Fractions are Hard!

3.1 Mixed numbers

A mixed number is a number of the form $$a+\dfrac{b}{c}$$, usually just written $$a\dfrac{b}{c}$$, with $$a$$, $$b$$, and $$c$$ integers.

Question: Is this right? Do textbooks allow $$2\dfrac{7}{3}$$ and $$3\dfrac{-5}{6}$$ and $$5\dfrac{0}{9}$$ as examples of mixed numbers? Or must these particular numbers be written as $$4\dfrac{1}{3}$$ and $$2\dfrac{1}{6}$$ and $$5$$, respectively?

Often fractions of the form $$\dfrac{a}{b}$$ with $$a>b$$ are dubbed “improper” and people prefer to express such fractions as mixed numbers.

Example: Write $$\dfrac{39}{8}$$ as a mixed number.

Answer: $$\dfrac{39}{8} = \dfrac{32+7}{8} = 5 + \dfrac{7}{8} = 5\dfrac{7}{8}$$.

Of course, we can convert mixed numbers into (improper) fractions too.

Example: Write $$20\dfrac{1}{20}$$ as fraction of the form $$\dfrac{a}{b}$$ with $$a$$ and $$b$$ whole numbers.

$$20\dfrac{1}{20}= \dfrac{20\dfrac{1}{20}}{1}$$.

Now multiply top and bottom by $$20$$ to get:

$$\dfrac{20\dfrac{1}{20}}{1} = \dfrac{20\dfrac{1}{20}\times 20}{1\times 20}= \dfrac{400+1}{20}=\dfrac{401}{20}$$.

Comment: Alternatively, $$20\dfrac{1}{20} = 20+\dfrac{1}{20}=\dfrac{400}{20}+\dfrac{1}{20}=\dfrac{401}{20}$$.

 Question: Write each of the following as a mixed number.   a) $$\dfrac{8}{5}$$          b) $$\dfrac{100}{13}$$          c)  $$\dfrac{200}{199}$$      d) $$\dfrac{199\frac{1}{2}}{199}$$   (Can the answer to part d) simply be $$1\dfrac{1/2}{199}$$?  Should we indeed have a strict definition of what a “mixed number” should be?)   Write each of the following in the form $$\dfrac{a}{b}$$ with $$a$$ and $$b$$ integers.   e) $$7\dfrac{2}{9}$$            f)   $$2\dfrac{3}{4} + 5\dfrac{2}{7}$$              g) $$300\dfrac{299}{300}$$

MULTIPLYING MIXED NUMBERS

The area model for multiplication can be of visual help here.

Example: Compute $$4\dfrac{2}{5}\times 7\dfrac{3}{8}$$.

Answer: This is the area of a rectangle, which naturally divides into four pieces:

We see that

$$4\dfrac{2}{5}\times 7\dfrac{3}{8} = 4\times 7 + 4 \times \dfrac{3}{8}+\dfrac{2}{5}\times 7 + \dfrac{2}{5}\times \dfrac{3}{8}$$

$$=48+\dfrac{12}{8}+\dfrac{14}{5}+\dfrac{6}{40}$$

$$=48+1+\dfrac{1}{2}+2+\dfrac{4}{5}+\dfrac{3}{20}$$

$$=51+\dfrac{10+16+3}{20}=51+\dfrac{29}{20}=52\dfrac{9}{20}$$.

Alternatively, $$4\dfrac{2}{5}\times 7\dfrac{3}{8}=\dfrac{22}{5}\times \dfrac{59}{8}=\dfrac{1298}{40}$$.

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