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Difficulty: Very hardTranslating and Solving Algebraic Word Problems

A charity concert sells floor tickets and balcony tickets. The price of a floor ticket is 55 dollars more than twice the price of a balcony ticket. If the concert organizers sell 8080 balcony tickets and 5050 floor tickets, their total revenue is RR dollars. Under a new promotional structure, the price of a balcony ticket is discounted by 25%25\%, the price of a floor ticket is increased by 10%10\%, and the organizers sell 120120 balcony tickets. In terms of the original price of a balcony ticket, bb, which of the following expressions represents the number of floor tickets they must sell under the new promotional structure to achieve the same total revenue, RR?

  1. A
    1500b+250022b+55\frac{1500b + 2500}{22b + 55}
  2. B
    900b+250022b−55\frac{900b + 2500}{22b - 55}
  3. 900b+250022b+55\frac{900b + 2500}{22b + 55}Answer
  4. D
    2500−1100b55−22b\frac{2500 - 1100b}{55 - 22b}
  5. E
    900b+500022b+110\frac{900b + 5000}{22b + 110}

Answer

900b+250022b+55\frac{900b + 2500}{22b + 55}
The correct expression is derived by first writing the original revenue in terms of the balcony ticket price bb, which yields R=180b+250R = 180b + 250. Under the new pricing, the balcony ticket costs 0.75b0.75b and the floor ticket costs 1.1(2b+5)=2.2b+5.51.1(2b + 5) = 2.2b + 5.5. Selling 120120 balcony tickets generates 90b90b in revenue. Setting the new total revenue equal to RR gives 90b+F(2.2b+5.5)=180b+25090b + F(2.2b + 5.5) = 180b + 250. Solving for the number of floor tickets, FF, results in F=90b+2502.2b+5.5F = \frac{90b + 250}{2.2b + 5.5}. Multiplying the numerator and denominator by 1010 to clear the decimals yields the correct expression.

Step-by-Step Solution

1
Express the original ticket prices and revenue in terms of bb.
Balcony ticket price = bb. Floor ticket price = 2b+52b + 5. Original revenue R=80b+50(2b+5)=180b+250R = 80b + 50(2b + 5) = 180b + 250.
To set up the baseline total revenue equation using the algebraic descriptions.
2
Determine the new pricing for both balcony and floor tickets.
New balcony ticket price = 0.75b0.75b. New floor ticket price = 1.1(2b+5)=2.2b+5.51.1(2b + 5) = 2.2b + 5.5.
To apply the 25%25\% discount and 10%10\% increase to the original ticket prices.
3
Set up the equation equating the new promotional revenue to the original revenue RR.
New Revenue = 120(0.75b)+F(2.2b+5.5)=180b+250120(0.75b) + F(2.2b + 5.5) = 180b + 250, where FF is the number of floor tickets.
To represent the condition that the total revenue remains the same under the new structure.
4
Simplify the equation and isolate the variable FF.
90b+F(2.2b+5.5)=180b+250  ⟹  F(2.2b+5.5)=90b+250  ⟹  F=90b+2502.2b+5.590b + F(2.2b + 5.5) = 180b + 250 \implies F(2.2b + 5.5) = 90b + 250 \implies F = \frac{90b + 250}{2.2b + 5.5}.
To solve for the number of floor tickets algebraically.
5
Clear decimals from the rational expression.
F=10(90b+250)10(2.2b+5.5)=900b+250022b+55F = \frac{10(90b + 250)}{10(2.2b + 5.5)} = \frac{900b + 2500}{22b + 55}.
To match the standard fraction format of the options by multiplying the numerator and denominator by 1010.

Key Concept

Translating verbal descriptions into multi-step algebraic systems and isolating a target variable from rational equations.
Estimated Time:3m 0s
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Topics

Translating and Solving Algebraic Word Problems

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