Mixed number fractions vs regular fractions? $3\frac{1}{6}-1\frac{11}{12}$

Question

I just passed Calculus 2 in college with an A and I'm rather embarrassed that I'm asking this question. My wife is taking an intermediate Algebra course in college and they gave her the below problem.

$$3\frac{1}{6}-1\frac{11}{12}$$

Well I guess I gave her bad advice because I told her that this problem is equivalent to multiplying the whole numbers $3$ and $1$ by the fraction simplifying the problem to:

$\frac{3}{6}-\frac{11}{12}$ or $\frac{1}{2}-\frac{11}{12}$

After solving I got an answer of $-\frac{5}{12}$ which is not the correct answer according to her book.

Is true that the mixed fraction gives a different result than a normal fraction multiplied by a number?

Answer 1

Mixed fractions are supposed to be additions. E.g. $3\frac{1}{6}=3+\frac16$. No wonder you were getting wrong results using mixed numbers, but now you are set. Of course, nobody uses this horrible notation in real life.

Answer 2

$3 \dfrac16$ denoted $3+\dfrac16 = \dfrac{19}6$ and not $3 \cdot \dfrac16$.

Similarly, $1\dfrac{11}{12}$ denotes $1+\dfrac{11}{12} = \dfrac{23}{12}$ and not $1 \cdot \dfrac{11}{12}$.

Hence, $$3 \dfrac16 - 1\dfrac{11}{12} = \dfrac{19}6 - \dfrac{23}{12} = \dfrac{38-23}{12} = \dfrac{15}{12} = \dfrac54$$