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Difficulty: MediumLinear Equations in One Variable

A non-profit organization hosted two fundraising events, Event A and Event B. Event A charged a ticket price of $30\$30 per person and collected an additional fixed donation of $250\$250 from a local sponsor. Event B charged a ticket price of $45\$45 per person and collected a fixed donation of $400\$400 from a corporate sponsor. The number of attendees at Event B was 1010 fewer than twice the number of attendees at Event A. If the total revenue raised from both events combined was $8,600\$8,600, how many people attended Event B?

  1. A
    60
  2. B
    70
  3. 130Answer
  4. D
    140
  5. E
    150

Answer

130 people attended Event B.
Setting up the linear equation for total revenue gives (30x+250)+(45(2x−10)+400)=8,600(30x + 250) + (45(2x - 10) + 400) = 8,600, where xx is the attendance at Event A. Expanding and combining like terms yields 120x+200=8,600120x + 200 = 8,600, which solves to x=70x = 70. Substituting x=70x = 70 into the expression for Event B attendance (2x−102x - 10) gives 2(70)−10=1302(70) - 10 = 130.

Step-by-Step Solution

1
Define variables for the unknown quantities.
Let xx be the number of attendees at Event A. The number of attendees at Event B is 2x−102x - 10.
Event B has 10 fewer attendees than twice Event A.
2
Express the revenue generated by each event in terms of xx.
Event A revenue: 30x+25030x + 250; Event B revenue: 45(2x−10)+400=90x−450+400=90x−5045(2x - 10) + 400 = 90x - 450 + 400 = 90x - 50.
Revenue equals ticket price times attendees plus fixed sponsor donations.
3
Set up and simplify the linear equation for total combined revenue.
(30x+250)+(90x−50)=8,600  ⟹  120x+200=8,600(30x + 250) + (90x - 50) = 8,600 \implies 120x + 200 = 8,600.
The sum of revenues from both events equals the total revenue of $8,600.
4
Solve the linear equation for xx.
120x=8,400  ⟹  x=70120x = 8,400 \implies x = 70.
Subtract 200 from both sides and divide by 120 to isolate xx.
5
Calculate the number of attendees at Event B.
Event B attendees =2(70)−10=140−10=130= 2(70) - 10 = 140 - 10 = 130.
Substitute x=70x = 70 into the expression 2x−102x - 10.

Key Concept

Linear Equations in One Variable
Estimated Time:1m 30s
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Topics

Linear Equations in One Variable

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