To simplify the expression \(\frac{(m + \frac{1}{2})}{(6m + 3) / (4m)}\) and show that it equals \(\frac{2m}{3}\), let's start step by step.
First, rewrite the expression for clarity:
\[ \frac{(m + \frac{1}{2})}{\frac{(6m + 3)}{(4m)}} \]
When dividing by a fraction, you multiply by the reciprocal of that fraction. Hence, we can rewrite the expression as:
\[ (m + \frac{1}{2}) \cdot \frac{4m}{6m + 3} \]
Next, we simplify \(6m + 3\):
\[ 6m + 3 = 3(2m + 1) \]
So now our expression looks like this:
\[ (m + \frac{1}{2}) \cdot \frac{4m}{3(2m + 1)} \]
Next, we need to deal with \(m + \frac{1}{2}\). We can write \(m + \frac{1}{2}\) as a single fraction:
\[ m + \frac{1}{2} = \frac{2m + 1}{2} \]
Now, substitute this back into the expression:
\[ \frac{2m + 1}{2} \cdot \frac{4m}{3(2m + 1)} \]
Now we can multiply these fractions:
\[ \frac{(2m + 1) \cdot 4m}{2 \cdot 3(2m + 1)} \]
Notice that \(2m + 1\) appears in both the numerator and the denominator, so we can cancel it out (as long as \(2m + 1 \neq 0\)):
\[ \frac{4m}{2 \cdot 3} = \frac{4m}{6} \]
Now simplify \(\frac{4m}{6}\):
\[ \frac{4m}{6} = \frac{2m}{3} \]
Thus, we have shown that:
\[ \frac{(m + \frac{1}{2})}{\frac{(6m + 3)}{(4m)}} = \frac{2m}{3} \]
So the final answer is:
\[ \frac{2m}{3} \]