The present ratio of students to teachers (OG2016)

boomgoesthegmat wrote:The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers?

A) 5
B) 8
C) 10
D) 12
E) 15

I like to use two variables for these "changing ratios" questions.

The PRESENT ratio of students to teachers at a certain school is 30 to 1
Let S = # of students PRESENTLY
Let T = # of teachers PRESENTLY
We can write: S/T = 30/1
Cross multiply to get: (1)(S) = (30)(T)
Simplify: S = 30T

If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1
In this scenario...
S + 50 = # of students
T + 5 = # of teachers
So, we get: (S + 50)/(T + 5) = 25/1
Cross multiply to get: (S + 50)(1) = (25)(T + 5)
Expand: S + 50 = 25T + 125
Simplify to get: S = 25T + 75

We now have 2 equations with 2 variables:
S = 30T
S = 25T + 75

To solve, replace S in the bottom equation with 30T
We get: 30T = 25T + 75
So: 5T = 75
Solve: T = 15
So, there are PRESENTLY 15 teachers

Answer: E

Cheers,
Brent

Hi All,

We're told that the present ratio of students to teachers at a certain school is 30 to 1 and IF the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, then the ratio of students to teachers would then be 25 to 1. We're asked for the CURRENT number of teachers. This question can be solved in a number of different ways, including by TESTing THE ANSWERS. Since most of the numbers involved are multiples of 5, it's likely that the correct answer will also be a multiple of 5. Let's Test Answer C first...

Answer C: 10
IF there are currently 10 teachers, then there are...
(30)(10) = 300 students
Increasing the students and teachers as described would give us....
350 students and 15 teachers.
This ratio is 350:15 = 70:3.... which is NOT a match (it's supposed to be 25 to 1). To increase the ratio, we need to increase the number of students (and the direct way to do THAT is to increase the STARTING number of teachers)....

Answer E: 15
IF there are currently 15 teachers, then there are...
(30)(15) = 450 students
Increasing the students and teachers as described would give us....
500 students and 20 teachers.
This ratio is 500:20 = 50:2 = 25:1
This is an exact match for what we were told, so this MUST be the answer.

Final Answer: E

GMAT assassins aren't born, they're made,
Rich