Word Problem HELP!?

Question

the Music Department at JSV were conducting a symphony orchestra at StoneCenter Auditorium. They charged $2 for staff· $5 for faculty; and $8 for the general public. Students get in free. The auditorium seats 1000 people. The general public were half as many staff and f acuity combined. They made a total of $5550 from the event. How many staff, faculty, general public attended? 

 

Answer 1

Let's assign the following variables...

s=staff

f=faculty

p=general public

 

Form the problem statement, we can write the following equations...

s+f+p=1000..........Eq1, total attendance, given

2s+5f+8p=5550...Eq2, total event profit, given

p=(1/2)(s+f)..........Eq3, general public total in

                               terms of staff and

                               faculty, given

 

Let's simplify Eq3....

p=(1/2)(s+f)........Eq3

  =0.5s+0.5f........distribute 1/2 to each

                             term

 

Let's substitute Eq3 into Eq1 to get two equations in two variables "s,f"...

               s+f+p=1000....Eq1

s+f+(0.5s+0.5f)=1000...substitute Eq3 for "p"

          1.5s+1.5f=1000...Eq4, combine like terms

 

Let's substitute Eq3 into each Eq2 to get two equations in two variables "s,f"...
               2s+5f+8p=5550...Eq2
2s+5f+8(0.5s+0.5f)=5550...substitute Eq3 for "p"

          2s+5f+4s+4f=5550...distribute 8 to each

                                              term

                     6s+9f=5550....Eq5, simplify

 

Now let's solve Eqs4,5 simultaneously....

1.5s+1.5f=1000....Eq4

      6s+9f=5550....Eq5

 

      -9s-9f=-6000...Eq4, multiply both sides by (-6)
       6s+9f=5550....Eq5, rewritten

       -----------------

           -3s=-450.....add Eqs4,5

          ∴  s=150.....divide both sides by (-3)

 

Let's substitute "s=150" back into Eq5 and solve for"f"...

       6s+9f=5550.....Eq5

6(150)+9f=5550.....substitute for "s"

    900+9f=5550.....simplify

            9f=4650......subtract 900, both sides

        ∴  f=516.67...divide both sides by 9

 

Let's substitute "s=150" and "f=516.67" back into Eq3 and solve for "p"...

     p=(1/2)(s+f)...................Eq3
     p=(1/2)(150+516.67)....substitute for "s, f"

       =(1/2)(666.67).............simplify

∴  p=333.33......................simplify

 

Let's check our solutions values by substituting back into Eqs1,2,3 generated from the problem statement and check for truth...

                     s+f+p=1000........Eq1

150+516.67+333.33=1000....substitute for

                                                 "s,f,p"

                       1000=1000.......true,√check

 

                   2s+5f+8p=5550...Eq2

2(150)+5(516.67)+

                   8(333.33)=5550...substitute for

                                                 "s,f,p"

300+2583.33+2666.67=5550

                           5550=5550...true,√check

 

         p=(1/2)(s+f)....................Eq3

333.33=(1/2)(150+516.67)....substitute for "s,f,p"

333.33=(1/2)(666.67).............simplify

333.33=333.33.......................true,√check

 

Our values check out against the information from the problem statement, so...

Staff totals 150

Faculty totals 516.67

General Public totals 333.33

 

NOTE:  AS YOU CAN SEE, WE CANNOT HAVE 1/3 OR 2/3 OF A PERSON FOR GENERAL PUBLIC AND FACULTY TOTALS!!  BUT WE HAVE SOLVED AND CHECKED FOR CORRECT VALUES WE ASSUMED.  A REASON FOR THIS IS WE DID NOT TAKE INTO ACCOUNT THE NUMBER OF STUDENT ATTENDEES!!!...AS STATED IN MY COMMENTS TO YOU!!!  SHOULD YOUR PROBLEM BE RESTATED TO INCLUDE THE STUDENT TOTAL, ADJUST THE ABOVE FORMULAS ACCORDINGLY BY SUBTRACTING THAT AMOUNT FROM 1000 IN EQ1.  EQNS 2,3 SHOULD REMAIN UNCHANGED.  THE STEP-BY-STEP PROCEDURE WILL REMAIN THE SAME WITH THE ADJUSTED TOTAL BUT WILL RESULT IN DIFFERENT CALCULATED VALUES FOR "s, f, and g".

 

Let me know how this problem gets resolved in your class discussion!

 

       *************************

 

Edit:

As follow-up to my solution above, we could round up the value of "f=516.67 to f=517", and round down the value of "p=333.33 to p=333".  Substituting these rounded values into Eq2 (total event profit), we have...

 

                    2s+5f+8p=5550.....Eq2

2(150)+5(517)+8(333)=5550....substitute for

                                                    "s,f,p"

          300+2585+2664=5550...simplify

                             5449≠5550..."not equal to",

                                                    though differs by

                                                    ONLY $1, say close 

                                                    enough