# There are 50 students in a middle school chorus. The ratio of boys to girls in the chorus is 2:3. What is the ratio of girls to the total number of chorus members?

Dec 15, 2016

Ratio of girls to the total number of chorus members is $3 : 5$

#### Explanation:

The ratio of boy to girls is $\textcolor{b l u e}{2} : \textcolor{red}{3}$

You can find the number of boys and girls by dividing the total number $\textcolor{b r o w n}{50}$ by the sum of $\textcolor{b l u e}{2}$ and $\textcolor{red}{3}$, and then multiply the quotient by $\textcolor{b l u e}{2}$ to find the numbers of boys, and $\textcolor{red}{3}$ to find the number of girls.

We need to find the number of girls to find the ratio of girls to the total number of chorus members $\to \textcolor{b r o w n}{50}$

$\frac{\textcolor{b r o w n}{50}}{\textcolor{b l u e}{2} + \textcolor{red}{3}} = \frac{50}{5} = 10$

Number of girls $= 10 \left(\textcolor{red}{3}\right) = 30$

The ratio of girls to the total number of chorus members is $30 : \textcolor{b r o w n}{50}$ which can be simplified to $3 : 5$

Dec 15, 2016

The answer is $3 : 5$.

#### Explanation:

We assume the chorus is made up of only boys and girls.

## The long way:

Scale the ratio $2 : 3$ so that the numbers on either side sum to $50$.

$2 + 3 = 5$, and $50 \div 5 = 10$, so we scale our ratio by a factor of $10$.

$\left(2 \times 10\right) : \left(3 \times 10\right) = 20 : 30$

So there are a total of $\textcolor{b l u e}{\text{20 boys}}$ and $\textcolor{p u r p \le}{\text{30 girls}}$ in the choir of $\textcolor{g r e e n}{\text{50 students}}$.

This means the ratio of girls-to-"all members" is $\textcolor{p u r p \le}{30} : \textcolor{g r e e n}{50}$. We then scale this ratio down so it's in smallest terms. The greatest common factor of $30$ and $50$ is $10$:

$\left(30 \div 10\right) : \left(50 \div 10\right) = 3 : 5$

Thus, the ratio of girls-to-everyone is $3 : 5$.

## The short way:

The ratio $\textcolor{b l u e}{\text{2 boys}}$ : $\textcolor{p u r p \le}{\text{3 girls}}$ means we can divide the chorus into groups of $2 + 3 = \textcolor{g r e e n}{\text{5 students}}$ where each group has $\textcolor{b l u e}{\text{2 boys}}$ and $\textcolor{p u r p \le}{\text{3 girls}}$ (with no students left over).

That means the ratio of boys-to-girls in each group will match the ratio of boys-to-girls in the whole chorus. Same goes for any ratio we may seek, including the ratio of girls-to-everyone.

Since each group has a ratio of $\textcolor{b l u e}{\text{2 boys}}$ : $\textcolor{p u r p \le}{\text{3 girls}}$, each group also has a ratio of $\textcolor{p u r p \le}{\text{3 girls}}$ : $\textcolor{g r e e n}{\text{5 students}}$, and so does the whole chorus.

And that's it. All we had to do was go from

$\textcolor{b l u e}{2} : \textcolor{p u r p \le}{3}$

to

$\textcolor{p u r p \le}{3} : \left(\textcolor{b l u e}{2} + \textcolor{p u r p \le}{3}\right) = \textcolor{p u r p \le}{3} : \textcolor{g r e e n}{5}$.