### Instruction: Find \[\dfrac{3}{\,\,\frac{6}{11}\,\,}.\].### Response: When I see a fraction divided by another fraction, I remember the rule that I can simplify this by multiplying the top fraction by the reciprocal of the bottom fraction.
The reciprocal of a fraction is just flipping the numerator and denominator, so the reciprocal of \[\frac{6}{11}\] is \[\frac{11}{6}\].
So, to find \[\dfrac{3}{\,\,\frac{6}{11}\,\,},\] I can multiply \[\frac{3}{1}\] by \[\frac{11}{6}\].
To multiply fractions, I just multiply the numerators and the denominators separately, so \[\frac{3}{1}\times\frac{11}{6}=\frac{3\times 11}{1\times 6}.\]
I can simplify this by doing the arithmetic, so \[\frac{3\times 11}{1\times 6}=\frac{33}{6}.\]
This fraction can be reduced further by dividing both the numerator and denominator by a common factor, which in this case is 3.
So, \[\frac{33}{6}=\frac{33\div 3}{6\div 3}=\frac{11}{2}.\]
This is the simplest form of the fraction, and I can also write it as a mixed number by dividing 11 by 2 and writing the remainder as a fraction, so \[\frac{11}{2}=5\frac{1}{2}.\]
Either way, this is the answer to the original problem.