---
tags:
  - prealgebra
  - fractions
---

# Mixed and improper fractions

## Improper fractions

- Proper fraction:
  - The numerator is smaller than the denominator
  - E.g. $\frac{2}{3}$, $-\frac{5}{10}$
- Improper fraction
  - The numerator is greater than the denominator
  - E.g. $\frac{3}{2}$, $-\frac{5}{4}$

## Mixed fractions

A mixed fraction is part whole number, part fraction. For example:
$5 \frac{3}{4}$.

It means, in effect: $5 + \frac{3}{4}$

## Converting mixed fractions into improper fractions

Mixed fractions are hard to calculate with. We need some way to convert them to
fractions. We can do this by converting them to improper fractions.

With the example $4 \frac{7}{8}$, we know this means $4 + \frac{7}{8}$. We need
to express the amount 4 in terms of eighths. It would be 4 lots of $\frac{8}{8}$
given that 4 is a whole number not a fractional amount. Thus the process would
be:

$$
    \frac{8}{8} + \frac{8}{8} + \frac{8}{8} + \frac{8}{8} + \frac{7}{8}
$$

But we know that when we
[add fractions with a common denominator](Add_Subtract_Fractions.md#adding-subracting-fractions-with-common-denominators),
we only add the numerators, not the denominators. Thus the calculation would
actually be:

$$
\frac{8 + 8 + 8 + 8 + 7}{8} = \frac{39}{8}
$$

Addition helps to explain the concepts underlying the procedure but it is more
efficient to use multiplication.

The procedure is as follows:

1. Multiply the whole number part by the denominator
2. Add the numerator
3. Place the result over the denominator

Thus:

$$
    4 \frac{7}{8} = \frac{(4 \cdot 8) + 7 }{8}
$$

## Converting improper fractions into mixed fractions

It is quite obvious how to reverse the process and turn an improper fraction
into a mixed fraction.

Take $\frac{27}{5}$. We work out how many times the numerator is divisible by
the denominator and make that the whole number. The remainder is then left as
the fractional part.

$$
\begin{split}
\frac{27}{5} = 27 \div 5  \\
= 5 r 2 \\
= 5 \frac{2}{5}
\end{split}
$$

## Multiplying and dividing by mixed fractions

Now that we know how to convert mixed fractions into improper fractions, it is
straight forward to multiply and divide with them. We convert the mixed fraction
into an improper fraction and then divide and multiply as we would with a proper
fraction.

### Demonstration of multiplication

Calculate $-2\frac{1}{12} \cdot 2 \frac{4}{5}$:

1. First convert each mixed fraction into an improper fraction:
   $$
     \begin{split}
     -2\frac{1}{12} = -2 \cdot -12  \\
     = 24 + 1 \\
     = - \frac{25}{12}
     \end{split}
   $$

$$
\begin{split}
  2 \frac{4}{5} =2 \cdot 5 \\
  = 10 + 4 \\
  = \frac{14}{5}
\end{split}
$$

2. Then carry out the multiplication
   [using factorization](Multiplying_fractions.md#prime-factorisation-in-place):

   $$
     \begin{split}
     - \frac{25}{12} \cdot \frac{14}{5} = \\
    - \frac{(5 \cdot 5) \cdot (7 \cdot 2)}{(3 \cdot 2 \cdot 2) \cdot (5)} = - \frac{5 \cdot 7 }{2 \cdot 3} \\
     \end{split}
   $$

3. Then simplify:

   $$
     - \frac{35}{6}
   $$

4. Finally, convert back into a mixed fraction:

$$
  \begin{split}
  - \frac{35}{6} = -35 \div 6 \\
  -  5 r 5 = \\
  - 5 \frac{5}{6}
  \end{split}
$$

## Demonstration of division

Again we convert the mixed fraction into an improper fraction and then follow
the requisite rule, in the case of division this is to
[invert and multiply](Dividing_fractions.md#formal-specification-of-how-to-divide-fractions').

Calculate $-4 \frac{4}{5} \div 5 \frac{3}{5}$.

1. Convert to improper fraction:
   $$
      \begin{split}
        -4 \cdot 5 = \\
    -20
   \end{split}
   $$

## Adding and subtracting mixed fractions
