Question

1. $6 div 3\frac{2}{9} = \frac{\quad}{\quad} div \frac{\quad}{\quad} = \frac{\quad}{\quad} \times \frac{\quad}{\quad} = \frac{\quad}{\quad} = \frac{\quad}{\quad}$

Explanation:

Step1: Convert mixed number to improper fraction

First, convert the mixed number \(3\frac{2}{9}\) 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 \(3\frac{2}{9}\), we have \(3\times9 + 2=29\), so \(3\frac{2}{9}=\frac{29}{9}\). And \(6\) can be written as \(\frac{6}{1}\). So the first two blanks are \(\frac{6}{1}\) and \(\frac{29}{9}\).
\[6\div3\frac{2}{9}=\frac{6}{1}\div\frac{29}{9}\]

Step2: Change division to multiplication (invert the divisor)

When dividing by a fraction, we multiply by its reciprocal. The reciprocal of \(\frac{29}{9}\) is \(\frac{9}{29}\). So we change the division to multiplication:
\[\frac{6}{1}\div\frac{29}{9}=\frac{6}{1}\times\frac{9}{29}\]

Step3: Multiply the fractions

Multiply the numerators together and the denominators together. The numerator is \(6\times9 = 54\) and the denominator is \(1\times29=29\), so:
\[\frac{6}{1}\times\frac{9}{29}=\frac{54}{29}\]

Step4: Convert improper fraction to mixed number (if needed)

Now, convert \(\frac{54}{29}\) to a mixed number. Divide \(54\) by \(29\): \(54\div29 = 1\) with a remainder of \(25\) (since \(29\times1 = 29\) and \(54 - 29=25\)). So \(\frac{54}{29}=1\frac{25}{29}\).

Answer:

\(6\div3\frac{2}{9}=\boldsymbol{\frac{6}{1}}\div\boldsymbol{\frac{29}{9}}=\boldsymbol{\frac{6}{1}}\times\boldsymbol{\frac{9}{29}}=\boldsymbol{\frac{54}{29}}=\boldsymbol{1\frac{25}{29}}\)