Ahealth club charges a one-time initiation fee and a monthly fee. john paid 100 dollars for 2 months of membership. however, peter paid 200 for 6 months of membership. how much will sylvia pay for 1 year of membership?

X=monthly fee ($)
y=one-time inititation fee
y+2x=100   (equation for John)
y+6x=200   (equation for Peter)
We have to compute the following system of equations:
y+2x=100
y+6x=200

We can solve this system by substitution systme:

y+2x=100 ⇒  y=100-2x

y+6x=200 
(100-2x)+6x=200
4x=200-100
4x=100
x=100/4=25

We find out the value of "y"
y=100-2x
y=100-2(25)
y=100-50
y=50

one-time initiation fee is $50 and monthly fee is $25

We find out how much will Sylvia pay for a year of mebership.
sylvia will pay=y+12x
sylvia will pay=$50+12*$25=$50+$300=$350

sylvia will pay $350. 

he last ratio is written incorrectly.

Mandy reversed the input and output in the numerator and denominator.

The correct ratio is 50/10.

The relationship is proportional.