding to a total multiplier of 1.121.12.
4
Solve for xx.
1.30x+1.00−1.00x=1.12⇒0.30x=0.12⇒x=0.40=40%1.30x + 1.00 - 1.00x = 1.12 \Rightarrow 0.30x = 0.12 \Rightarrow x = 0.40 = 40\%.
Isolating xx gives the proportion of the fund allocated to Enterprise X.

Anahtar Kavram

Combining simple interest and successive percentage change in weighted portfolio problems.
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Konular

Percent Change and Interest

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