Question

divide. write the answer in simplest form.
$2\frac{1}{6} \div \frac{3}{4} = \square$

Explanation:

Step1: Convert mixed number to improper fraction

A mixed number \(2\frac{1}{6}\) can be converted to an improper fraction by multiplying the whole number (2) by the denominator (6) and adding the numerator (1). So, \(2\frac{1}{6}=\frac{2\times6 + 1}{6}=\frac{13}{6}\).

Step2: Divide by multiplying by reciprocal

Dividing by a fraction \(\frac{3}{4}\) is the same as multiplying by its reciprocal. The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). So, \(\frac{13}{6}\div\frac{3}{4}=\frac{13}{6}\times\frac{4}{3}\).

Step3: Multiply the fractions

Multiply the numerators together and the denominators together: \(\frac{13\times4}{6\times3}=\frac{52}{18}\).

Step4: Simplify the fraction

Simplify \(\frac{52}{18}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, \(\frac{52\div2}{18\div2}=\frac{26}{9}\). We can also write this as a mixed number: \(2\frac{8}{9}\), but since the question says "simplest form", an improper fraction or a mixed number is acceptable, but \(\frac{26}{9}\) (or \(2\frac{8}{9}\)) is the simplified form. Wait, actually, let's check the multiplication again. Wait, \(\frac{13}{6}\times\frac{4}{3}=\frac{13\times4}{6\times3}=\frac{52}{18}\), then divide numerator and denominator by 2: \(\frac{26}{9}\), which is \(2\frac{8}{9}\). Wait, but let's check the steps again. Wait, \(2\frac{1}{6}\) is \(\frac{13}{6}\), correct. Dividing by \(\frac{3}{4}\) is multiplying by \(\frac{4}{3}\), correct. Then \(\frac{13\times4}{6\times3}=\frac{52}{18}\), simplify: divide numerator and denominator by 2, get \(\frac{26}{9}\), which is \(2\frac{8}{9}\). Alternatively, we can simplify before multiplying. Let's see, 4 and 6 have a common factor of 2. So, \(\frac{13}{6}\times\frac{4}{3}=\frac{13}{3}\times\frac{2}{3}\) (divided 4 and 6 by 2), then multiply: \(\frac{13\times2}{3\times3}=\frac{26}{9}\), same result.

Answer:

\(\frac{26}{9}\) (or \(2\frac{8}{9}\))