At the start of a fiscal year, a venture capital fund divided its total investment capital… — GRE General Test

At the start of a fiscal year, a venture capital fund divided its total investment capital between Asset Class $X$ and Asset Class $Y$ . Over the course of the year, the value of Asset Class $X$ increased by $25\%$ , while the value of Asset Class $Y$ decreased by $40\%$ . At the end of the year, the total value of the fund was exactly equal to its total value at the start of the year. Which of the following statements must be true? Select all that apply.

At the start of the year, the value of Asset Class $X$ was $60\%$ greater than the value of Asset Class $Y$ .Cevap
B
At the start of the year, Asset Class $X$ accounted for exactly $60\%$ of the total capital of the fund.
At the end of the year, Asset Class $X$ represented more than $75\%$ of the total value of the fund.Cevap
D
The ratio of the final value of Asset Class $X$ to the final value of Asset Class $Y$ was $5$ to $2$ .
The dollar amount by which Asset Class $X$ increased was equal to the dollar amount by which Asset Class $Y$ decreased.Cevap

Cevap

The statements that must be true are: at the start of the year, the value of Asset Class X was 60% greater than the value of Asset Class Y; at the end of the year, Asset Class X represented more than 75% of the total value of the fund; and the dollar amount by which Asset Class X increased was equal to the dollar amount by which Asset Class Y decreased.

The problem establishes that

0.25X = 0.40Y

, which yields

X = 1.60Y

. This confirms that at the start of the year, the value of Asset Class

X

was

60\%

greater than Asset Class

Y

. Furthermore, the final value of Asset Class

X

becomes

2.00Y

out of a total final fund value of

2.60Y

, giving it a share of

\frac{10}{13} \approx 76.92\%

, which is strictly greater than

75\%

. Finally, because the total portfolio value is unchanged, the exact dollar increase in Asset Class

X

(

0.25X

) must balance the dollar decrease in Asset Class

Y

(

0.40Y

Adım Adım Çözüm

Set up an equation using initial variables

X

and

Y

for the two asset classes.

Initial total value =

X + Y

. Final value of

X = 1.25X

, final value of

Y = 0.60Y

. Final total value =

1.25X + 0.60Y

Percentage increases and decreases apply directly as multiplier factors to their respective initial base values.

Equate the initial and final total fund values to find the relationship between

X

and

Y

1.25X + 0.60Y = X + Y \implies 0.25X = 0.40Y \implies X = 1.60Y

The problem states the net total value of the fund remained unchanged over the year.

Evaluate the initial relationship between Asset Class

X

and Asset Class

Y

Since

X = 1.60Y = (1 + 0.60)Y

, Asset Class

X

was

60\%

greater than Asset Class

Y

at the start of the year.

A multiplier of

1.60

corresponds to a

60\%

increase over the base value

Y

Determine the initial and final percentage allocations of Asset Class

X

relative to the total fund.

Initial share =

\frac{1.60Y}{2.60Y} = \frac{8}{13} \approx 61.54\%

. Final share =

\frac{1.25(1.60Y)}{1.25(1.60Y) + 0.60Y} = \frac{2.00Y}{2.60Y} = \frac{10}{13} \approx 76.92\%

Evaluating part-to-whole fractions demonstrates that the final share of Asset Class

X

(

76.92\%

) exceeds

75\%

Analyze dollar change equality between the assets.

Dollar gain in

X = 0.25X

. Dollar loss in

Y = 0.40Y

. Since

0.25X = 0.40Y

, the dollar gain in

X

equals the dollar loss in

Y

Zero net change in total fund value guarantees that positive dollar changes cancel negative dollar changes exactly.

Anahtar Kavram

Weighted Percentage Changes and Base Value Shifting

Tahmini Süre:3m 0s

Soru

Cevap

Adım Adım Çözüm

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