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In a sports league each team has $27$ players and $6$ coaches. There are also a number of team assistants. The ratio of team assistants to players is $2:3$ . What is the ratio of coaches to assistants? $\newline$ A. $1/3$ $\newline$ B. $2/3$ $\newline$ C. $3/2$ $\newline$ D. $3$

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GCF and LCM: word problems

Full solution

Q. In a sports league each team has

27

players and

6

coaches. There are also a number of team assistants. The ratio of team assistants to players is

2:3

. What is the ratio of coaches to assistants?

\newline

A.

1/3

\newline

B.

2/3

\newline

C.

3/2

\newline

D.

3

Calculate assistants per player: First, let's find out how many assistants there are for each player. The ratio given is $2$ assistants for every $3$ players.
Set up proportion: Since each team has $27$ players, we can calculate the number of assistants by setting up a proportion: $(\frac{2 \text{ assistants}}{3 \text{ players}}) = (\frac{x \text{ assistants}}{27 \text{ players}})$ .
Cross-multiply to solve: Cross-multiply to solve for $x$ : $3x = 2 \times 27$ .
Perform the math: Now, do the math: $3x = 54$ .
Divide to find $x$ : Divide both sides by $3$ to find $x$ : $x = \frac{54}{3}$ .
Find total assistants: So, $x = 18$ . This means there are $18$ team assistants for every $27$ players.
Determine coaches to assistants ratio: Now, we need to find the ratio of coaches to assistants. There are $6$ coaches and $18$ assistants.
Simplify the ratio: The ratio of coaches to assistants is $6$ coaches / $18$ assistants.
Final ratio determination: Simplify the ratio by dividing both numbers by $6$ : $(\frac{6}{6})$ coaches / $(\frac{18}{6})$ assistants.
Final ratio determination: Simplify the ratio by dividing both numbers by $6$ : $(\frac{6}{6})$ coaches / $(\frac{18}{6})$ assistants.After simplifying, we get $1$ coach / $3$ assistants.
Final ratio determination: Simplify the ratio by dividing both numbers by $6$ : $(6/6)$ coaches / $(18/6)$ assistants.After simplifying, we get $1$ coach / $3$ assistants.So, the ratio of coaches to assistants is $1/3$ , which corresponds to option $A$ .

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