To simplify the expression \((m + \frac{1}{2}) / (6m + 3) / 4m\), we can rewrite it for clarity:
\[ \frac{(m + \frac{1}{2})}{(6m + 3)} \div (4m) \]
Dividing by a fraction is the same as multiplying by its reciprocal. Thus, we can rewrite the expression as:
\[ \frac{(m + \frac{1}{2})}{(6m + 3)} \times \frac{1}{4m} \]
This simplifies to:
\[ \frac{(m + \frac{1}{2})}{(6m + 3) \cdot 4m} \]
Next, we can simplify the denominator:
\[ 6m + 3 = 3(2m + 1) \]
So, the denominator becomes:
\[ (3(2m + 1)) \cdot (4m) = 12m(2m + 1) \]
Now our expression looks like this:
\[ \frac{(m + \frac{1}{2})}{12m(2m + 1)} \]
Next, we can simplify the numerator \(m + \frac{1}{2}\) by expressing it as:
\[ \frac{2m + 1}{2} \quad \text{(multiplying by } \frac{2}{2} \text{)} \]
So our expression now becomes:
\[ \frac{\frac{2m + 1}{2}}{12m(2m + 1)} \]
This simplifies to:
\[ \frac{2m + 1}{2 \cdot 12m(2m + 1)} = \frac{1}{24m} \]
(Note that \(2m + 1\) in the numerator and denominator cancels out.)
Thus, the simplified expression is:
\[ \frac{1}{24m} \]
So your final answer is:
\[ \frac{1}{24m} \]