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Difficulty: HardPercentages, Percent Change, and Interest

At the beginning of Year 1, a commercial logistics company allocated its total storage space between two facilities, Facility X and Facility Y, such that Facility X contained 60%60\% of the total space and Facility Y contained the remaining 40%40\%. During Year 1, the storage space of Facility X was increased by x%x\%, while the storage space of Facility Y was increased by 25%25\%. During Year 2, the storage space of Facility X was decreased by 20%20\% from its Year 1 level, while the storage space of Facility Y was increased by x%x\% from its Year 1 level. If the total combined storage space of both facilities at the end of Year 2 was 17.6%17.6\% greater than the total combined storage space at the beginning of Year 1, what is the value of xx?

  1. A
    15
  2. B
    18.5
  3. C
    19.6
  4. 20Answer
  5. E
    25

Answer

The value of xx is 20.
Let the initial total space be AA. Facility X initially has 0.60A0.60A space and Facility Y has 0.40A0.40A space. After Year 1, Facility X space becomes 0.60A(1+x/100)0.60A(1 + x/100) and Facility Y space becomes 0.40A(1.25)=0.50A0.40A(1.25) = 0.50A. At the end of Year 2, Facility X space decreases by 20%20\% to 0.48A(1+x/100)0.48A(1 + x/100), while Facility Y space increases by x%x\% to 0.50A(1+x/100)0.50A(1 + x/100). Factoring out (1+x/100)(1 + x/100), the total final combined space is (0.48A+0.50A)(1+x/100)=0.98A(1+x/100)(0.48A + 0.50A)(1 + x/100) = 0.98A(1 + x/100). Setting this equal to 1.176A1.176A (a 17.6%17.6\% increase over AA) yields 1+x/100=1.176/0.98=1.201 + x/100 = 1.176 / 0.98 = 1.20, which gives x=20x = 20.

Step-by-Step Solution

1
Define initial storage spaces for Facility X and Facility Y in terms of total initial area AA.
Facility X initial space = 0.60A0.60A, Facility Y initial space = 0.40A0.40A.
Establishing explicit algebraic representations based on the given 60%/40%60\% / 40\% ratio.
2
Calculate the storage space of each facility after Year 1 changes.
Facility X after Year 1 = 0.60A(1+x100)0.60A \left(1 + \frac{x}{100}\right); Facility Y after Year 1 = 0.40A(1+0.25)=0.50A0.40A (1 + 0.25) = 0.50A.
Facility X increased by x%x\% and Facility Y increased by 25%25\% of its original base.
3
Calculate the storage space of each facility at the end of Year 2.
Facility X final = 0.60A(1+x100)(1−0.20)=0.48A(1+x100)0.60A \left(1 + \frac{x}{100}\right) (1 - 0.20) = 0.48A \left(1 + \frac{x}{100}\right); Facility Y final = 0.50A(1+x100)0.50A \left(1 + \frac{x}{100}\right).
Facility X decreased by 20%20\% from its Year 1 amount, and Facility Y increased by x%x\% from its Year 1 amount.
4
Sum the final spaces of both facilities and set equal to the total space given (17.6%17.6\% overall increase over AA).
Total final space = 0.48A(1+x100)+0.50A(1+x100)=0.98A(1+x100)=1.176A0.48A \left(1 + \frac{x}{100}\right) + 0.50A \left(1 + \frac{x}{100}\right) = 0.98A \left(1 + \frac{x}{100}\right) = 1.176A.
The total space at the end of Year 2 is 100%+17.6%=117.6%100\% + 17.6\% = 117.6\% of initial total space AA.
5
Solve the linear equation for xx.
1+x100=1.1760.98=1.20  ⟹  x100=0.20  ⟹  x=201 + \frac{x}{100} = \frac{1.176}{0.98} = 1.20 \implies \frac{x}{100} = 0.20 \implies x = 20.
Dividing both sides by 0.98A0.98A isolates the percentage factor.

Key Concept

Successive Percent Change and Multi-Base Percentage Problems
Estimated Time:2m 30s
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Topics

Percentages, Percent Change, and Interest

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