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Zorluk: OrtaPercentages, Percent Change, and Interest

In Year 1, a technology firm allocated 40%40\% of its total operating budget to Research and Development (R&D) and the remaining 60%60\% to Business Operations. In Year 2, the amount allocated to R&D was increased by 25%25\% over its Year 1 amount, while the amount allocated to Business Operations was decreased by 15%15\% from its Year 1 amount. If the firm's total operating budget in Year 2 was $505,000\$505,000, what was its total operating budget in Year 1?

  1. A
    $454,500\$454,500
  2. B
    $480,952\$480,952
  3. $500,000\$500,000Cevap
  4. D
    $499,950\$499,950
  5. E
    $510,050\$510,050

Cevap

The total operating budget in Year 1 was $500,000\$500,000.
The budget allocation in Year 1 assigns 40%40\% to R&D and 60%60\% to Operations. In Year 2, R&D becomes 0.40×1.25=0.500.40 \times 1.25 = 0.50 of the Year 1 budget, and Operations becomes 0.60×0.85=0.510.60 \times 0.85 = 0.51 of the Year 1 budget. Adding these yields a Year 2 total equal to 1.011.01 times the Year 1 budget. Setting 1.01×Year 1 Budget=$505,0001.01 \times \text{Year 1 Budget} = \$505,000 and dividing gives $500,000\$500,000.

Adım Adım Çözüm

1
Define Year 1 components in terms of total Year 1 budget BB.
R&D allocation = 0.40B0.40B; Operations allocation = 0.60B0.60B.
The problem specifies a 40%/60%40\% / 60\% split of the total budget BB in Year 1.
2
Calculate Year 2 allocations after percentage changes.
Year 2 R&D = 0.40B×1.25=0.50B0.40B \times 1.25 = 0.50B; Year 2 Operations = 0.60B×0.85=0.51B0.60B \times 0.85 = 0.51B.
R&D increases by 25%25\% (1+0.25=1.251 + 0.25 = 1.25) and Operations decreases by 15%15\% (1−0.15=0.851 - 0.15 = 0.85).
3
Sum Year 2 allocations to express total Year 2 budget in terms of BB.
Total Year 2 Budget = 0.50B+0.51B=1.01B0.50B + 0.51B = 1.01B.
Combining both component allocations gives the total budget for Year 2.
4
Solve for the Year 1 budget BB using the given Year 2 total.
1.01B=$505,000  ⟹  B=505,0001.01=$500,0001.01B = \$505,000 \implies B = \frac{505,000}{1.01} = \$500,000.
Dividing the Year 2 budget by the total growth multiplier yields the original Year 1 budget.

Anahtar Kavram

Weighted Percent Change and Initial Base Calculation
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Konular

Percentages, Percent Change, and Interest

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