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Difficulty: Very hardPercentages, Percent Change, and Interest

An investment fund managed two separate accounts, Account X and Account Y, over a two-year period.

- At the start of Year 1, Account X held $100,000\$100,000 and Account Y held $150,000\$150,000.
- During Year 1, Account X increased in value by 20%20\%, while Account Y decreased in value by x%x\%.
- During Year 2, Account X decreased in value by x%x\%, while Account Y increased in value by 20%20\%.
- At the end of Year 2, the combined total value of both accounts was $243,000\$243,000.

Which of the following statements must be true? Select all such statements.

  1. The value of xx is equal to 1919.Answer
  2. At the end of Year 2, Account X experienced a net decrease of 2.8%2.8\% relative to its initial value.Answer
  3. At the end of Year 1, Account Y had a greater dollar value than Account X.Answer
  4. D
    At the end of Year 2, the overall percentage decrease in the value of Account Y from its initial value was greater than that of Account X.
  5. E
    The overall percent change of the combined portfolio over the two-year period was a decrease of 7.0%7.0\%.

Answer

The correct statements are those asserting that x=19x = 19, that Account X experienced a net decrease of 2.8%2.8\% relative to its initial value, and that at the end of Year 1 Account Y had a greater dollar value than Account X.
The statement specifying that x=19x = 19 is correct because solving the combined account equation $300,000(1−x/100)=$243,000\$300,000(1 - x/100) = \$243,000 yields 1−x/100=0.811 - x/100 = 0.81, giving x=19x = 19. The statement regarding Account X experiencing a net decrease of 2.8%2.8\% is correct because 1.20×0.81=0.9721.20 \times 0.81 = 0.972, which corresponds to a 2.8%2.8\% decrease from initial value. The statement asserting that Account Y had a greater dollar value at the end of Year 1 than Account X is correct because Account Y was worth $121,500\$121,500 compared to $120,000\$120,000 for Account X.

Step-by-Step Solution

1
Express the value of each account at the end of Year 1 in terms of xx.
Account X value = $100,000×(1+0.20)=$120,000\$100,000 \times (1 + 0.20) = \$120,000. Account Y value = $150,000×(1−x/100)\$150,000 \times (1 - x/100).
Calculate the intermediate values after Year 1 percentage changes.
2
Express the value of each account at the end of Year 2 and sum them to form an equation for the combined total.
Account X end of Year 2 = $120,000(1−x/100)\$120,000(1 - x/100). Account Y end of Year 2 = $150,000(1−x/100)(1+0.20)=$180,000(1−x/100)\$150,000(1 - x/100)(1 + 0.20) = \$180,000(1 - x/100). Combined total = ($120,000+$180,000)(1−x/100)=$300,000(1−x/100)=$243,000(\$120,000 + \$180,000)(1 - x/100) = \$300,000(1 - x/100) = \$243,000.
Use the given final total value of $243,000\$243,000 to solve for xx.
3
Solve for xx.
1−x/100=243,000/300,000=0.81  ⟹  x/100=0.19  ⟹  x=191 - x/100 = 243,000 / 300,000 = 0.81 \implies x/100 = 0.19 \implies x = 19.
Find the numerical value of xx.
4
Evaluate each statement against the calculated values.
1. x=19x = 19 is true.
2. Net change multiplier for Account X = 1.20×0.81=0.9721.20 \times 0.81 = 0.972, representing a 2.8%2.8\% decrease (true).
3. End of Year 1 values: Account X = $120,000\$120,000, Account Y = $150,000×0.81=$121,500\$150,000 \times 0.81 = \$121,500. Since $121,500>$120,000\$121,500 > \$120,000, Account Y was greater (true).
4. Account Y net change multiplier = 0.81×1.20=0.9720.81 \times 1.20 = 0.972, identical to Account X, so percentage decreases are equal (false).
5. Overall combined percent change = (243,000−250,000)/250,000=−7,000/250,000=−2.8%(243,000 - 250,000)/250,000 = -7,000/250,000 = -2.8\% (false).
Determine which statements are mathematically true.

Key Concept

Successive Percentage Changes and Base Shifts in Multi-Asset Portfolios
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Topics

Percentages, Percent Change, and Interest

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