A private equity firm allocated its total investment capital between two tech startups, Co… — GMAT

A private equity firm allocated its total investment capital between two tech startups, Company X and Company Y. Over the first year, the value of the investment in Company X increased by $60\%$ , while the value of the investment in Company Y increased by $20\%$ . Over the second year, the value of Company X decreased by $25\%$ from its Year 1 value, while the value of Company Y increased by $25\%$ from its Year 1 value. If the total combined value of the two investments at the end of the second year was $38\%$ greater than the original total investment capital allocated, what percent of the original total investment capital was allocated to Company X?

Cevap: 40 %

Cevap

40%

Evaluating successive percent changes gives growth multipliers of

1.60 \times 0.75 = 1.20

for Company X and

1.20 \times 1.25 = 1.50

for Company Y. Setting the weighted sum of final values equal to

1.38(X + Y)

yields

1.20X + 1.50Y = 1.38X + 1.38Y

, which simplifies to

0.18X = 0.12Y

Y = 1.5X

. The proportion allocated to Company X is

\frac{X}{X + 1.5X} = \frac{1}{2.5} = 40\%

Adım Adım Çözüm

Calculate the cumulative growth multiplier for Company X over the two-year period.

Company X's final value is

1.20X

, representing a net increase of

20\%

Successive percent changes are calculated by multiplying the growth factors:

(1 + 0.60) \times (1 - 0.25) = 1.60 \times 0.75 = 1.20

Calculate the cumulative growth multiplier for Company Y over the two-year period.

Company Y's final value is

1.50Y

, representing a net increase of

50\%

Successive percent changes are calculated by multiplying the growth factors:

(1 + 0.20) \times (1 + 0.25) = 1.20 \times 1.25 = 1.50

Set up an equation for the combined final portfolio value relative to initial capital.

1.20X + 1.50Y = 1.38(X + Y)

The problem states that the combined final value is

38\%

greater than the total initial capital

X + Y

Simplify the equation to express

Y

in terms of

X

0.12Y = 0.18X \implies Y = 1.5X

Expanding

1.38X + 1.38Y

and collecting like terms yields

1.50Y - 1.38Y = 1.38X - 1.20X

Determine Company X's proportion of the total initial allocation.

\frac{X}{X + 1.5X} = \frac{1}{2.5} = 40\%

The initial percentage allocated to Company X is

\frac{X}{X + Y} \times 100\%

Anahtar Kavram

Weighted Successive Percent Change

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Adım Adım Çözüm

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