## Modeling Improper Fractions and Mixed Numbers

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In an example in the previous lesson, you were given eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, \(\frac{1}{5},\) so altogether you had eight fifths, which we can write as \(\frac{8}{5}.\) The fraction \(\frac{8}{5}\) is one whole, \(1,\) plus three fifths, \(\frac{3}{5},\) or \(1\frac{3}{5},\) which is read as *one and three-fifths*.

The number \(1\frac{3}{5}\) is called a mixed number. A mixed number consists of a whole number and a fraction.

### Mixed Numbers

A **mixed number** consists of a whole number \(a\) and a fraction \(\frac{b}{c}\) where \(c\ne 0.\) It is written as follows.

\(a\frac{b}{c}\phantom{\rule{2em}{0ex}}c\ne 0\)

Fractions such as \(\frac{5}{4},\frac{3}{2},\frac{5}{5},\) and \(\frac{7}{3}\) are called improper fractions. In an improper fraction, the **numerator** is greater than or equal to the **denominator**, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as \(\frac{1}{2},\frac{3}{7},\) and \(\frac{11}{18}\) are proper fractions.

### Proper and Improper Fractions

The fraction \(\frac{a}{b}\) is a **proper fraction** if \(a<b\) and an **improper fraction** if \(a\ge b.\)

## Example

Name the improper fraction modeled. Then write the improper fraction as a mixed number.

### Solution

Each circle is divided into three pieces, so each piece is \(\frac{1}{3}\) of the circle. There are four pieces shaded, so there are four thirds or \(\frac{4}{3}.\) The figure shows that we also have one whole circle and one third, which is \(1\frac{1}{3}.\) So, \(\frac{4}{3}=1\frac{1}{3}.\)

## Example

Draw a figure to model \(\frac{11}{8}.\)

### Solution

The denominator of the improper fraction is \(8.\) Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have \(11\) eighths. We must shade three of the eight parts of another circle.

So, \(\frac{11}{8}=1\frac{3}{8}.\)

## Example

Use a model to rewrite the improper fraction \(\frac{11}{6}\) as a mixed number.

### Solution

We start with \(11\) sixths \(\left(\frac{11}{6}\right).\) We know that six sixths makes one whole.

That leaves us with five more sixths, which is

\(\frac{5}{6}\phantom{\rule{0.2em}{0ex}}\left(11\phantom{\rule{0.2em}{0ex}}\text{sixths minus}\phantom{\rule{0.2em}{0ex}}6\phantom{\rule{0.2em}{0ex}}\text{sixths is}\phantom{\rule{0.2em}{0ex}}5\phantom{\rule{0.2em}{0ex}}\text{sixths}\right).\)

So, \(\frac{11}{6}=1\frac{5}{6}.\)

## Example

Use a model to rewrite the mixed number \(1\frac{4}{5}\) as an improper fraction.

### Solution

The mixed number \(1\frac{4}{5}\) means one whole plus four fifths. The denominator is \(5,\) so the whole is \(\frac{5}{5}.\) Together five fifths and four fifths equals nine fifths.

So, \(1\frac{4}{5}=\frac{9}{5}.\)

### Optional Video: Identify Fractions Using Pattern Blocks

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