To find the quotient of [tex]\( 6 \frac{3}{4} \)[/tex] and [tex]\( \frac{5}{6} \)[/tex], we can follow these steps:
1. Convert the mixed number to an improper fraction:
[tex]\( 6 \frac{3}{4} \)[/tex] can be written as [tex]\( 6 + \frac{3}{4} \)[/tex].
- [tex]\( 6 + \frac{3}{4} \)[/tex] is equal to [tex]\( 6 + 0.75 = 6.75 \)[/tex].
2. Write both values as fractions:
- [tex]\( 6 \frac{3}{4} \)[/tex] is equal to [tex]\( 6.75 \)[/tex].
- [tex]\( 6.75 \)[/tex] as a fraction is [tex]\( \frac{27}{4} \)[/tex] (since [tex]\( 6.75 \times 4 = 27 \)[/tex]).
- The other number given is [tex]\( \frac{5}{6} \)[/tex].
3. Divide the two fractions:
To divide [tex]\( \frac{27}{4} \)[/tex] by [tex]\( \frac{5}{6} \)[/tex], multiply [tex]\( \frac{27}{4} \)[/tex] by the reciprocal of [tex]\( \frac{5}{6} \)[/tex]:
[tex]\[
\frac{27}{4} \div \frac{5}{6} = \frac{27}{4} \times \frac{6}{5}
\][/tex]
4. Simplify the multiplication:
[tex]\[
\frac{27}{4} \times \frac{6}{5} = \frac{27 \times 6}{4 \times 5} = \frac{162}{20}
\][/tex]
5. Simplify the resulting fraction:
Simplify [tex]\( \frac{162}{20} \)[/tex]:
- Both 162 and 20 can be divided by their Greatest Common Divisor (GCD) which is 2:
[tex]\[
\frac{162 \div 2}{20 \div 2} = \frac{81}{10}
\][/tex]
6. Convert the improper fraction to a mixed number:
- Divide 81 by 10 to get the whole number part:
[tex]\[
81 \div 10 = 8 \, (\text{remainder } 1)
\][/tex]
- This gives us the mixed number [tex]\( 8 \frac{1}{10} \)[/tex].
Therefore, the quotient of [tex]\( 6 \frac{3}{4} \)[/tex] and [tex]\( \frac{5}{6} \)[/tex] is [tex]\( 8 \frac{1}{10} \)[/tex].
Thus, the correct answer is [tex]\( \boxed{D} \)[/tex].
