A school board has determined that there should be 4 teachers for every 50 students. How many teachers are needed for an enrollment of 2,650 students?

Proportions are used when we are asked about finding one unknown value, knowing the value of others that keep a ratio among all of them, including the unknown value.

In this case, we know that a school board estimated that 4 teachers are needed for every 50 students:

[tex]\\ \frac{4_{teachers} }{50_{students}}=ratio[/tex]

If we divide 4 teachers by 50 students, we obtained a ratio, or a constant relationship that remains between teachers and students, that is, how many teachers are needed for the number of students, or more generally "how many times the first number contains the second" [Wikipedia, 2019].

We can also notice that more students require more teachers (which is a relation of direct proportionality), that is, if we require more students, more teachers are also required in the same ratio or constant already mentioned.

Then, we can make a proportion among these values because they keep the same ratio of teachers to students (but can also be students to teachers).

So, we can pose the following question:

If four (4) teachers are needed for every 50 students, how many teachers are needed to attend 2,650 students? Then,

[tex]\\ \frac{4_{teachers}}{50_{students}} = \frac{X_{teachers}}{2650_{students}} [/tex] = the same ratio or constant.

To know the amount of teachers needed for 2,650 students:

[tex]\\ X_{teachers} = \frac{2650_{students} * 4_{teachers} }{50_{students}}[/tex].

or,

[tex]\\ X_{teachers} = \frac{2650 * 4_{teachers} }{50}[/tex].

[tex]\\ X_{teachers} = 212_{teachers}[/tex].

So, for an enrollment of 2,650 students there should be 212 teachers, according to the ratio (teachers/students) previously determined by the school board.