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Zorluk: OrtaAlgebraic Word Problems and Modeling

A theater sold a total of 500500 tickets for an evening performance, consisting of VIP tickets priced at $80\$80 each and General Admission tickets priced at $50\$50 each. On the day of the show, a promotional discount of 20%20\% was applied to all General Admission tickets, while VIP ticket prices remained unchanged. If the total revenue collected from ticket sales was $31,000\$31,000, which of the following statements must be true? Select all such statements.

  1. More than 70% of the total revenue was generated from VIP ticket sales.Cevap
  2. The number of VIP tickets sold was 50 greater than the number of General Admission tickets sold.Cevap
  3. C
    The ratio of VIP tickets sold to General Admission tickets sold was 11 to 7.
  4. D
    Applying the 20% discount to General Admission tickets reduced the total potential revenue by 20%.
  5. E
    The number of General Admission tickets sold was 250.

Cevap

The statements confirming that more than 70% of total revenue came from VIP ticket sales and that 50 more VIP tickets were sold than General Admission tickets are correct.
The system of equations V+G=500V + G = 500 and 80V+40G=31,00080V + 40G = 31,000 yields V=275V = 275 VIP tickets and G=225G = 225 General Admission tickets. The VIP revenue is 22,000,whichisapproximately70.9722,000, which is approximately 70.97% of the total 31,000 revenue (greater than 70%). Additionally, the difference 275−225=50275 - 225 = 50 confirms that 50 more VIP tickets were sold than General Admission tickets.

Adım Adım Çözüm

1
Define variables and determine the discounted price of General Admission tickets.
Let VV be the number of VIP tickets and GG be the number of General Admission tickets. The discounted price for General Admission tickets is $50×(1−0.20)=$40\$50 \times (1 - 0.20) = \$40.
Establishing correct variable representations and effective unit prices is required to build the revenue model.
2
Set up and solve the system of linear equations.
V+G=500V + G = 500 and 80V+40G=31,00080V + 40G = 31,000. Substituting G=500−VG = 500 - V gives 80V+40(500−V)=31,000  ⟹  40V=11,000  ⟹  V=27580V + 40(500 - V) = 31,000 \implies 40V = 11,000 \implies V = 275. Thus, G=225G = 225.
Solving the linear system determines the exact quantity of each ticket type sold.
3
Calculate revenue shares and verify each statement.
VIP revenue = 275×$80=$22,000275 \times \$80 = \$22,000. Revenue share of VIP = 22,00031,000≈70.97%>70%\frac{22,000}{31,000} \approx 70.97\% > 70\%. Ticket difference = 275−225=50275 - 225 = 50. Ratio V:G=275:225=11:9V:G = 275:225 = 11:9.
Evaluating each calculated metric against the given statements determines which statements must be true.

Anahtar Kavram

Linear word problems involving systems of equations and percentage modifications.
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Algebraic Word Problems and Modeling

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