To determine which of the expressions is NOT equal to \( \frac{5}{6} \), we will calculate the value of each option.
Option A: \( \frac{7}{6} - \frac{1}{2} \) First, find a common denominator. The least common denominator of 6 and 2 is 6.
\[ \frac{1}{2} = \frac{3}{6} \]
Now substitute:
\[ \frac{7}{6} - \frac{3}{6} = \frac{7 - 3}{6} = \frac{4}{6} = \frac{2}{3} \]
Option B: \( \frac{1}{3} + \frac{1}{2} \) Again, find a common denominator. The least common denominator of 3 and 2 is 6.
\[ \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6} \]
Now substitute:
\[ \frac{2}{6} + \frac{3}{6} = \frac{2 + 3}{6} = \frac{5}{6} \]
Option C: \( 1 - \frac{1}{6} \) This can also be rewritten using a common base:
\[ 1 = \frac{6}{6} \]
Now substitute:
\[ \frac{6}{6} - \frac{1}{6} = \frac{6 - 1}{6} = \frac{5}{6} \]
Option D: \( \frac{1}{6} \div \frac{1}{5} \) Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{1}{6} \div \frac{1}{5} = \frac{1}{6} \times \frac{5}{1} = \frac{5}{6} \]
After evaluating all options:
- Option A: \( \frac{2}{3} \) (NOT equal to \( \frac{5}{6} \))
- Option B: \( \frac{5}{6} \) (equal)
- Option C: \( \frac{5}{6} \) (equal)
- Option D: \( \frac{5}{6} \) (equal)
Thus, the expression that is NOT equal to \( \frac{5}{6} \) is:
\[ \boxed{A} \]