Question

after 3/4 of the students in the auditorium leave, fifteen students enter the same auditorium. the number of students who are now in the auditorium, assuming no other students enter or leave, is 1/3 of the original number of students who were in the auditorium. how many students left the auditorium?

Answer 1

Let's say there are x people in the auditorium to start. 3/4 of them leave, which means we have 1/4 of the original number of people left in the auditorium. This gives us the expression (1/4)x.

15 people enter the auditorium. So we add 15 to our previous expression to get (1/4)x+15.

This is the SAME AS 1/3 of the original number of students. This is a big tip-off that we need an equal sign. 1/3 of the original number of students is expressed as (1/3)x. Let's put this together to get
\[(1/4)x+15=(1/3)x\]
Everybody hates fractions. Let's get rid of the fractions by multiplying EVERYTHING by 4.
\[x+60=(4/3)x\]One of our fractions is gone. Let's get rid of the other one by multiplying everything by 3.
\[3x+180=4x\]Much better.

Now we have to solve for x. We have x-terms on both sides of the equation, so let's fix this by subtracting 3x from both sides.
\[180=x\]

This means that the original number of students in the auditorium was 180. We want to know how many are left at the end. We know that only 1/3 of the original number are remaining, so we divide 180 by 3 to get 60 students left in the auditorium.
Check by substituting this back into the equation.