Visualize Improper Fractions

In Example 4.4 (b), you had 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.

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.

Source: Rice University, https://openstax.org/books/prealgebra/pages/4-1-visualize-fractions
This work is licensed under a Creative Commons Attribution 4.0 License.

EXAMPLE 4.5

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

EXAMPLE 4.6

Draw a figure to model $\dfrac{11}{8}$.

EXAMPLE 4.7

Use a model to rewrite the improper fraction $\dfrac{11}{6}$ as a mixed number.

EXAMPLE 4.8

Use a model to rewrite the mixed number $1 \dfrac{4}{5}$ as an improper fraction.

TRY IT 4.9

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

TRY IT 4.10

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

TRY IT 4.11

Draw a figure to model $\dfrac{7}{6}$.

TRY IT 4.12

Draw a figure to model $\dfrac{6}{5}$.

TRY IT 4.13

Use a model to rewrite the improper fraction as a mixed number: $\dfrac{9}{7}$.

TRY IT 4.14

Use a model to rewrite the improper fraction as a mixed number: $\dfrac{7}{4}$.

TRY IT 4.15

Use a model to rewrite the mixed number as an improper fraction: $1 \dfrac{3}{8}$.

TRY IT 4.16

Use a model to rewrite the mixed number as an improper fraction: $1 \dfrac{5}{6}$.

EXAMPLE 4.5

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

EXAMPLE 4.6

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, $\dfrac{11}{8}=1 \dfrac{3}{8}$.

EXAMPLE 4.7

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

$\dfrac{6}{6}=1$

That leaves us with five more sixths, which is $\dfrac{5}{6}$ $\text{(11 sixths minus 6 sixths is 5 sixths)}$. So, $\dfrac{11}{6}=1 \dfrac{5}{6}$.

EXAMPLE 4.8

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

So, $1 \dfrac{4}{5}=\dfrac{9}{5}$.

TRY IT 4.9

TRY IT 4.10

$\dfrac{13}{8}=1 \dfrac{5}{8}$

TRY IT 4.11

TRY IT 4.12

TRY IT 4.13

$1 \dfrac{2}{7}$

TRY IT 4.14

$1 \dfrac{3}{4}$

TRY IT 4.15

$\dfrac{11}{8}$

TRY IT 4.16

$\dfrac{11}{6}$