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A movie theater is filled to its capacity of 350. The theater charges $4.50 for children, $7.50 for students, and $12.50 for adults. There are half as many adults as there are students. If the total ticket sales was $2415, how many children, students, and adults attended. Write your answer as an ordered triple in the form (# of children, # of students, # of adults). For example, (1,2,3).

Respuesta :

Answer:

Step-by-step explanation:

Answer:

(170;120;60).

Step-by-step explanation:

We know that:

Capacity of the theater: 350 people.

Adults is half the students.

Children's cost: $4.50

Student's cost: $7.50

Adults' cost: $12.50

Total Sales: $2415

We are gonna call x the children, y the students and z the adults:

So, the equation to express the capacity of the theater would be:

[tex]x+y+z=350[/tex]

But, [tex]z=\frac{y}{2}[/tex] (adults are have students)

So, the expression would be: [tex]x+y+\frac{y}{2} =350[/tex]

Solving y's and Isolating x :

[tex]x+\frac{3y}{2} =350\\x=350-\frac{3y}{2}[/tex]

Now, we need a expression for costs. We have:

[tex]4.50x+7.50y+12.50z=2415[/tex]

Replacing the x and z equation:

[tex]4.50(350-\frac{3y}{2} )+7.50y+12.50\frac{y}{2}=2415[/tex]

Now, we solve for y:

[tex]1575-6.75y+7.50y+6.25y=2415\\07y=2415-1575\\7y=840\\y=\frac{840}{7} =120[/tex]

But, we know that adults are half as many students, so:

[tex]z=\frac{y}{2}=\frac{120}{2} =60[/tex]

Lastly,

[tex]x+y+z=350\\x+120+60=350\\x=350-120-60\\x=170[/tex]

Therefore, there are 170 children, 120 students and 60 adults. Expressing the results as an ordered triple would be (170;120;60)

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