To determine if the ratios are equivalent, we can compare them mathematically.
The first ratio is 27 teachers to 30 students, which can be expressed as:
\[ \frac{27}{30} \]
The second ratio is 18 teachers to 20 students, which can be expressed as:
\[ \frac{18}{20} \]
Now, let's simplify both ratios:
- For the first ratio \(\frac{27}{30}\):
\[ \frac{27}{30} = \frac{9}{10} \quad \text{(dividing both the numerator and denominator by 3)} \]
- For the second ratio \(\frac{18}{20}\):
\[ \frac{18}{20} = \frac{9}{10} \quad \text{(dividing both the numerator and denominator by 2)} \]
Since both ratios simplify to \(\frac{9}{10}\), we can conclude that:
\[ \frac{27}{30} = \frac{18}{20} \]
Therefore, the ratios 27 teachers to 30 students and 18 teachers to 20 students are equivalent.