Divide. Write your answer as a mixed number if needed.
$\frac{5}{6} \div \frac{2}{9}$
We are tasked with dividing two fractions: \(\frac{5}{6}\) and \(\frac{2}{9}\). The goal is to compute \(\frac{5}{6} \div \frac{2}{9}\) and express the resulting value as a mixed number. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal (or "flipping" the second fraction). The divisor is \(\frac{2}{9}\), so its reciprocal is \(\frac{9}{2}\).
To divide the fractions, we set up the multiplication:
\[\frac{5}{6} \div \frac{2}{9} = \frac{5}{6} \times \frac{9}{2}\]
Performing the multiplication of the numerators and the denominators:
\[\frac{5 \times 9}{6 \times 2} = \frac{45}{12}\]
The resulting fraction is \(\frac{45}{12}\). To simplify this, we identify the greatest common divisor of the numerator (45) and the denominator (12), which is 3. Dividing both by 3, we get:
\[\frac{45 \div 3}{12 \div 3} = \frac{15}{4}\]
Since the resulting fraction \(\frac{15}{4}\) is an improper fraction (because 15 > 4), we convert it into a mixed number. We divide 15 by 4:
\[15 \div 4 = 3 \text{ with a remainder of } 3.\]
This means that 15 contains 3 whole units and a remainder of 3/4. Therefore, the mixed number is \(3\frac{3}{4}\). The final answer is \(\boxed{3\frac{3}{4}}\).
Imagine you have a large container with \(5/6\) of a gallon of juice and you want to pour it into smaller containers, each holding \(2/9\) of a gallon. To find out how many full containers you can fill and what fraction of a container you will have left, you perform a division of these fractions. Mastering this operation allows you to efficiently distribute resources, cook using recipe adjustments, or manage materials in construction projects where fractional parts are common.