# Finite math

A theater has a seating capacity of 900 and charges \$5 for children, \$7 for students, and \$9 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled \$6100. How many children attended the show?

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3. 👁 1,606
1. 5 c + 7 s + 9 a = 6100

c + s + a = 900

a = .5 (c+s) or 2a - c - s = 0

============================
3 a = 900
so
a = 300

c+s = 600 so s = 600 - c

3 c + 7 s + 2700 = 6100

3 c + 7(600-c) = 3400
3 c + 4200 - 7 c = 3400
4 c = 800
c = 200

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c = childrens

s = students

With a full audience:

a + c + s = 900

There were half as many adults as children and students combined mean:

a = ( s + c ) / 2

Replace this in equation:

a + c + s = 900

( s + c ) / 2 + c + s = 900 Multiply both sides by 2

s + c + 2 c + 2 s = 1,800

3 c + 3 s = 1,800 Divide both sides by 3

c + s = 600

The receipts totaled \$6,100 mean:

\$9 * adults + \$7 * students + \$5 * chidrens= \$6,100

\$9 * a + \$7 * s + \$5 * c = \$6,100

9 a + 7 s + 5 c = 6,100

Replace a = ( s + c ) / 2 in this equation.

9 * ( s + c ) / 2 + 7 s + 5 c = 6,100 Multiply both sides by 2

9 ( s + c ) + 2 * 7 s + 2 * 5 c = 2 * 6,100

9 s + 9 c + 14 s + 10 c = 12,200

23 s + 19 c = 12,200

Now you must solve 2 equations with 2 unknow:

c + s = 600

23 s + 19 c = 12,200

Try that.

The solutions are : c = 400 , s = 200

400 childrens

200 students

Proof:

a = ( s + c ) / 2

a = ( 400 + 200 ) / 2 = 600 / 2 = 300

\$9 * a + \$ 7 * s + \$5 * c = \$6,100

\$9 * 300 + \$ 7 * 200 + \$5 * 400 = \$6,100

\$2,700 + \$1,400 + \$2,000 = \$6,100

\$6,100 = \$6,100

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3. A theater is presenting a program on drinking and driving for students and their parents or other responsible adults. The proceeds will be donated to a local alcohol information center. Admission is \$ 10.00 for adults and \$ 5.00 for students.​ However, this situation has two​ constraints: The theater can hold no more than 210 people and for every two​ adults, there must be at least one student. How many adults and students should attend to raise the maximum amount of​ money?
To raise the maximum amount of​ money,
nothing students should attend.

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