Question

A Broadway theater has 500500 ​seats, divided into​ orchestra, main, and balcony seating. Orchestra seats sell for $ 60 comma$60, main seats for $ 45 comma$45, and balcony seats for $ 30.$30. If all the seats are​ sold, the gross revenue to the theater is $ 21 comma 600.$21,600. If all the main and balcony seats are​ sold, but only half the orchestra seats are​ sold, the gross revenue is $ 18 comma 600.$18,600. How many are there of each kind of​ seat?

Answer 1

The number of orchestra seats=100

The number of main seats=240

The number of balcony seats=160

Explanation:

Total number of seats in the theater=500

Let the number of orchestra seats=x

Let the number of main seats=y

Let the number of balcony seats=z

x+y+z=500

Cost of Orchestra Seats =$60

Cost of Main Seats =$45

Cost of Balcony Seats =$30

If all the seats are​ sold, the gross revenue to the theater is $ 21,600.

60x+45y+30z=21,600.

If all the main and balcony seats are​ sold, but only half the orchestra seats are​ sold, the gross revenue is $18,600.

45y+30z+x(60)=18,600

We solve the three equations simultaneously

x+y+z=500

60x+45y+30z=21,600.

30x+45y+30z=18,600

Subtract the third equation from the second

30x=3000

x=100

Substitute x=100 into the equations x+y+z=500

100+y+z=500

y+z=500-100

y=400-z

Substitute y=400-z and x=100 into 30x+45y+30z=18,600

30x+45y+30z=18,600

30(100)+45(400-z)+30z=18600

3000+18000-45z+30z=18600

-15z=18600-21000

-15z=-3600

z=240

y=400-z=400-240=160

Therefore: x=100, y=240, z=160

The number of orchestra seats=100

The number of main seats=240

The number of balcony seats=160

Answer 2

number of orchestra seats is 100

number of main seats is 240

number of balcony seats is 160

Explanation:

Let's assume that the number of orchestra seats to be x,the number of main seats to be y and that of balcony seats to be z.

From the first statement,

X + y + z = 500______ equation 1

From the second statement,

60x + 45y + 30z= 21600_____ equation 2

From the third statement,

30x + 45y + 30z = 18600______ equation 3

From equation 1,we make x the subject of the formula

X = 500 - y - z

Apply the above in equation 2 and 3

60(500 - y - z) + 45y + 30z = 21600

30000 - 60y - 60z + 45y + 30z = 21600

15y +30z = 8400____ equation 4

And

30x + 45y + 30z = 18600

30(500 - y -z) + 45y + 30z = 18600

15000 - 30y - 30z + 45y + 30z = 18600

15y = 3600

Y = 240

Replace y = 240 in equation 4

15y +30z = 8400

15(240) + 30z = 8400

3600 + 30z = 8400

30z = 4800

Z = 160

Now remember that x + y + z = 500

X + 240 + 160 = 500

X = 100

X which is number of orchestra seats is 100

Y which is number of main seats is 240

And x which is number of balcony seats is 160