The problem is to evaluate the expression $\frac{5}{6} \times (\frac{2}{3} - \frac{1}{4}) + \frac{1}{2}$.

1. Problem Description

The problem is to evaluate the expression $\frac{5}{6} \times (\frac{2}{3} - \frac{1}{4}) + \frac{1}{2}$.

2. Solution Steps

First, we need to evaluate the expression inside the parenthesis.

We have $\frac{2}{3} - \frac{1}{4}$.

To subtract these two fractions, we need to find a common denominator. The least common multiple of 3 and 4 is

1

2. So, we rewrite the fractions with a common denominator of 12:

$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Now, we can subtract the fractions:

$\frac{8}{12} - \frac{3}{12} = \frac{8-3}{12} = \frac{5}{12}$

So, the expression becomes:

$\frac{5}{6} \times \frac{5}{12} + \frac{1}{2}$

Next, we multiply $\frac{5}{6}$ and $\frac{5}{12}$:

$\frac{5}{6} \times \frac{5}{12} = \frac{5 \times 5}{6 \times 12} = \frac{25}{72}$

Now, the expression becomes:

$\frac{25}{72} + \frac{1}{2}$

To add these fractions, we need to find a common denominator. The least common multiple of 72 and 2 is

7

2. So, we rewrite the fractions with a common denominator of 72:

$\frac{1}{2} = \frac{1 \times 36}{2 \times 36} = \frac{36}{72}$

Now, we can add the fractions:

$\frac{25}{72} + \frac{36}{72} = \frac{25+36}{72} = \frac{61}{72}$

3. Final Answer

The final answer is $\frac{61}{72}$.