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Zorluk: ZorPercentages, Percent Change, and Interest

At the start of 2024, a technology company allocated a total research budget of BB dollars between Department X and Department Y, such that Department X received 60%60\% of the total budget and Department Y received the remaining 40%40\%. In 2025, Department X's budget was increased by 25%25\%, while Department Y's budget was decreased by 20%20\%. In 2026, the combined budget of both departments was reduced by a uniform 10%10\%. If the total combined budget of the two departments in 2026 was $288,900\$288,900, what was the initial total research budget BB at the start of 2024?

  1. A
    $270,000\$270,000
  2. B
    $297,000\$297,000
  3. $300,000\$300,000Cevap
  4. D
    $321,000\$321,000
  5. E
    $340,000\$340,000

Cevap

$300,000\$300,000
The initial budget BB is divided into 0.60B0.60B for Department X and 0.40B0.40B for Department Y. In 2025, Department X's budget becomes 0.60B×1.25=0.75B0.60B \times 1.25 = 0.75B and Department Y's budget becomes 0.40B×0.80=0.32B0.40B \times 0.80 = 0.32B, yielding a combined 2025 budget of 0.75B+0.32B=1.07B0.75B + 0.32B = 1.07B. In 2026, a 10%10\% reduction results in a combined budget of 1.07B×0.90=0.963B1.07B \times 0.90 = 0.963B. Setting 0.963B=288,9000.963B = 288,900 and dividing gives B=$300,000B = \$300,000.

Adım Adım Çözüm

1
Express the 2024 budget allocations for each department in terms of BB.
Department X received 0.60B0.60B and Department Y received 0.40B0.40B.
Department X was allocated 60%60\% of the total initial budget BB, leaving 40%40\% for Department Y.
2
Calculate the budget of each department in 2025 after their respective percentage changes.
Department X: 0.60B×(1+0.25)=0.75B0.60B \times (1 + 0.25) = 0.75B; Department Y: 0.40B×(1−0.20)=0.32B0.40B \times (1 - 0.20) = 0.32B.
Department X increased by 25%25\% (1.251.25 multiplier) and Department Y decreased by 20%20\% (0.800.80 multiplier).
3
Determine the combined budget for 2025 and apply the 2026 reduction.
Combined 2025 budget = 0.75B+0.32B=1.07B0.75B + 0.32B = 1.07B. Combined 2026 budget = 1.07B×(1−0.10)=0.963B1.07B \times (1 - 0.10) = 0.963B.
The two department budgets sum to 1.07B1.07B, which then undergoes a uniform 10%10\% reduction (0.900.90 multiplier).
4
Solve for the initial budget BB using the given 2026 total of $288,900\$288,900.
0.963B=288,900  ⟹  B=288,9000.963=300,0000.963B = 288,900 \implies B = \frac{288,900}{0.963} = 300,000.
Dividing the 2026 total by the net multiplier 0.9630.963 gives the original budget BB.

Anahtar Kavram

Successive percentage changes with initial weighted allocations
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Konular

Percentages, Percent Change, and Interest

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