# The number of football players is 4 times the number of basketball players, and the number of baseball players is 9 more than basketball players. If the total number of players is 93 and each one plays a single sport, how many are on each team?

Sep 28, 2015

$56$ football players
$14$ basketball players
$23$ baseball players

#### Explanation:

Define:
$\textcolor{w h i t e}{\text{XXX}} f :$number of football players
$\textcolor{w h i t e}{\text{XXX}} b :$number of basketball players
$\textcolor{w h i t e}{\text{XXX}} d :$number of baseball players

We are told:
color(white)("XXX"color(red)(f=4b)
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{d = b + 9}$
$\textcolor{w h i t e}{\text{XXX}} f + b + d = 93$

Substituting (from ) $\textcolor{red}{4 b}$ for $\textcolor{red}{f}$ and (from ) $\textcolor{b l u e}{b + 9}$ for $\textcolor{b l u e}{d}$ in 
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{4 b} + b + \textcolor{b l u e}{b + 9} = 93$

Simplifying
$\textcolor{w h i t e}{\text{XXX}} 6 b + 9 = 93$
$\textcolor{w h i t e}{\text{XXX}} 6 b = 84$
$\textcolor{w h i t e}{\text{XXX}} b = 14$

Substituting $14$ for $b$ in 
$\textcolor{w h i t e}{\text{XXX}} d = 14 + 9 = 23$

Substituting $14$ for $b$ in 
$\textcolor{w h i t e}{\text{XXX}} f = 4 \cdot 14 = 56$

Sep 28, 2015

56 football players, 14 basketball players, and 23 baseball players.

#### Explanation:

Let the number of football players be x
Let the number of basketball players be y
Let the number of baseball players be z

Now rewrite all the sentences in algebraic form in terms of x, y and z. Doing so we get :

$x = 4 y$
$z = y + 9$
$x + y + z = 93$

Now we can substitute both x and z (which we have in terms of y) into the last equation and then solve for y. This yields

$4 y + y + \left(y + 9\right) = 93$

$\therefore 6 y = 84 \implies y = 14$ and so there 14 basketball players.

Now substitute the value of y back into the first 1 equations to determine x and z.
$\therefore x = 4 \times 14 = 56 \mathmr{and} z = 14 + 9 = 23$
This means there are then 56 football players and 23 baseball players.