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Difficulty: Very hardFractions, Decimals, and Percentages

A company allocates its annual budget to three departments: Research, Marketing, and Operations. Initially, the Research department receives 14\frac{1}{4} of the total budget. Marketing receives 40%40\% of the remaining budget, and Operations receives the rest. Mid-year, the budget is adjusted: Research's budget is decreased by 16%16\% of its initial allocation, and Marketing's budget is increased by 112\frac{1}{12} of its initial allocation. If the total company budget remains unchanged, by what percentage must the Operations department's budget increase relative to its initial allocation?

  1. A
    1.5%1.5\%
  2. 313%3\frac{1}{3}\%Answer
  3. C
    513%5\frac{1}{3}\%
  4. D
    723%7\frac{2}{3}\%
  5. E
    1449%14\frac{4}{9}\%

Answer

The Operations department's budget must increase by 313%3\frac{1}{3}\% relative to its initial allocation.
The correct answer is 313%3\frac{1}{3}\%. Under a total budget of 1,0001,000 units, Research initially receives 250250 units, Marketing receives 300300 units (40%40\% of the remaining 750750), and Operations receives the remaining 450450 units. The mid-year decrease for Research is 4040 units (16%16\% of 250250), and the increase for Marketing is 2525 units (112\frac{1}{12} of 300300). To keep the total budget constant, the Operations budget must increase by 1515 units (40−2540 - 25). Relative to its initial budget of 450450 units, this represents an increase of 15450×100%=313%\frac{15}{450} \times 100\% = 3\frac{1}{3}\%.

Step-by-Step Solution

1
Set up a total initial budget and find initial allocations for each department.
Let the total initial budget be 1,0001,000 units. Research receives 14\frac{1}{4} of the budget, which is 250250 units. The remaining budget is 1,000−250=7501,000 - 250 = 750 units. Marketing receives 40%40\% of this remaining budget, which is 0.40×750=3000.40 \times 750 = 300 units. Operations receives the rest, which is 1,000−250−300=4501,000 - 250 - 300 = 450 units.
Establishing the initial values of each department's budget provides a base to calculate subsequent changes.
2
Calculate the mid-year budget changes for Research and Marketing.
Research decreases by 16%16\%: −0.16×250=−40-0.16 \times 250 = -40 units. Marketing increases by 112\frac{1}{12}: +112×300=+25+\frac{1}{12} \times 300 = +25 units.
Finding the individual change values shows the net budget surplus or deficit before adjusting Operations.
3
Determine the required budget change for the Operations department.
To keep the total budget unchanged, the net sum of changes must be zero: −40+25+ΔO=0⇒ΔO=+15-40 + 25 + \Delta O = 0 \Rightarrow \Delta O = +15 units.
The change in the Operations budget must balance out the changes in the other two departments.
4
Calculate the percentage increase for the Operations department relative to its initial budget.
Percentage increase = 15450×100%=130×100%=103%=313%\frac{15}{450} \times 100\% = \frac{1}{30} \times 100\% = \frac{10}{3}\% = 3\frac{1}{3}\%.
Dividing the change in Operations by its initial budget yields the relative percentage increase.

Key Concept

Calculating percentage changes across weighted parts of a whole
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Topics

Fractions, Decimals, and Percentages

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