150 lines
3.6 KiB
Markdown
150 lines
3.6 KiB
Markdown
---
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tags:
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- prealgebra
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- fractions
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---
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# Mixed and improper fractions
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## Improper fractions
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- Proper fraction:
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- The numerator is smaller than the denominator
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- E.g. $\frac{2}{3}$, $-\frac{5}{10}$
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- Improper fraction
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- The numerator is greater than the denominator
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- E.g. $\frac{3}{2}$, $-\frac{5}{4}$
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## Mixed fractions
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A mixed fraction is part whole number, part fraction. For example:
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$5 \frac{3}{4}$.
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It means, in effect: $5 + \frac{3}{4}$
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## Converting mixed fractions into improper fractions
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Mixed fractions are hard to calculate with. We need some way to convert them to
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fractions. We can do this by converting them to improper fractions.
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With the example $4 \frac{7}{8}$, we know this means $4 + \frac{7}{8}$. We need
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to express the amount 4 in terms of eighths. It would be 4 lots of $\frac{8}{8}$
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given that 4 is a whole number not a fractional amount. Thus the process would
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be:
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$$
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\frac{8}{8} + \frac{8}{8} + \frac{8}{8} + \frac{8}{8} + \frac{7}{8}
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$$
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But we know that when we
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[add fractions with a common denominator](Add_Subtract_Fractions.md#adding-subracting-fractions-with-common-denominators),
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we only add the numerators, not the denominators. Thus the calculation would
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actually be:
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$$
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\frac{8 + 8 + 8 + 8 + 7}{8} = \frac{39}{8}
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$$
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Addition helps to explain the concepts underlying the procedure but it is more
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efficient to use multiplication.
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The procedure is as follows:
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1. Multiply the whole number part by the denominator
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2. Add the numerator
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3. Place the result over the denominator
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Thus:
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$$
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4 \frac{7}{8} = \frac{(4 \cdot 8) + 7 }{8}
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$$
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## Converting improper fractions into mixed fractions
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It is quite obvious how to reverse the process and turn an improper fraction
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into a mixed fraction.
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Take $\frac{27}{5}$. We work out how many times the numerator is divisible by
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the denominator and make that the whole number. The remainder is then left as
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the fractional part.
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$$
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\begin{split}
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\frac{27}{5} = 27 \div 5 \\
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= 5 r 2 \\
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= 5 \frac{2}{5}
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\end{split}
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$$
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## Multiplying and dividing by mixed fractions
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Now that we know how to convert mixed fractions into improper fractions, it is
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straight forward to multiply and divide with them. We convert the mixed fraction
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into an improper fraction and then divide and multiply as we would with a proper
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fraction.
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### Demonstration of multiplication
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Calculate $-2\frac{1}{12} \cdot 2 \frac{4}{5}$:
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1. First convert each mixed fraction into an improper fraction:
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$$
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\begin{split}
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-2\frac{1}{12} = -2 \cdot -12 \\
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= 24 + 1 \\
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= - \frac{25}{12}
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\end{split}
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$$
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$$
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\begin{split}
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2 \frac{4}{5} =2 \cdot 5 \\
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= 10 + 4 \\
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= \frac{14}{5}
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\end{split}
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$$
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2. Then carry out the multiplication
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[using factorization](Multiplying_fractions.md#prime-factorisation-in-place):
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$$
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\begin{split}
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- \frac{25}{12} \cdot \frac{14}{5} = \\
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- \frac{(5 \cdot 5) \cdot (7 \cdot 2)}{(3 \cdot 2 \cdot 2) \cdot (5)} = - \frac{5 \cdot 7 }{2 \cdot 3} \\
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\end{split}
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$$
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3. Then simplify:
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$$
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- \frac{35}{6}
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$$
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4. Finally, convert back into a mixed fraction:
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$$
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\begin{split}
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- \frac{35}{6} = -35 \div 6 \\
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- 5 r 5 = \\
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- 5 \frac{5}{6}
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\end{split}
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$$
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## Demonstration of division
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Again we convert the mixed fraction into an improper fraction and then follow
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the requisite rule, in the case of division this is to
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[invert and multiply](Dividing_fractions.md#formal-specification-of-how-to-divide-fractions').
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Calculate $-4 \frac{4}{5} \div 5 \frac{3}{5}$.
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1. Convert to improper fraction:
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$$
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\begin{split}
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-4 \cdot 5 = \\
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-20
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\end{split}
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$$
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## Adding and subtracting mixed fractions
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