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

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?

• Yeah, the conventional notation taught in our elementary schools is that $3\frac16$ means $3+\frac16$, not $3\cdot\frac16$. – Lubin Dec 17 '13 at 0:42
• $3\frac{1}{6}=3+\frac{1}{6}=\frac{19}{6}$. I think you can take it from here. I agree, though, the notation is a bit unfortunate, luckily it's not used in most (any?) higher math courses. – mojambo Dec 17 '13 at 0:42
• Haha, thanks guys. It didn't make any sense whatsoever to me. – hax0r_n_code Dec 17 '13 at 0:44

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.

• Thanks, I thought it was something weird like this. – hax0r_n_code Dec 17 '13 at 0:45
• it's not weird. It's the best way to write a fraction without losing accuracy through decimals or making your reader calculate a division with a full fraction. – Red Alert Dec 17 '13 at 1:12
• My most profound apologies for being a "nobody", then. In truth, although the number of engineers still using mixed fractions to express measurements is in decline it is not yet zero, and the financial markets in the US use mixed fractions when expressing prices of stocks. – ClickRick Jun 15 '14 at 7:59
• @ClickRick don't be mad, that was just a hyperbole. I have seen it in the wild once or twice too, but I do think this notation is bad. – Ian Mateus Jun 15 '14 at 11:55
• @Ian Your use of emotive language such as "horrible" and "bad" are at odds with mathematical principles. That you "have seen it in the wild once or twice" is all very well, but it's far removed from experiences of others, who may have used it "in the wild" on a daily basis for many years. – ClickRick Jun 15 '14 at 20:18

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

Having read the other answers, which have correctly established the principles at work and identified where you had gone wrong, I quickly worked out the answer in my head using approximations, thus:

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

is approximately

$$3-2$$

with the error being

$$\frac{1}{6}-(-\frac{1}{12})$$

which rearranges easily enough as

$$\frac{2}{12}+\frac{1}{12}$$

$$(3-2)+\frac{3}{12}$$
$$1\frac{1}{4}$$