Determine the Number of People that attended the game in Each Category Cost and the Total Revenue.

Question

A Football stadium has 100,000 seats. In a game with full capacity people with the following ticket and associated cost attended the game:

Cost:

Student - 25

Alumni - 50

Faculty - 75

Public - 100

Veterans - 30

Guests - 20

Note:

There were 11,000 more alumni than faculty, the number of public plus alumni together was 10 times the number of veterans, the number of faculty plus alumni together was the same as the number of students, and the number of faculty plus the number of students together was four time larger than the number of guests and veterans together.

Answer 1

Hi Austin,

As stated, this problem is indeterminate. There are 6 unknown variables (number of attendees from each category), and only 5 pieces of information that can be used to generate an equation.

The general solution approach to solve a problem like this would be to:

Step 1) Identify the unknown variables

Step 2) Generate equations from the given information

Step 2a) Manipulate the equations to get coefficients of all unknown variables on one side

Step 3) Create a matrix representation of the system of equations (A*B = C)

1. Note: In this representation, B would be a column vector of the unknowns, A would contain all coefficients for those unknown variables, and C would contain all scalars on the other side of each equation.

Step 4) Solve for the unknown variables by doing B = A^-1*C

For this problem:

Step 1) Identify the unknowns

Our 6 unknown variables are the number of 1) Students 2) Alumni 3) Faculty 4) Public 5) Veterans and 6) Guests that attended the game

So, we will need 6 equations!!

Step 2) Create equations from the given information

Given info 1: There were 11,000 more alumni than faculty

Equation 1: Alumni + 11,000 = Faculty

Given info 2: The number of public plus alumni together was 10 times the number of veterans

Equation 2: Public + Alumni = 10*Veterans

Given info 3: The number of faculty plus alumni together was the same as the number of students

Equation 3: Faculty + Alumni = Students

Given info 4: The number of faculty plus the number of students together was four time larger than the number of guests and veterans together

Equation 4: Faculty + Alumni = 4*(Guests + Veterans)

Given info 5: The stadium (with 100,000 seats) was full

Equation 5: Students + Alumni + Faculty + Public + Veterans + Guests = 100,000

We are one equation short of being able to solve this problem, so let's just say that theoretically we knew the total revenue was $500,000, then we would know:

Given info 6: The sum of all tickets was $500,000

Equation 6: 25*Students + 50*Alumni + 75*Faculty + 100*Public + 30*Veterans + 20*Guests = 500,000

Step 2a) Move all unknown variable coefficients to the left side

Equation 1: Alumni + (-1)*Faculty = 11,000

Equation 2: Public + Alumni + (-10)*Veterans = 0

Equation 3: Faculty + Alumni + (-1)*Students = 0

Equation 4: Faculty + Alumni + (-4)*Guests + (-4)*Veterans = 0

Equation 5: Students + Alumni + Faculty + Public + Veterans + Guests = 100,000

Equation 6: 25*Students + 50*Alumni + 75*Faculty + 100*Public + 30*Veterans + 20*Guests = 500,000

Step 3) Create a matrix representation of the system

We will now create a matrix representation for this system of equations (A*B = C), where the entry in columns 1->6 of the coefficient matrix A will represent the coefficients from our equations for 1) Students, 2) Alumni, 3) Faculty, 4) Public, 5) Veterans, and 6) Guests.

Take some time to really understand how these matrices were formed:

A = [0 1 -1 0 0 0 ;

0 1 0 1 -10 0;

-1 1 1 0 0 0;

1 0 1 0 -4 -4;

1 1 1 1 1 1;

25 50 75 100 30 20]

B = [Students ;

Alumni ;

Faculty ;

Public ;

Veterans ;

Guests];

C = [11000 ;

0 ;

0;

0;

100000;

500000];

Step 4) Solve for vector B

We know that we can solve A*B = C by computing B = A^-1*C. This can be done easily in MATLAB with the following command:

C = inv(A)*B;

From there, you just need to do some simple multiplication to calculate the total revenue per category.

I hope that helps! Please let me know if you have questions.