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Difficulty: HardAlgebraic Word Problems and Equation Modeling

A software provider offers two cloud subscription plans for enterprise clients: Tier A and Tier B. The monthly fee per user is $x\$x for Tier A and $y\$y for Tier B. In 2024, a corporate customer purchased 50 user licenses for Tier A and 30 user licenses for Tier B, resulting in a total monthly cost of $13,500\$13,500.

In 2025, the provider adjusted its rates by increasing the Tier A monthly fee per user by 20%20\% and increasing the Tier B monthly fee per user by 10%10\%. Additionally, the customer altered its order by increasing its Tier A licenses by 10%10\% and decreasing its Tier B licenses by 20%20\%. If the customer's total monthly cost in 2025 was $15,180\$15,180, what was the original monthly fee per user for Tier A in 2024?

  1. A
    $125\$125
  2. $150\$150Answer
  3. C
    $175\$175
  4. D
    $200\$200
  5. E
    $225\$225

Answer

The original monthly fee per user for Tier A in 2024 was $150\$150.
The correct option correctly models the two-variable linear system. By expressing the 2024 total cost as 5x+3y=1,3505x + 3y = 1,350 and the 2025 cost as 66x+26.4y=15,18066x + 26.4y = 15,180, solving the simultaneous equations yields x=150x = 150 for Tier A and y=200y = 200 for Tier B.

Step-by-Step Solution

1
Formulate the linear equation for the 2024 monthly cost.
50x+30y=13,50050x + 30y = 13,500, which simplifies to 5x+3y=1,3505x + 3y = 1,350.
The total cost is the sum of the products of license quantities and their respective per-user prices.
2
Determine the 2025 price per user and license quantities for both tiers.
Tier A: New price = 1.20x1.20x, New quantity = 50×1.10=5550 \times 1.10 = 55.
Tier B: New price = 1.10y1.10y, New quantity = 30×0.80=2430 \times 0.80 = 24.
Price and quantity changes must be applied independently as percentage increases or decreases.
3
Formulate the linear equation for the 2025 total monthly cost.
55(1.20x)+24(1.10y)=15,180  ⟹  66x+26.4y=15,18055(1.20x) + 24(1.10y) = 15,180 \implies 66x + 26.4y = 15,180.
Multiply the updated unit price by the updated quantity for each tier.
4
Simplify the 2025 equation and solve the system of linear equations.
Multiply 66x+26.4y=15,18066x + 26.4y = 15,180 by 55 to eliminate decimals: 330x+132y=75,900330x + 132y = 75,900, which simplifies by dividing by 66 to 55x+22y=12,65055x + 22y = 12,650.
Multiply 5x+3y=1,3505x + 3y = 1,350 by 1111: 55x+33y=14,85055x + 33y = 14,850.
Subtract the two equations: (55x+33y)−(55x+22y)=14,850−12,650  ⟹  11y=2,200  ⟹  y=200(55x + 33y) - (55x + 22y) = 14,850 - 12,650 \implies 11y = 2,200 \implies y = 200.
Substitute y=200y = 200 into 5x+3(200)=1,350  ⟹  5x=750  ⟹  x=1505x + 3(200) = 1,350 \implies 5x = 750 \implies x = 150.
Eliminate variable xx to solve for yy, then back-substitute to find xx.

Key Concept

Multi-variable system modeling with multi-step percentage modifications
Estimated Time:2m 30s
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Topics

Algebraic Word Problems and Equation Modeling

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