The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by

Gmat_mission wrote: ↑

Fri Jan 08, 2021 1:59 am

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

Answer: E

Source: Official Guide

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

Gmat_mission wrote: ↑

Fri Jan 08, 2021 1:59 am

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

Answer: E

Source: Official Guide

Solution:

We are given that the ratio of students to teacher is 30 to 1. We can rewrite this using variable multipliers.

students : teachers = 30x : x

We are next given that student enrollment increases by 50 and the number of teachers increases by 5. With this change the new ratio becomes 25 to 1. We can put all this into an equation:

(30x + 50) / (x + 5) = 25/1

Cross multiplying, we have:

25(x + 5) = 30x + 50

25x + 125 = 30x + 50

75 = 5x

15 = x

Since x is the present number of teachers, currently there are 15 teachers.

Answer: E