Question

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?

Answer 1

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

Explanation:

The ratio of boy to girls is `color(blue)2:color(red)3`

You can find the number of boys and girls by dividing the total number `color(brown)50` by the sum of `color(blue)2` and `color(red)3`, and then multiply the quotient by `color(blue)2` to find the numbers of boys, and `color(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 `->color(brown)50`

`color(brown)50/(color(blue)2+color(red)3)=50/5=10`

Number of girls `= 10(color(red)3)=30`

The ratio of girls to the total number of chorus members is `30:color (brown)50` which can be simplified to `3:5`

Answer 2

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-:5=10`, so we scale our ratio by a factor of `10`.

`(2 times 10) : (3 times 10)=20:30`

So there are a total of `color(blue)"20 boys"` and `color(purple)"30 girls"` in the choir of `color(green)"50 students"`.

This means the ratio of girls-to-"all members" is `color(purple)30:color(green)50`. We then scale this ratio down so it's in smallest terms. The greatest common factor of `30` and `50` is `10`:

`(30-:10):(50-:10)=3:5`

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

The short way:

The ratio `color(blue)"2 boys"` : `color(purple)"3 girls"` means we can divide the chorus into groups of `2+3=color(green)"5 students"` where each group has `color(blue)"2 boys"` and `color(purple)"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 `color(blue)"2 boys"` : `color(purple)"3 girls"`, each group also has a ratio of `color(purple)"3 girls"` : `color(green)"5 students"`, and so does the whole chorus.

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

`color(blue)2:color(purple)3`

to

`color(purple)3:(color(blue)2+color(purple)3)=color(purple)3:color(green)5`.