Question

comparing the total members of two clubs
the swim and diving clubs at riverdale high school have a total of 55 members and no student is a member of both teams. \\(\frac{1}{3}\\) of the swim team members are seniors and \\(\frac{1}{5}\\) of the diving team members are seniors. if there are 13 seniors in the two clubs, how many members does each club have? let \\(x\\) represent the total number of swim club members and let \\(y\\) represent the total number of diving club members.
the equation that represents the total number of members is
the equation that represents the total number of seniors is
the diving club has more members than the swim club
\\(x + y = 13\\)
\\(x + y = 55\\)
\\((1/3)x + (1/5)y = 13\\)
\\((1/3)x + (1/5)y = 55\\)

Explanation:

Step1: Total members equation

The sum of swim and diving club members is 55, so:
$x + y = 55$

Step2: Senior members equation

1/3 of swim seniors plus 1/5 of diving seniors equals 13:
$\frac{1}{3}x + \frac{1}{5}y = 13$

Step3: Solve the system of equations

First, multiply the second equation by 15 to eliminate denominators:
$15\times(\frac{1}{3}x + \frac{1}{5}y) = 15\times13$
$5x + 3y = 195$
From the first equation, express $x = 55 - y$, substitute into the above:
$5(55 - y) + 3y = 195$
$275 - 5y + 3y = 195$
$-2y = 195 - 275$
$-2y = -80$
$y = 40$

Step4: Find x value

Substitute $y=40$ into $x = 55 - y$:
$x = 55 - 40 = 15$

Step5: Find member difference

Calculate $y - x$:
$40 - 15 = 25$

Answer:

The equation that represents the total number of members is $\boldsymbol{x + y = 55}$
The equation that represents the total number of seniors is $\boldsymbol{\frac{1}{3}x + \frac{1}{5}y = 13}$
The diving club has $\boldsymbol{25}$ more members than the swim club
Swim club: 15 members, Diving club: 40 members