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Difficulty: MediumAlgebraic Word Problems and Equation Modeling

A software enterprise sells two annual licensing tiers, Tier 1 and Tier 2. In January, the company sold a combined total of 200200 licenses for the two tiers. The price of a Tier 1 license was $40\$40 and the price of a Tier 2 license was $60\$60. In February, the price of a Tier 1 license increased by 25%25\% and the number of Tier 1 licenses sold increased by 50%50\%, while both the price and the number of Tier 2 licenses sold remained unchanged from January. If the total monthly revenue from these two tiers increased by $2,800\$2,800 in February compared to January, how many Tier 1 licenses were sold in January?

  1. A
    60
  2. B
    70
  3. 80Answer
  4. D
    100
  5. E
    120

Answer

80 Tier 1 licenses were sold in January.
The correct answer is 80. Modeling the sales volume of Tier 1 in January as xx, Tier 1 generated 40x40x dollars in January. In February, with a 25%25\% price increase (to $50\$50) and a 50%50\% volume increase (to 1.5x1.5x), Tier 1 generated 50(1.5x)=75x50(1.5x) = 75x dollars. Because Tier 2 price and sales volume remained constant, the total revenue increase of $2,800\$2,800 is equal to the change in Tier 1 revenue: 75x−40x=35x=2,80075x - 40x = 35x = 2,800, which gives x=80x = 80.

Step-by-Step Solution

1
Define variables for January sales volume and set up equations for each tier.
Let xx be the number of Tier 1 licenses sold in January. The number of Tier 2 licenses sold in January is 200−x200 - x.
The total number of licenses sold in January was given as 200.
2
Calculate January revenue from Tier 1.
\text{Revenue}_{\text{Tier 1, Jan}} = 40x.
Tier 1 licenses cost $40\$40 each in January.
3
Determine February price and quantity for Tier 1, and calculate February Tier 1 revenue.
\text{Price}_{\text{Tier 1, Feb}} = 40 \times 1.25 = \5050; \text{Quantity}_{\text{Tier 1, Feb}} = 1.5x$. Thus, \text{Revenue}_{\text{Tier 1, Feb}} = 50(1.5x) = 75x.
Tier 1 price increased by 25%25\% and volume increased by 50%50\%.
4
Express the total revenue change between February and January.
Since Tier 2 price and volume did not change, net revenue change comes entirely from Tier 1: 75x−40x=35x75x - 40x = 35x.
Unchanged terms cancel out when taking the difference (February Revenue minus January Revenue).
5
Solve for xx using the given revenue increase of $2,800\$2,800.
35x = 2,800 \implies x = 80.
Dividing both sides by 35 gives the January quantity for Tier 1 licenses.

Key Concept

Algebraic Word Problems and Equation Modeling
Estimated Time:2m 0s
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Topics

Algebraic Word Problems and Equation Modeling

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