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Difficulty: HardPercentages, Percent Change, and Interest

The market value of a rare piece of artwork increased by x%x\% during its first decade in a gallery and then decreased by x%x\% during its second decade. If the artwork's market value at the end of the second decade was $19,200\$19,200, which represented a net decrease of $800\$800 from its initial market value prior to the two decades, what is the value of xx?

  1. A
    44
  2. B
    1010
  3. C
    1616
  4. 2020Answer
  5. E
    2525

Answer

The value of xx is 2020.
The initial value of the artwork is $19,200+$800=$20,000\$19,200 + \$800 = \$20,000. Applying an increase of x%x\% followed by a decrease of x%x\% gives a net value of $20,000×(1+x100)(1−x100)=$20,000(1−x210,000)\$20,000 \times (1 + \frac{x}{100})(1 - \frac{x}{100}) = \$20,000 \left(1 - \frac{x^2}{10,000}\right). Setting this equal to $19,200\$19,200 yields 1−x210,000=0.961 - \frac{x^2}{10,000} = 0.96, which simplifies to x210,000=0.04\frac{x^2}{10,000} = 0.04, so x2=400x^2 = 400 and x=20x = 20.

Step-by-Step Solution

1
Determine the initial market value before the two decades.
Initial Value =$19,200+$800=$20,000= \$19,200 + \$800 = \$20,000.
The final value of $19,200\$19,200 reflects a net decrease of $800\$800 from the original starting value.
2
Express the successive percentage changes in terms of xx.
Final Value =Initial Value×(1+x100)×(1−x100)=Initial Value×(1−x210,000)= \text{Initial Value} \times \left(1 + \frac{x}{100}\right) \times \left(1 - \frac{x}{100}\right) = \text{Initial Value} \times \left(1 - \frac{x^2}{10,000}\right).
An increase of x%x\% followed by a decrease of x%x\% on the new base yields the difference of squares formula for percent multipliers.
3
Set up the equation with known numerical values and solve for x2x^2.
20,000(1−x210,000)=19,200  ⟹  1−x210,000=19,20020,000=0.96  ⟹  x210,000=0.04  ⟹  x2=40020,000 \left(1 - \frac{x^2}{10,000}\right) = 19,200 \implies 1 - \frac{x^2}{10,000} = \frac{19,200}{20,000} = 0.96 \implies \frac{x^2}{10,000} = 0.04 \implies x^2 = 400.
Dividing the final value by the initial value gives the net multiplier of 0.960.96, meaning a 4%4\% overall decrease.
4
Take the positive square root to find xx.
x=400=20x = \sqrt{400} = 20.
Since xx represents a positive percentage rate, x=20x = 20.

Key Concept

Successive percentage changes and base shifts using the difference of squares (1+k)(1−k)=1−k2(1 + k)(1 - k) = 1 - k^2.
Estimated Time:2m 0s
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Topics

Percentages, Percent Change, and Interest

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