Question

A middle school has clubs and sports teams as it’s after school activities. The ratio of clubs to sports teams is 3:2. What is the ratio of sports teams to all after school activities

Answer 1

Let the number of clubs be represented by \( C \) and the number of sports teams be represented by \( S \). According to the given ratio of clubs to sports teams, we can express this as:

\[
\frac{C}{S} = \frac{3}{2}
\]

From this ratio, we can say that:

\[
C = \frac{3}{2}S
\]

Now, the total number of after school activities \( T \) is the sum of the number of clubs and the number of sports teams:

\[
T = C + S = \frac{3}{2}S + S
\]

To combine these, we convert \( S \) to a fraction with a common denominator:

\[
T = \frac{3}{2}S + \frac{2}{2}S = \frac{5}{2}S
\]

Now, we want to find the ratio of sports teams to all after school activities:

\[
\frac{S}{T} = \frac{S}{\frac{5}{2}S}
\]

Since \( S \) is in both the numerator and the denominator, we can simplify:

\[
\frac{S}{\frac{5}{2}S} = \frac{1}{\frac{5}{2}} = \frac{2}{5}
\]

Thus, the ratio of sports teams to all after school activities is:

\[
\boxed{\frac{2}{5}}
\]