Question

A Broadway theater has 600 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $50, main seats for $35, and balcony seats for $35. If all the seats are sold, the gross revenue to the theater is $20,700. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,700. How many are there of each kind of seat?

Answer 1

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A Broadway theater has 600 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $50, main seats for $35, and balcony seats for $25. If all the seats are sold, the gross revenue to the theater is $20,700. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,700. How many are there of each kind of seat?

Answer:

Orchestra = X = 270 seats

Main = Y = 350 seats

Balcony = Z

Step-by-step explanation:

Let's represent number of sits in:

Orchestra = X

Main = Y

Balcony = Z

A Broadway theater has 600 seats, divided into orchestra, main, and balcony seating.

X + Y + Z = 600

X = 600 - Y - Z

Orchestra seats sell for $50, main seats for $35, and balcony seats for $25. If all the seats are sold, the gross revenue to the theater is $20,700.

Hence,

50× X + 35 × Y + $25 × Z = $20,700

50X + 35Y + 25Z = 20,700....... Equation 1

If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,700.

50×1/2(X) + 35Y + 35Z = $17,700

25X + 35Y + 25Z = $17,700........ Equation 2

600 - Y - Z for X in Equation 1 and 2

Equation 1

50(600 - Y - Z) + 35Y + 25Z = 20,700

30,000 - 50Y - 50Z + 35Y + 25Z = 20,700

Collect like terms

30,000 - 20,700 = 50Y + 50Z - 35Y - 25Z

9,300 = 15Y + 25Z ......... Equation 3

From Equation 2

25(600 - Y - Z) + 35Y + 25Z= $17,700

15,000 -25Y - 25Z + 35Y + 25Z = 17,700

35Y - 25Y + 25Z - 25Z = 17,700 - 15,000

10Y + 0Z = 2,700.......

10Y = 2,700

Y = 2700/10

Y = 270

Hence:

15Y + 15Z = 9,300......... Equation 3

15(270) + 15Z = 9,300

4050 + 15Z = 9,300

15Z = 9300 - 4050

15Z = 5250

Z = 5250/15

Z = 350

X + Y + Z = 600

270 + 350 + Z = 600

Z = 600 - 270 - 350

Z =