CPhill

Let John start  with   x  magnets  and Liam start  with (7/4) x  magnets

Note  that 200% as  many  must  be  twice as many  (100% as many  would  be  the  same number given away)

Then  John ended up with  v/4  magnets  and  gave away  (3/4)x

Liam   gave  away  2 ( 3/4)x = 6/4x  magnets  and  was  left  with  x/4  magnets

Notice  that  they  end up with  the  same number....so  John  has 100%  of Liam's  magnets

Thisis known as  a  piece-wise  function....this one  will   just  plot  points  depending upon  the   parity of  n    (either even or odd )

Excellent, BF  !!!!

First one  can be

y =  ( x - 4.5)^2  - 8  =  x^2 - 9x  + 12.25

For  the  second.....we  can  just  change it to

y=  -( x - 4.5)^2   -  8     = -x^2  + 9x  - 28.25

Each  will have  the  same vertex....the  first turns upward....the  second turns downward

Excellent  answer   !!!!

6 it is   !!!!!....by  majority consensus......

Ryan had some blue and green buttons. 3/7 of the buttons are blue and the rest were green. He gave away 2/5 of his buttons. 80 of the buttons given away were blue. 2/3 of the buttons left were green. How many buttons did Ryan have altogether at first?

Let  the  total buttons = T

If (2/3)  of  the  buttons  left  were  green,  then (1/3) left  were blue

So

Blue  Buttons left  =  (1/3)  (of  what  remains)  

(3/7)T   - 80  =   (1/3)  [ T - (2/5)T]        simplify

(3/7)T  - 80  = (1/3 ) (3/5)T

(3/7)T    - 80   =  (1/5)T

(3/7)T - (1/5)T  = 80

[ 15  - 7 ] T  / 35  = 80

8/35 T  =  80

T =  80  (35/8)  =  350  buttons  at first

Proof

(2/5)(350) =    140  were given   away

210  were  left

140 were  green

70  were  blue

Total  blue =  80  + 70  =  (3/7)(350)   =150

Total green =  (4/7) (350)  = 200

150 + 200  =  350

One possibility......

y = -2 ( x -- 100)^2  +  200 

Expand

y  = -2( x^2  - 200x + 10000)  +  200

y = -2x^2  + 400x  - 20000  + 200

y =  -2x^2  + 400x -  19800 

Buttons left =

36  - (2/3)36  =    36  - 24  = 12

And  then

12 - (1/2)(12)  =    6 left

Just as TW  found   !!!

THX   !!!