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Zorluk: ZorFractions, Decimals, and Percents Arithmetic

An investment fund allocated its total initial capital among three venture portfolios: Healthcare, Clean Tech, and Artificial Intelligence. Initially, 25\frac{2}{5} of the capital was allocated to Healthcare and 0.350.35 was allocated to Clean Tech, with the remaining fraction allocated to Artificial Intelligence. During a mid-year portfolio rebalancing, the amount in Healthcare was increased by 25%25\%, the amount in Clean Tech was decreased by 20%20\%, and the amount in Artificial Intelligence remained unchanged. If the total capital across all three portfolios after rebalancing was $5,150,000\$5,150,000, what was the initial amount, in dollars, allocated to Artificial Intelligence?

  1. $1,250,000Cevap
  2. B
    $1,287,500
  3. C
    $1,400,000
  4. D
    $1,750,000
  5. E
    $2,000,000

Cevap

The initial amount allocated to Artificial Intelligence was $1,250,000 dollars.
The correct option of 1,250,000isobtainedbyfirstcalculatingtheinitialproportionallocatedtoArtificialIntelligenceas1,250,000 is obtained by first calculating the initial proportion allocated to Artificial Intelligence as 1 - 0.40 - 0.35 = 0.25 .ApplyingthepercentageadjustmentsyieldsHealthcareat. Applying the percentage adjustments yields Healthcare at 0.50C ,CleanTechat, Clean Tech at 0.28C ,andArtificialIntelligenceat, and Artificial Intelligence at 0.25C .Thesumoftheseupdatedallocationsgives. The sum of these updated allocations gives 1.03C = \5,150,0005,150,000, which implies C=$5,000,000C = \$5,000,000. Taking 25%25\% of $5,000,000\$5,000,000 yields $1,250,000\$1,250,000.

Adım Adım Çözüm

1
Determine the initial decimal fraction allocated to each portfolio.
Healthcare = 25=0.40\frac{2}{5} = 0.40; Clean Tech = 0.350.35; Artificial Intelligence = 1−(0.40+0.35)=0.251 - (0.40 + 0.35) = 0.25.
The sum of all initial portfolio fractions must equal 11.
2
Express the post-rebalancing portfolio values in terms of the initial total capital CC.
Healthcare = 0.40C×(1+0.25)=0.50C0.40C \times (1 + 0.25) = 0.50C; Clean Tech = 0.35C×(1−0.20)=0.28C0.35C \times (1 - 0.20) = 0.28C; Artificial Intelligence = 0.25C0.25C.
A 25%25\% increase scales a value by 1.251.25, and a 20%20\% decrease scales a value by 0.800.80.
3
Sum the new portfolio values to set up an equation for total capital CC.
0.50C+0.28C+0.25C=1.03C=5,150,0000.50C + 0.28C + 0.25C = 1.03C = 5,150,000.
The sum of all rebalanced portfolios equals the given final capital of $5,150,000\$5,150,000.
4
Solve for initial total capital CC and calculate the Artificial Intelligence allocation.
C=5,150,0001.03=5,000,000C = \frac{5,150,000}{1.03} = 5,000,000; Artificial Intelligence = 0.25×5,000,000=1,250,0000.25 \times 5,000,000 = 1,250,000.
Multiplying the initial total capital by 0.250.25 yields the original Artificial Intelligence allocation.

Anahtar Kavram

Combining sequential percentage changes and decimal/fraction conversions to solve multi-step financial allocation word problems.
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Konular

Fractions, Decimals, and Percents Arithmetic

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