Reason Abstractly Complete the Venn diagram to compare and contrast the division and multiplication problems.

Identify Structure Find two positive fractions with a quotient of \(\frac{5}{6}\). Give the equivalent multiplication sentence.

Identify Repeated Reasoning Is the quotient \(\frac{2}{3}\div\frac{1}{2}\) greater than or less than 1? Is the quotient \(\frac{1}{2}\div\frac{2}{3}\) of greater than or less than 1? Explain your reasoning.

Persevere with Problems Complete the steps to demonstrate why you multiply by the reciprocal when dividing fractions. Find \(\frac{1}{4}\div\frac{3}{8}\)  .      

Step 1 Rewrite it as \(\large\frac{\frac{1}{4}}{\frac{3}{8}}\)

Step 2 Multiply the numerator and the denominator by the reciprocal of \(\frac{\Box}{\Box }\). \(\large\frac{\frac{1}{4}}{\frac{3}{8}}=\frac{\frac{1}{4}\times\frac{\Box}{\Box}}{\frac{3}{8}\times\frac{\Box}{\Box}}\)       

Step 3 Simplify the denominator.  \(\large\frac{\frac{1}{4}\times\frac{8}{3}}{\Box}\)

Step 4 Simplify the fraction. \(\frac{\Box}{\Box}\times\frac{\Box}{\Box}\) 

Reason Inductively In cooking, 1 drop is equal to \(\frac{1}{6}\)  of a dash. If a recipe calls for \(\frac{2}{3}\)  of a dash, which expression would give the number of drops that are needed?