Is converting a mixed-fraction to an improper fraction is just adding the remaining part to the whole?

Question

For example, if you're simplifying $3 \frac{1}{2}$, the conventional way is to do $3 \cdot 2$ and then add the numerator $1$.
Isn't this the same as adding $3/1$ and $1/2$?

Answer 1

Yes, obviously: the meaning of $3 \frac{1}{2}$ is $3 + \frac{1}{2}$, which comes down to $\frac{2 \cdot 3}{2} + \frac{1}{2}$, which is equal to $\frac{2 \cdot 3 + 1}{2}$, which is equal to $\frac{7}{2}$.

Answer 2

Yes. The reason you multiply the 3 by the 2 is that you are really forming the numerator of a fraction with a common denominator, since $3\frac12$ really means $3 + \frac12$. So: $$3 + \frac12 = \frac31 + \frac12$$ $$= \frac{3\cdot 2}{1\cdot2} + \frac12$$ $$=\frac{3\cdot 2}2 + \frac12$$ $$=\frac{3\cdot2 + 1}2$$ $$=\frac72$$ The "shortcut" skips from line 1 to line 4: $$3 +\dfrac12 =\frac{3\cdot2 + 1}2 =\frac72$$