To divide the fractions \( \frac{3}{4} \) by \( 9 \), we first need to express \( 9 \) as a fraction. We can write \( 9 \) as \( \frac{9}{1} \).
Now the expression becomes:
\[ \frac{3}{4} \div \frac{9}{1} \]
Dividing by a fraction is the same as multiplying by its reciprocal. So, we multiply \( \frac{3}{4} \) by the reciprocal of \( \frac{9}{1} \), which is \( \frac{1}{9} \):
\[ \frac{3}{4} \times \frac{1}{9} \]
Now, we multiply the numerators and the denominators:
\[ \frac{3 \times 1}{4 \times 9} = \frac{3}{36} \]
Next, we simplify \( \frac{3}{36} \):
To simplify, we can divide both the numerator and the denominator by their greatest common divisor, which is \( 3 \):
\[ \frac{3 \div 3}{36 \div 3} = \frac{1}{12} \]
So, the simplified form of the quotient is:
\[ \frac{1}{12} \]
Therefore,
\[ \frac{3}{4} \div 9 = \frac{1}{12} \]