A high-end audio equipment manufacturer evaluated the market value of a specialized manufa… — GMAT

A high-end audio equipment manufacturer evaluated the market value of a specialized manufacturing unit over a three-year period. In the first year, the unit's value increased by $25\%$ . In the second year, its value increased by $20\%$ relative to its value at the end of the first year. In the third year, its value decreased by $20\%$ relative to its value at the end of the second year. If the value of the unit at the end of the third year was $\$144,000$ , what was its original value at the beginning of the first year?

A
$\$80,000$
B
$\$96,000$
C
$\$103,680$
D
$\$115,200$
$\$120,000$ Cevap

Cevap

The original value at the beginning of the first year was

\$120,000

To determine the original value, find the overall multiplier for successive changes: an increase of

25\%

(factor

1.25

), an increase of

20\%

(factor

1.20

), and a decrease of

20\%

(factor

0.80

). The overall cumulative factor is

1.25 \times 1.20 \times 0.80 = 1.20

. Dividing the final value of

\$144,000

1.20

yields the correct initial value of

\$120,000

Adım Adım Çözüm

Express each year's percentage change as a growth/decay multiplier.

Year 1 factor

= 1 + 0.25 = 1.25 = \frac{5}{4}

. Year 2 factor

= 1 + 0.20 = 1.20 = \frac{6}{5}

. Year 3 factor

= 1 - 0.20 = 0.80 = \frac{4}{5}

Successive percentage changes must be calculated relative to each new intermediate value.

Calculate the cumulative multiplier over the three years.

Cumulative Multiplier

= \frac{5}{4} \times \frac{6}{5} \times \frac{4}{5} = \frac{6}{5} = 1.20

Multiplying individual period factors yields the net relationship between initial and final values.

Solve for the initial value

V_0

V_0 \times 1.20 = 144,000 \implies V_0 = \frac{144,000}{1.20} = 120,000

Dividing the final value by the overall cumulative factor yields the original price.

Anahtar Kavram

Successive Percent Change and Base Multipliers

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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