Is the teacher-to-student ratio in your school proportional to the one in my school?

Question

Homi · Accepted Answer

Understand the Question

Two ratios are proportional if their simplest forms are equal.
To determine if the teacher-to-student ratios are proportional, express both ratios in simplest form and compare them.

Step 1 · Simplify the Given School's Ratio

The ratio of teachers to students in the given school is $5 : 170$ . Diagram 1

Dividing both terms by their greatest common factor ( $5$ )

\begin{aligned} \text{Given school's ratio} &= \text{Teachers} : \text{Students} \\[0.6em] &= 5 : 170 \\[0.6em] &= (5 \div 5) : (170 \div 5) \\[0.6em] &= 1 : 34 \end{aligned}

Step 2 · Simplify My School's Ratio

The ratio of teachers to students in my school is

\begin{aligned} \text{My school's ratio} &= \text{Teachers} : \text{Students} \\[0.6em] &= 1 : 30 \end{aligned}

This ratio is already in its simplest form.

Step 3 · Compare the Ratios

Comparing the two simplified ratios $\text{Given school's ratio} = 1 : 34$ $\text{My school's ratio} = 1 : 30$

Since $1 : 34 \neq 1 : 30$ , the two ratios are not equal.

Answer

No, the teacher-to-student ratio in my school is not proportional to the one in the given school ( $1 : 34 \neq 1 : 30$ ).

Common Mistakes

Comparing Unsimplified Ratios: Attempting to compare $5 : 170$ and $1 : 30$ directly without reducing both to unit form or simplest form.
Order of Terms: Inverting the terms (students to teachers, $34 : 1$ ) instead of teachers to students ( $1 : 34$ ).

Question 5