An asset management firm divides its total portfolio into three distinct funds: Fund X, Fu… — GRE General Test

An asset management firm divides its total portfolio into three distinct funds: Fund X, Fund Y, and Fund Z. Initially, $\frac{1}{3}$ of the total portfolio value is in Fund X, and $\frac{2}{5}$ is in Fund Y, with the remaining fraction in Fund Z. Over the course of one year, the value of Fund X increases by $\frac{1}{5}$ , the value of Fund Y decreases by $\frac{1}{4}$ , and the value of Fund Z increases by $\frac{1}{2}$ . At the end of the year, what fraction of the portfolio's total value is contained in Fund Z?

$\frac{4}{11}$ Cevap
B
$\frac{2}{5}$
C
$\frac{4}{7}$
D
$\frac{4}{15}$
E
$\frac{15}{16}$

Cevap

The fraction of the portfolio's total final value contained in Fund Z is

\frac{4}{11}

Fund Z grows to

\frac{2}{5}

of the original portfolio value, while the overall portfolio value increases to

\frac{11}{10}

of the original total. Comparing Fund Z's final value to the new total yields

\frac{2/5}{11/10} = \frac{4}{11}

Adım Adım Çözüm

Calculate the initial fraction of the total portfolio allocated to Fund Z.

The initial allocation for Fund Z is

1 - \left(\frac{1}{3} + \frac{2}{5}\right) = 1 - \frac{11}{15} = \frac{4}{15}

The sum of all initial fund allocations must equal 1.

Determine the final value of each fund relative to the initial total portfolio value

V

Fund X:

\frac{1}{3}V \times \left(1 + \frac{1}{5}\right) = \frac{2}{5}V

. Fund Y:

\frac{2}{5}V \times \left(1 - \frac{1}{4}\right) = \frac{3}{10}V

. Fund Z:

\frac{4}{15}V \times \left(1 + \frac{1}{2}\right) = \frac{2}{5}V

Multiply each initial fund share by its respective growth or reduction factor.

Calculate the new total portfolio value relative to the initial value

V

Total final value =

\frac{2}{5}V + \frac{3}{10}V + \frac{2}{5}V = \frac{4}{10}V + \frac{3}{10}V + \frac{4}{10}V = \frac{11}{10}V

Sum the final values of all three individual funds.

Compute the final fraction of the portfolio contained in Fund Z.

Fraction =

\frac{\frac{2}{5}V}{\frac{11}{10}V} = \frac{2}{5} \times \frac{10}{11} = \frac{4}{11}

Divide the final value of Fund Z by the new total portfolio value.

Anahtar Kavram

Rational number operations and shifting base values in multi-step fraction problems

Tahmini Süre:2m 0s

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