At the beginning of a fiscal year, a technology company allocated its total operating budg… — GMAT

At the beginning of a fiscal year, a technology company allocated its total operating budget among three divisions: Research & Development, Marketing, and Operations. Research & Development received $0.45$ of the total budget, Marketing received $\frac{1}{3}$ of the total budget, and Operations received the remaining portion. Mid-year, the company adjusted the budget by increasing the Research & Development allocation by $20\%$ of its initial amount and reducing the Marketing allocation by $\frac{1}{4}$ of its initial amount, while the Operations allocation remained unchanged. If these mid-year adjustments resulted in a net increase of $\$30,000$ in the company's total operating budget, what was the company's total operating budget at the beginning of the fiscal year?

A
$187,500
B
$600,000
C
$2,700,000
D
$3,600,000
$4,500,000Cevap

Cevap

$4,500,000

The option specifying

4,500,000 is correct. The increase in the Research & Development division is

20\%

of

0.45 B

, which equals

0.09 B

(or

\frac{9}{100} B

). The reduction in Marketing is

\frac{1}{4}

of

\frac{1}{3} B

, which equals

\frac{1}{12} B

. Combining these using a common denominator gives a net increase of

\frac{27}{300} B - \frac{25}{300} B = \frac{2}{300} B = \frac{1}{150} B

. Setting

\frac{1}{150} B = \

30,000

yields

B = \$4,500,000

Adım Adım Çözüm

Express initial allocations as fractions of total budget B

Research & Development initial share =

0.45 B = \frac{9}{20} B

; Marketing initial share =

\frac{1}{3} B

Converting decimals to fractions simplifies exact combined calculations.

Calculate the monetary change in each division relative to total budget B

R&D increase =

20\% \times 0.45 B = 0.20 \times 0.45 B = 0.09 B = \frac{9}{100} B

. Marketing decrease =

\frac{1}{4} \times \frac{1}{3} B = \frac{1}{12} B

. Operations change =

0

Each change applies specifically to the division's initial allocation, not the total budget.

Find the net change in the total budget by taking the difference between the increase and the decrease

Net change =

\left(\frac{9}{100} - \frac{1}{12}\right) B = \left(\frac{27}{300} - \frac{25}{300}\right) B = \frac{2}{300} B = \frac{1}{150} B

Determining the net fraction allows direct equation setup with the given dollar increase.

Solve for the initial total operating budget B using the given net increase of $30,000

\frac{1}{150} B = 30,000 \implies B = 30,000 \times 150 = 4,500,000

Multiplying both sides by 150 yields the original total budget.

Anahtar Kavram

Combining decimal and fraction arithmetic to determine net percentage changes across component allocations.

Alternatif Yöntem

Convert all components to a common denominator out of 300 parts initially. Total budget = 300 parts. R&D = 135 parts (0.45). Marketing = 100 parts (1/3). Operations = 65 parts. R&D increase = 20% of 135 = +27 parts. Marketing decrease = 1/4 of 100 = -25 parts. Net change = +2 parts =

30,000. Therefore 1 part =

15,000, and 300 parts = $4,500,000.

Tahmini Süre:1m 45s

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