Question

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 1

The correct option is E $15$

Let the number of student and teacher be represented by S and T respectively.

$\because S:T=30:1⟹\frac{S}{T}=\frac{30}{1}⟹S:30T⟶\left(1\right)$

If number of students increases by $50$ and number of teachers increases by $5$

$50+S:T+5=25:1$

$\Rightarrow \frac{S+50}{T+S}=\frac{25}{1}\Rightarrow S+50=25(T+5)=25T+125$

$\Rightarrow 30T+50=25T+125$ [From $\left(1\right)$]

$⟹30T-25T=125-50$

$⟹5T=75$

$⟹T=\frac{75}{5}=15$

$\therefore$ No of teachers=$T=15$