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Difficulty: MediumNumerical, Proportion, and Statistical Inferences

In 2021, digital audiobook licenses accounted for exactly 20 percent of all new catalog acquisitions made by the City Library System, with physical printed books accounting for the remaining 80 percent. By 2025, digital audiobook licenses represented 40 percent of all new catalog acquisitions, with physical printed books making up the remaining 60 percent. The total number of physical printed books acquired by the library system in 2025 was exactly the same as the total number of physical printed books acquired in 2021.

If the statements above are true, which of the following must also be true on the basis of them?

  1. The total number of new catalog acquisitions made by the City Library System was greater in 2025 than in 2021.Answer
  2. B
    The total number of physical printed books acquired by the library system decreased steadily each year between 2021 and 2025.
  3. C
    Physical printed books made up a smaller percentage of the library system's total cumulative collection in 2025 than they did in 2021.
  4. D
    The library system spent more money on acquiring digital audiobook licenses in 2025 than it spent on physical printed books in 2021.
  5. E
    The increased proportion of digital audiobook acquisitions caused a corresponding decline in physical book borrowings by library patrons.

Answer

The total number of new catalog acquisitions made by the City Library System was greater in 2025 than in 2021.
The conclusion that total acquisitions were greater in 2025 than in 2021 is logically inescapable. In 2021, physical printed books constituted 80 percent of total acquisitions (P=0.80T2021P = 0.80 T_{2021}). In 2025, physical printed books constituted 60 percent of total acquisitions (P=0.60T2025P = 0.60 T_{2025}). Because the absolute number of physical printed books (PP) was identical in both years, 0.60T2025=0.80T20210.60 T_{2025} = 0.80 T_{2021}, which simplifies to T2025=43T2021T_{2025} = \frac{4}{3} T_{2021}. Thus, total acquisitions in 2025 were 33 percent higher than in 2021.

Step-by-Step Solution

1
Express physical book acquisitions as a function of total acquisitions for both years
In 2021, P2021=0.80×T2021P_{2021} = 0.80 \times T_{2021}. In 2025, P2025=0.60×T2025P_{2025} = 0.60 \times T_{2025}, where PP represents physical books and TT represents total acquisitions.
Establishing algebraic relationships allows comparison between proportions and absolute amounts.
2
Apply the given condition that physical book volume remained constant
Since P2021=P2025P_{2021} = P_{2025}, set the equations equal: 0.80×T2021=0.60×T20250.80 \times T_{2021} = 0.60 \times T_{2025}.
Directly uses the premise stating that physical book acquisition counts were identical.
3
Solve for the ratio between total acquisitions in 2025 and 2021
T2025=0.800.60×T2021=43×T2021≈1.33×T2021T_{2025} = \frac{0.80}{0.60} \times T_{2021} = \frac{4}{3} \times T_{2021} \approx 1.33 \times T_{2021}.
Proves mathematically that total acquisitions in 2025 must be larger than in 2021.

Key Concept

Numerical, Proportion, and Statistical Inferences
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Topics

Numerical, Proportion, and Statistical Inferences

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