Question

divide. write your answer as a fraction or a mixed number in simplest form. \\( 5\frac{3}{4} \div \frac{5}{6} \\)

Explanation:

Step1: Convert mixed number to improper fraction

To divide a mixed number by a fraction, first convert the mixed number \(5\frac{3}{4}\) to an improper fraction. The formula for converting a mixed number \(a\frac{b}{c}\) to an improper fraction is \(\frac{a\times c + b}{c}\). So for \(5\frac{3}{4}\), we have \(a = 5\), \(b = 3\), \(c = 4\). Then \(\frac{5\times4+3}{4}=\frac{20 + 3}{4}=\frac{23}{4}\).

Step2: Multiply by reciprocal

Dividing by a fraction is the same as multiplying by its reciprocal. So \(\frac{23}{4}\div\frac{5}{6}=\frac{23}{4}\times\frac{6}{5}\).

Step3: Multiply numerators and denominators

Multiply the numerators together and the denominators together: \(\frac{23\times6}{4\times5}=\frac{138}{20}\).

Step4: Simplify the fraction

Simplify \(\frac{138}{20}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. \(\frac{138\div2}{20\div2}=\frac{69}{10}\).

Step5: Convert to mixed number (optional, but the problem allows mixed numbers)

Convert \(\frac{69}{10}\) to a mixed number. \(69\div10 = 6\) with a remainder of 9, so \(\frac{69}{10}=6\frac{9}{10}\).

Answer:

\(6\frac{9}{10}\) (or \(\frac{69}{10}\))