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Difficulty: Very hardPercentages, Percent Change, and Interest

At the beginning of 2025, Company X's allocation for Marketing was 40%40\% greater than its allocation for Research. Over the course of the year, the Marketing allocation was decreased by 25%25\%, while the Research allocation was increased by p%p\%. At the end of 2025, the revised Research allocation was 20%20\% greater than the revised Marketing allocation. What is the value of pp?

  1. A
    20
  2. B
    25
  3. 26Answer
  4. D
    38
  5. E
    50

Answer

The value of pp is 26.
The correct option correctly applies successive percentage changes to each variable. Starting with Research as RR, Marketing is 1.40R1.40R. Reducing Marketing by 25%25\% yields 1.40R×0.75=1.05R1.40R \times 0.75 = 1.05R. Increasing this result by 20%20\% gives the final Research amount of 1.20×1.05R=1.26R1.20 \times 1.05R = 1.26R, corresponding to a 26%26\% increase over the initial Research allocation RR.

Step-by-Step Solution

1
Define variables for the initial allocations of both departments.
Let RR represent the initial Research allocation. Then the initial Marketing allocation is 1.40R1.40R.
Marketing was 40%40\% greater than Research at the start of the year.
2
Calculate the revised Marketing allocation at the end of the year.
Revised Marketing allocation =1.40R×(1−0.25)=1.40R×0.75=1.05R= 1.40R \times (1 - 0.25) = 1.40R \times 0.75 = 1.05R.
The Marketing allocation decreased by 25%25\% from its initial value.
3
Determine the revised Research allocation using the end-of-year relationship.
Revised Research allocation =1.20×1.05R=1.26R= 1.20 \times 1.05R = 1.26R.
The revised Research allocation was 20%20\% greater than the revised Marketing allocation.
4
Find the percentage increase pp for the Research allocation.
R(1+p100)=1.26R  ⟹  1+p100=1.26  ⟹  p=26R \left(1 + \frac{p}{100}\right) = 1.26R \implies 1 + \frac{p}{100} = 1.26 \implies p = 26.
The change from RR to 1.26R1.26R represents a 26%26\% increase.

Key Concept

Successive percent changes and base shifts in multi-variable algebraic scenarios
Estimated Time:2m 30s
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Topics

Percentages, Percent Change, and Interest

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