Visualize Fractions

Indeed, we can convert every mixed number into an improper fraction. We will discuss this conversion trick in the next section. Read this text, complete the practice problems, and check your answers.

Model Improper Fractions and Mixed Numbers

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.


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

a \frac{b}{c} \quad c \neq 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.


The fraction \frac{a}{b} is a proper fraction if a and an improper fraction if a \geq b.

Source: Rice University,
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