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Difficulty: HardFractions, Decimals, and Percents Arithmetic

At a technology firm, 38\frac{3}{8} of the total annual operating budget was initially allocated to Research & Development, 0.350.35 of the budget was allocated to Marketing, and the remainder was allocated to Operations. Mid-year, the Marketing allocation was increased by 20%20\% of its initial value, and the Operations allocation was decreased by 40%40\% of its initial value, while the Research & Development allocation remained unchanged. By what net percentage did the firm's total annual operating budget change?

  1. A
    A decrease of 20%20\%
  2. B
    A decrease of 6%6\%
  3. A decrease of 4%4\%Answer
  4. D
    An increase of 4%4\%
  5. E
    A decrease of 15%15\%

Answer

A net decrease of 4%4\%
Converting 38\frac{3}{8} to 37.5%37.5\% leaves 27.5%27.5\% for Operations (100%−37.5%−35%100\% - 37.5\% - 35\%). A 20%20\% increase on the 35%35\% Marketing budget adds 7%7\% to the overall total (0.20×35%=+7%0.20 \times 35\% = +7\%). A 40%40\% decrease on the 27.5%27.5\% Operations budget reduces the overall total by 11%11\% (0.40×27.5%=−11%0.40 \times 27.5\% = -11\%). Summing these changes gives +7%−11%=−4%+7\% - 11\% = -4\%, representing a net decrease of 4%4\%.

Step-by-Step Solution

1
Convert initial fractional and decimal allocations to percentages of the total budget
Research & Development allocation = 38=37.5%\frac{3}{8} = 37.5\%; Marketing allocation = 0.35=35%0.35 = 35\%
Converting all components to a common percentage format enables straightforward comparison and arithmetic.
2
Calculate the initial percentage allocated to Operations
Operations allocation = 100%−(37.5%+35%)=100%−72.5%=27.5%100\% - (37.5\% + 35\%) = 100\% - 72.5\% = 27.5\%
The remainder of the budget after R&D and Marketing constitutes the Operations department share.
3
Determine the net percentage change contributed by each department's adjustment
Marketing change = +20% of 35%=+7%+20\% \text{ of } 35\% = +7\% of total budget; Operations change = −40% of 27.5%=−11%-40\% \text{ of } 27.5\% = -11\% of total budget; R&D change = 0%0\%
Each departmental percentage change must be weighted by that department's portion of the overall budget.
4
Sum the net contributions to find the overall budget change
Net overall change = +7%−11%=−4%+7\% - 11\% = -4\% (a decrease of 4%4\%)
Combining the positive and negative adjustments yields the overall net percentage change relative to the initial budget.

Key Concept

Weighted Percent Change across Mixed Fractional and Decimal Sub-components
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Topics

Fractions, Decimals, and Percents Arithmetic

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