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Difficulty: HardRates, Ratios, Percentages, and Applied Word Problems

A private equity firm allocated its initial capital between two portfolio ventures: Venture X and Venture Y. In 2025, Venture X yielded a profit equal to 20%20\% of its initial investment, while Venture Y incurred a loss equal to 10%10\% of its initial investment. What was the firm's overall percentage profit or loss across both ventures combined?

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrasing the question target shows that the overall percentage return depends strictly on the ratio of the investment amounts, XY\frac{X}{Y}. Statement (1) directly gives X=1.5YX = 1.5Y, establishing a fixed ratio XY=32\frac{X}{Y} = \frac{3}{2}, which uniquely yields an overall profit of 8%8\%. Statement (2) reveals the dollar investment in Venture X (200,000200,000) but leaves the investment in Venture Y completely unconstrained, making it impossible to compute a unique overall percentage.

Step-by-Step Solution

1
Rephrase the target question algebraically.
Let XX be the initial investment in Venture X and YY be the initial investment in Venture Y. The total net profit is 0.20X−0.10Y0.20X - 0.10Y, and the total investment is X+YX + Y. The overall percentage return is given by 0.20X−0.10YX+Y×100%=(0.20(XX+Y)−0.10(YX+Y))×100%\frac{0.20X - 0.10Y}{X + Y} \times 100\% = \left(0.20 \left(\frac{X}{X+Y}\right) - 0.10 \left(\frac{Y}{X+Y}\right)\right) \times 100\%. Thus, finding the ratio XY\frac{X}{Y} is sufficient to determine the overall return.
Data Sufficiency stems asking for a combined percentage rate depend only on the ratio of the component base values, not their absolute amounts.
2
Evaluate Statement (1): The amount invested in Venture X was 50%50\% greater than the amount invested in Venture Y.
This implies X=1.5Y=32YX = 1.5Y = \frac{3}{2}Y, so XY=32\frac{X}{Y} = \frac{3}{2}. Substituting X=1.5YX = 1.5Y into the overall return formula yields 0.20(1.5Y)−0.10Y1.5Y+Y=0.30Y−0.10Y2.5Y=0.20Y2.5Y=0.202.5=0.08=8%\frac{0.20(1.5Y) - 0.10Y}{1.5Y + Y} = \frac{0.30Y - 0.10Y}{2.5Y} = \frac{0.20Y}{2.5Y} = \frac{0.20}{2.5} = 0.08 = 8\% profit.
Since a unique overall percentage (8% profit) is determined, Statement (1) alone is sufficient.
3
Evaluate Statement (2): The dollar amount of profit generated by Venture X was $40,000\$40,000.
This gives 0.20X=40,000  ⟹  X=200,0000.20X = 40,000 \implies X = 200,000. However, no information is provided about the value of YY. If Y=100,000Y = 100,000, overall profit is 40,000−10,000300,000=10%\frac{40,000 - 10,000}{300,000} = 10\%. If Y=400,000Y = 400,000, overall profit is 40,000−40,000600,000=0%\frac{40,000 - 40,000}{600,000} = 0\%.
Multiple overall percentage returns are possible depending on YY, so Statement (2) alone is not sufficient.

Key Concept

Weighted Average Percentage Yields and Stem Simplification in Data Sufficiency
Estimated Time:2m 0s
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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