Proper Fractions

Fractions are an alternative way to denote division between two numbers and are generally used to express proportions, for example, to express three-quarters of a quantity we write $\frac{3}{4}$ or to denote half of a cake we simply write $\frac{1}{2}$ . It is possible to represent the fractions graphically to make them easier to understand.

Formally, if we consider two integers $a$ and $b neq 0$ , then we will say that $a$ is the numerator of the fraction and $b$ is the denominator of the fraction, and so, the division $a div b$ will be represented by the following expression

$\frac{a}{b}$

The number above is called numerator and the number below is called denominator.

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Fraction properties

When we work with fractions, we will find very particular expressions that we can identify when we want to simplify mathematical operations. Let’s consider $a$ an non-zero integer and see below what these fractions are.

One divided by one equals one. In general, if we consider any non-zero real, the division of this number by itself, is equal to one, then,

$\dfrac{1}{1} = 1$ .

$\dfrac{a}{a} = 1$ .

Any integer number can be expressed as the division of itself with one, this information will be useful when we are presented with operations between numbers expressed in fractions and integers.

$\dfrac{a}{1} = a$ .

When dividing zero by any non-zero real number, the result will always be the same, zero.

$\dfrac{0}{a} = 0$

On the contrary, if we take any real number, it cannot be divided by zero because this operation is not defined, that is, division by zero is not defined.

$\dfrac{a}{0}$

not defined.

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The Law of Signs for Fractions

Since fractions represent divisions, we can also establish the law of signs for division, if $a$ and $b$ are integers such that $b$ is non-zero, then

$\dfrac{a}{b} = \dfrac{a}{b}$ .

$\dfrac{-a}{b} = -\dfrac{a}{b}$ .

$\dfrac{a}{-b} = -\dfrac{a}{b}$ .

$\dfrac{-a}{-b} = \dfrac{a}{b}$ .

The advantage in the use of fractions is that they provide rigidity in the results and thus avoid approximation or rounding errors when making divisions, which is why it is necessary to master the operations of sum, subtraction, multiplication and division between the fractions.

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Proper and improper fractions

One way to classify fractions is by considering the size of their numerator and denominator, because these will determine the portion they really represent. If $a$ and $b$ are two integers such that $b neq 0$ , we have to

If $a < b$ , we will say that the fraction is $\frac{a}{b}$ is proper, that is, if the numerator is smaller than the denominator.
If $a \geq b$ , we will say that the fraction is $\frac{a}{b}$ is improper, that is, if the numerator is greater or equal than the denominator.

To clarify this idea, let’s see some examples.

Examples

Example 1

The fraction $\frac{1}{2}$ , is a proper fraction, because its numerator is less than its denominator.

Example 2

The fraction $\frac{7}{15}$ , is an proper fraction, because its numerator is smaller than its denominator.

Example 3

The fraction $\frac{4}{9}$ , is an proper fraction, because its numerator is smaller than its denominator.

Example 4

The fraction $\frac{6}{20}$ , is an proper fraction, because its numerator is smaller than its denominator.

Example 5

The fraction $\frac{5}{3}$ , is an improper fraction, because its numerator is greater than its denominator.

Example 6

The fraction $\frac{10}{4}$ , is an improper fraction, because its numerator is bigger than its denominator.

Example 7

The fraction $\frac{20}{12}$ , is an improper fraction, because its numerator is bigger than its denominator.

Example 8

The fraction $\frac{75}{44}$ , is an improper fraction, because its numerator is bigger than its denominator.

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Mixed fractions

When reading a recipe it is common to find measurements for ingredients such as one and a half cup of sugar or, that is why we can find containers with measurements of $\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{1}{4}$ or $\frac{1}{8}$ . This also occurs when buying foods that must be weighed, such as one kilograms and a quarter cheese or three and a half kilograms of meat.

Fractions are ideal to express this type of measurements, they are designed to measure portions, for example, to write a cup and a half you can write $1 + \frac{1}{2}$ which in turn is equal to $\frac{3}{2}$ . However, the way they are written may not present comfort or clarity in practice, that is why the mixed fractions (or mixed numbers) are defined, then, that instead of writing $1 + \frac{1}{2}$ , one writes

$1 \tfrac{1}{2}$

In this way, we define mixed fractions to separate the whole part from its non-integer part, the latter usually represented with a fraction of its own. Any mixed fraction can be rewritten as an improper fraction, because if $a$ , $b$ and $c$ are positive integers, then the following mixed fraction