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avatar+2417 

Chloe charged for admission to her play on three different nights. Each night, a different number of people were in attendance, but remarkably, Chloe collected $541 each night. If the admission charges for each child and each adult were $9 and $17, respectively, how many people in total came to the three showings?

 

I solved this tediously by guessing and checking half a year ago in the Mathcounts Competition. I finished just the problem in time with no time to spare for checking my other answers!

 

Please help, I need a faster strategy to solve this.

 Jun 22, 2019
edited by CalculatorUser  Jun 22, 2019
 #1
avatar
0

You posted the same question about 4 weeks ago. See here for the answers:

 

https://web2.0calc.com/questions/money-problem_5

 Jun 22, 2019
 #2
avatar+2417 
+2

Sorry I posted again I was confused by the explanation... I dont understand why it works.

 

It is locked now I couldnt reply to the answer today.

CalculatorUser  Jun 22, 2019
 #3
avatar+104962 
+1

Here's the solution ....

 

Note that we should have this equation

 

9x + 17y  =  541           which we can manipulate as follows

 

9x  =     541  - 17y

 

x  =        541  - 17y

             ________

                     9

 

x  =  541   -  (18 - 1)y

       _____________

                   9

 

x  =  541 + y                18y

       _______     -        ____

             9                      9

 

x  =  540  + 1 +  y                18y

        ___________   -         ____

                   9                         9

 

x =  540           1 + y             18y

      ___   +       _____  -       ___

        9                  9                 9

 

Note that the first and last terms will always be divisible by 9

 

And.....the middle term will be divisible by 9  when y  =  8, 17  or 26

 

 

So   we have the following possibilities for x

 

x  =  541 - 17(8)        =      45

        _________   

               9

 

x =   541 - 17*17    =    28

        __________

                9

 

x =   541  -   17*26      =   11

        ____________

                  9

 

Note   if y =  35,we get the next multiple of 9...but  x  = -6 if this is so....    so....these are the only possible solutions

 

So....the solutions  (x, y)   = ( 11,26)  (28, 17)  (45, 8)

 

Just add these coordinates and you will get your answer, CU

 

 

cool cool cool

 Jun 22, 2019
edited by CPhill  Jun 22, 2019
edited by CPhill  Jun 22, 2019
edited by CPhill  Jun 22, 2019
edited by CPhill  Jun 22, 2019
 #4
avatar+2417 
+2

thank you so much! that is a very beatiful solution I will make sure in the future to use the same strategy!

 Jun 22, 2019
 #5
avatar+104962 
+2

OK, CU..you're welcome... [ I just happened to hit on it by some luck.....LOL!!!!  ]

 

 

cool cool cool

CPhill  Jun 22, 2019

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