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Difficulty: Very hardDrawing Inferences and Valid Conclusions

Prior to 2022, technology firm X generated over 80 percent of its annual revenue from software licensing, with the remainder derived from consulting services. In 2022, Firm X introduced a cloud platform, offering it exclusively via usage-based subscriptions. By 2025, Firm X's total annual revenue was twice that of its 2021 revenue. However, its 2025 software licensing revenue had dropped by exactly 50 percent relative to 2021, while its consulting revenue remained unchanged. All 2025 cloud platform subscription revenues were generated from clients who had never previously purchased software licenses or consulting services from Firm X.

If the statements above are true, which of the following must also be true about Firm X in 2025?

  1. A
    The average subscription fee paid per client for the cloud platform in 2025 was higher than the average fee paid per client for software licensing in 2021.
  2. B
    In 2025, revenue from consulting services represented a larger percentage of Firm X's total annual revenue than it did in 2021.
  3. Clients who had never purchased products or services from Firm X prior to 2022 accounted for more than half of the firm's total annual revenue in 2025.Answer
  4. D
    At least some clients who purchased software licenses from Firm X in 2021 canceled their licenses to switch to competitor software by 2025.
  5. E
    Every client who purchased consulting services from Firm X in 2025 had also purchased software licenses from Firm X in 2021.

Answer

In 2025, clients who had never purchased products or services from Firm X prior to 2022 accounted for more than half of the firm's total annual revenue.
The correct choice must strictly follow from combining the mathematical premises. In 2021, licensing was over 80% of revenue and consulting was under 20%. In 2025, total revenue doubled (2R2R). Licensing revenue fell to half its 2021 level (under 40% of RR, or under 20% of 2R2R), while consulting revenue remained unchanged (under 20% of RR, or under 10% of 2R2R). Thus, all non-cloud revenue combined accounted for under 30% of 2025 total revenue (2R2R). Since all cloud platform revenue came from new clients who had never purchased from Firm X prior to 2022, these new clients generated over 70% of 2025 total revenue, which is strictly greater than half.

Step-by-Step Solution

1
Express the 2021 revenue breakdown algebraically.
Let RR be the total revenue in 2021. Software licensing revenue L2021>0.80RL_{2021} > 0.80R and consulting revenue C2021<0.20RC_{2021} < 0.20R, with L2021+C2021=RL_{2021} + C_{2021} = R.
Establishing 2021 baseline figures allows precise comparison with 2025 figures.
2
Calculate the maximum revenue generated in 2025 from clients who purchased products prior to 2022.
In 2025, total revenue is 2R2R. Licensing revenue L2025=0.50×L2021<0.40RL_{2025} = 0.50 \times L_{2021} < 0.40R. Consulting revenue C2025=C2021<0.20RC_{2025} = C_{2021} < 0.20R. Total revenue from returning client streams is L2025+C2025<0.40R+0.20R=0.60RL_{2025} + C_{2025} < 0.40R + 0.20R = 0.60R.
Since cloud subscription clients are strictly new clients, returning client revenue consists solely of licensing and consulting.
3
Calculate the minimum contribution of new cloud platform clients to 2025 total revenue.
Cloud platform subscription revenue P2025=2R−(L2025+C2025)>2R−0.60R=1.40RP_{2025} = 2R - (L_{2025} + C_{2025}) > 2R - 0.60R = 1.40R. As a fraction of 2025 total revenue (2R2R), cloud revenue is P20252R>1.40R2R=70%\frac{P_{2025}}{2R} > \frac{1.40R}{2R} = 70\%.
Comparing new client revenue to the 2025 total (2R2R) proves that new clients accounted for over 70% of total revenue, which is strictly greater than 50%.

Key Concept

Quantitative and Proportional Deductions in Logical Arguments
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Topics

Drawing Inferences and Valid Conclusions

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