Question 73828: A total of 434 people attended a community theatre performance. The admission prices were $9.00 for adults, $7.50 for students, and $8.00 for senior citizens. The ticket sales totaled $3712. At the theatre's next board meeting, the finance manager proposed that for the next play they raise prices to $11.00 for adults, $8.50 for students, and $8.50 for senior citizens. He said that if the exact same number of people attend the next performance, the ticket sales would be $4374. How many adults, students, and senior citizens attended the play?
Found 2 solutions by Edwin McCravy, ankor@dixie-net.com:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website! A total of 434 people attended a community theatre performance. The admission prices were $9.00 for adults, $7.50 for students, and $8.00 for senior citizens. The ticket sales totaled $3712. At the theatre's next board meeting, the finance manager proposed that for the next play they raise prices to $11.00 for adults, $8.50 for students, and $8.50 for senior citizens. He said that if the exact same number of people attend the next performance, the ticket sales would be $4374. How many adults, students, and senior citizens attended the play?
Let A = number of Adults who attended
Let S = number of Students who attended
Let C = number of senior Citizens who
attended
A total of 434 people attended a community
theatre performance.
A + S + C = 434
The admission prices were $9.00 for
adults, $7.50 for students, and $8.00 for
senior citizens. The ticket sales totaled
$3712.
9A + 7.5S + 8C = 3712
At the theatre's next board meeting, the
finance manager proposed that for the next
play they raise prices to $11.00 for adults,
$8.50 for students, and $8.50 for senior
citizens. He said that if the exact same
number of people attend the next performance,
the ticket sales would be $4374.
11A + 8.5S + 8.5C = 4374
So we have this system of three
equations in three unknowns.
A + S + C = 434
9A + 7.5S + 8C = 3712
11A + 8.5S + 8.5C = 4374
You can simplify the 2nd and 3rd
equations:
Simplifying the 2nd equation:
9A + 7.5S + 8C = 3712
First clear of decimals by multiplying
through by 10
90A + 75S + 80C = 37120
Now divide every term through by 5
18A + 15S + 16C = 7424
Simplifying the 3rd equation:
11A + 8.5S + 8.5C = 4374
First clear of decimals by multiplying
through by 10
110A + 85S + 85C = 43740
Now divide every term through by 5
22A + 17S + 17C = 8748
Now the system is
A + S + C = 434
18A + 15S + 16C = 7424
22A + 17S + 17C = 8748
That's a little easier to solve.
Can you solve it? If not post
again asking how to.
Answer:
A = 274 S = 68, C = 92,
that is, 274 adults, 68 students,
and 92 senior citizens attended
the play.
Edwin
Answer by ankor@dixie-net.com(22740)  (Show Source):
You can put this solution on YOUR website! Write an equation for each statement; a=adults, b=kids, c=old folks
:
"A total of 434 people attended a community theatre performance."
:
eq1: a + b + c = 434
:
"The admission prices were $9.00 for adults, $7.50 for students, and $8.00 for senior citizens. The ticket sales totaled $3712."
:
eq2: 9a + 7.5b + 8c = 3712
:
"At the theatre's next board meeting, the finance manager proposed that for the next play they raise prices to $11.00 for adults, $8.50 for students, and $8.50 for senior citizens. Ticket sales would be 4374"
:
eq3: 11a + 8.5b + 8.5c = 4374
:
He said that if the exact same number of people attend the next performance, the ticket sales would be $4374. How many adults, students, and senior citizens attended the play?
:
A matrix could be used it would be:
1 + 1 + 1 + 434
9 + 7.5 + 8 + 3712
11 + 8.5 + 8.5 + 4374
:
However elimination can also be used, here is that method.
:
Subtract eq 2 from eq 3
Then subtract eq 1 from that result:
:
11a + 8.5b + 8.5c =4374
9a + 7.5b + 8c = 3712
------------------------ subtract:
2a + 1b + .5c = 662
1a + 1b + 1c = 434
---------------------- subtracting eliminates b:
1a + 0b - .5c = 228
a - .5c = 228; a two unknown equation
:
Multiply equation 1 by 7.5 and subtract it from equation 2:
9a + 7.5b + 8c = 3712
7.5a + 7.5b + 7.5c = 3255
---------------------------subtracting eliminates b again:
1.5a + 0b + .5c = 457
1.5a + .5c = 457; another two unknown equation
:
Add our "two unknown" equations and find a:
a - .5c = 228
1.5a + .5c = 457
---------------------adding eliminates c
2.5a + 0c = 685
a = 685/2.5
a = 274 adults
:
Use a - .5c = 228 to find c, substitute 274 for a
274 - .5c = 228
-.5c = 228 - 274
-.5c = -46
c =-46/-.5
c = +92 old folks
:
I'm sure you can find the number of kids, equation 1 will work well:
Check the solutions in eq 2 or 3.
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