Definition of Proper and Improper Fractions

Proper fractions comprise a positive property numerator and denominator, where the numerator is less than the denominator, and always with a value less than 1, whose symbolic language is expresses:
The fraction \(\frac{a}{b}\), with 0 < a < b, is proper and its values ​​are less than 1.

On the other hand, in the improper fraction, the numerator and denominator are positive, to which the numerator is greater or equal to the denominator, and with a value that can be greater than or equal to 1, whose symbolic language is establishes:
The fraction \(\frac{a}{b}\), with 0 < a \(\le\) b, is improper and with values ​​greater than or equal to 1.

Mathematical and conceptual principles of the fraction

The fraction of the object arises from dividing and taking it in equal parts, which constitutes the intuitive idea of ​​the concept of fraction, not However, the formal definition states that: a number is a fraction if it is obtained by dividing an integer \(a\) by an integer \(b\ne 0\), which is write as:

\(\frac{a}{b},~{}^{a}\!\!\diagup\!\!{}_{b}\;,~a\div b\)

The above is one of the numerical representations of a fraction.

The interpretation of the fraction \(\frac{a}{b},~b\ne 0,\) is that an object has been divided into \(b\) equal parts and \(a\) is taken from them.

For example, the fraction \(\frac{3}{8}\) means that an object has been divided into 8 equal parts and 3 of them are taken.

Essentially, a fraction is governed by two elements: numerator (indicates the number of equal parts that have been taken) and denominator (number into which the object has been divided and must always be different from zero). Thus in the fraction \(\frac{4}{7}\) the numerator is 4 and the denominator is seven and the fraction is read as four sevenths or 4 divided by 7.

In general, the fraction is of the form:

\(\frac{\text{numerator}}{\text{denominator}}\)

Different representations of a fraction

geometric representation

The Rectangle has been divided into 12 equal parts; the blue area represents \(\frac{5}{12}~\) and the yellow area represents \(\frac{7}{12}.\)

In the circle, it represents that \(\frac{1}{3}~\)(one third) will be extracted and \(\frac{2}{3}\) will remain.

verbal representation

We have already used verbal language to express a fraction as five sixths to refer to \(\frac{5}{6};~\)but it is common for various media to present us with information about the following way:

In the world, approximately 9 out of 10 people, over the age of 15, know how to read and write, which is numerically interpreted as \(\frac{9}{10}\).

Another example is

"In Mexico, 13 out of 24 people are female, while worldwide, 381 out of 770 people are of the female gender” numerically the above means \(\frac{13}{24}~~\)y \(\frac{381}{770}\), respectively.

Representation with percentages

Businesses usually offer discounts and express it in percentages to tell you how much less you are going to pay for every $100 you buy for For example, a discount of 30% indicates that for every $100 they will discount $30 and an alternative way of expressing 30% is with the fraction \(\frac{30}{100}.\)

Many economic variables are expressed in percentage such as interest rate, inflation, GDP increase (Gross Domestic Product) for example, if a bank offers you a 5% interest rate when investing with they; what it is promising you is that for every $100 they will give you $5, so \(5%~\) is also represented by \(\frac{5}{100}\).

decimal representation

The number \(0.4\) is read as 4 tenths; which is represented with \(\frac{4}{10},\) that is:

\(0.4=\frac{4}{10}\)

The number \(0.625\) is interpreted as \(625\) thousandths, and we can guarantee the following equality:

\(0.625=\frac{625}{1000}\)

To find the decimal representation of a fraction, it is necessary to perform the division manually or with a calculator. Here are some examples

\(\frac{5}{8}=0.625\)

\(\frac{8}{5}=1.6\)

\(\frac{2}{3}=0.\bar{6}\)

\(\frac{1}{7}=0.\overline{142857}\)

proper fractions

Next, we will show several examples of proper fractions in their different representations.

\(\frac{1}{8},~\frac{4}{5},~\frac{13}{16},\frac{17}{24}\) are proper fractions.

The illuminated part of the previous figures are proper fractions and both represent \(\frac{3}{4}\).

The numbers \(0.5,~0.375,\text{ }\!\!~\!\!\text{ y}~0.1\bar{6}\) are the decimal representation of the proper fractions \(\frac{1}{2},\frac{3}{8}~\text{y }\!\!~\!\!\text{ }\frac{1}{6},\ ) respectively.

The percentages 30%, 25%, and 50% can be represented by the fractions \(\frac{3}{10},\frac{1}{4},~\text{y}~\frac{1}{2 }\)

improper fractions

Next, we will show several examples of improper fractions in their different representations.

\(\frac{5}{4},\frac{19}{7},\frac{11}{9}~\) are improper fractions.

The illuminated part of the previous figures represents the same improper fraction, namely, \(\frac{6}{4}.\)

The numbers \(1.5,~3.375,\text{ }\!\!~\!\!\text{ y}~6.1\bar{6}\) are the decimal representation of the proper fractions \(\frac{3}{2},\frac{27}{8}~\text{y }\!\!~\!\!\text{ }\frac{37}{6},\ ) respectively.

The percentages 130%, 105%, and 150% can be represented by the fractions \(\frac{130}{100},\frac{105}{100},~\text{y}~\frac{150}{100 }\)