Types of Fraction: Definition, Types and Examples

The fractions are elements of mathematics representing the ratio of two numbers. Proper fraction, improper fraction, and mixed fraction are three basic types of fraction. This is precisely why the fraction is completely associated with the operation of division, in fact, a fraction can be said to be a division or a quotient between two numbers

The writing of the fractions follows the following pattern: there are two written numbers, one above the other and separated by a hyphen, or separated by a diagonal line, similar to the one written when representing a percentage (%). The number that is above is known as the numerator,  the one below is the denominator; the latter is the one that acts as a divisor.

For example, the fraction 5/8 represents 5 divided by 8, so it equals 0.625. If the numerator is greater than the denominator it means that the fraction is greater than unity, so it can be re-expressed as an integer value plus a fraction less than 1 (for example, 50/12 is equal to 48/12 plus 2 / 12, i.e. 4 + 2/12).

In this sense it is easy to see that the same number can be re-expressed by an infinite number of fractions; in the same way that 5/8 will be equal to 10/16, 15/24, and 5000/8000, always equivalent to 0.625. These fractions are called equivalents and always maintain a directly proportional relationship.

In everyday life, fractions are usually expressed in the smallest number possible, for this we look for the smallest integer denominator that makes the numerator also an integer. In the example of the previous fractions, there is no way to reduce it further, since there is no integer less than 8 that is at the same time divisor of 5.

Fractions and Mathematical Operations

Regarding basic mathematical operations between fractions, it should be noted that for addition and subtraction, the denominators must coincide and, therefore, the least common multiple must be sought by means of equivalence (for example, 4/9 + 11/6 is 123/54, since 4/9 is 24/54 and 11/6 is 99/54).

For multiplications and divisions, the process is somewhat simpler: in the first case, multiplication between numerators is used over multiplication between denominators; in the second, a ‘cross’ multiplication is performed.

Fractions in everyday life

It should be said that fractions are one of the elements of mathematics that appear most frequently in everyday life. A huge number of products are sold expressed as fractions, be it kilo, liter, or even arbitrary and historically established units for certain items, such as eggs or invoices, that go by the dozen.

Thus we have ‘half a dozen’, ‘a quarter of a kilo’, ‘five percent discount’, ‘three percent interest, etc., but all of them involve understanding the idea of ​​a fraction.

Examples of fractions

1. 4/5
2. 21/13
3. 61/2
4. 1/3
5. 40/13
6. 44/9
7. 31/22
8. 177/17
9. 30/88
10. 51/2
11. 505/2
12. 140/11
13. 1/10 ⁸
14. 6/7
15. 1/7
16. 33/9
17. 7/29
18. 101/100
19. 49/7
20. 69/21

How Many Types of fraction?

Here are the Three Types of fractions..

• Improper Fraction
• Proper Fraction
• Mixed Fraction

Improper Fraction

Considering fractions as proportional relationships between two numbers, a distinction is made between those that exceed unity, called improper fractions, and those that do not, which are their own.

In improper fractions the numerator (the number that is above in the fraction) is always greater than its denominator (the one that is below), so it can also be expressed as the combination between an integer and another fractional number and less than 1.

There is talk of ‘combination’ because in writing they appear like this: the whole number and to its right the fractional number. Although a ‘+’ sign should be formally written between the two, this is usually not done.

Those numbers made up of an integer and a fraction are called mixed numbers, and they are often seen on posters of businesses that sell products by weight.

For example, in an ice cream parlor, hardly anyone chooses to order 5/2 of a kilo of ice cream (much less in a higher ratio, such as 25/10), but they will surely request 2 ½, that is, “two and a half kilos” of frozen.

The exercise of transforming an improper fraction into a mixed number is simple: you have to decompose the numerator so that it is divisible by the denominator, resulting in an integer (in the example, 4/2 = 2), the remaining fraction ( in this case ½) will be the fraction.

For the purposes of mathematical analysis, it is useless to express an improper fraction as the number of units it has and the smallest quotient of one, because what matters is each number separately: the operations between fractions, as well as those that combine fractions and integers, are much simpler as you work with improper fractions.

Although the operations between proper and improper fractions are carried out in the same way, there are certain differential characteristics in both cases, such as the fact that multiplication between improper fractions results in an improper fraction.

While the division between improper fractions depends precisely on which number is located as a dividend (numerator) and which as a divisor (denominator): if the first is greater than the second, then it will be an improper fraction, while if the second is the largest will be a fraction of its own.

A particular case of improper fractions is those that result in a division in which there is no remainder, that is, one in which the numerator is a multiple of the denominator and then it is an integer: these are known as apparent fractions.

Examples of improper fractions

Here are some examples of improper fractions:

• 4/3
• 11/21
• 50/18
• 100/17
• 10/9
• 23/8
• 4/3
• 9/21
• 72/33
• 41/8
• 11/10
• 3/2
• 7/17
• 6/5
• 41/5
• 100/99
• 414/200
• 121/100
• 77/10
• 32/9

Mixed Fraction

A mixed fraction is the combination of a whole number and a fraction. Every fraction is made up of two numbers, written one above the other separated by a line:

• The numerator ( above ): is the number of parts taken from the unit. Eg if a person takes two servings of that cake, he takes 2/5. That is, the numerator is 2.
• The denominator ( below ): is the number of parts that make up the entire unit. Eg if a cake is divided into five portions, the denominator is 5.

When the numerator is greater than the denominator it means that there is more than one complete unit. In these cases, the quantity can be expressed through an improper fraction (a fraction with a numerator greater than the denominator) or through a mixed fraction. A proper fraction can never be expressed as a mixed fraction.

To convert improper fractions to mixed fractions :

• Divide the numerator by the denominator.
• Write the quotient as a whole number
• The rest is the new numerator of the fraction (with the same denominator).

To convert mixed fractions to improper :

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• The result of the sum is the new numerator of the fraction (with the same denominator).

See also: Examples of Own Fractions

Examples of mixed fractions

• 3 2/5 (three integers and two fifths)
• 1 2/3 (One whole and 2 thirds)
• 45 74/100 (forty-five integers and seventy-four hundredths)
• 62 3/8 (sixty-two integers and three-eighths)
• 2 5/6 = (Two integers and five-sixths).
• 5 4/7 = (Five integers and four sevenths).
• 8 3/10 = (Eight integers and three tenths).
• 11 2/6 = (Eleven fifths and two sixths).
• 7 4/10 = (Seven integers and four-tenths).
• 261 10/14 = (Two hundred sixty-one integers and ten fourteen).
• 8 7/16 = (Eight integers and seven sixteenths).
• 16 3/16 = (Sixteen integers and 3 sixteenths).
• 6 5/6 = (six integers and five sixths).
• 5 2/7 = (Five integers and two sevenths).
• 4 2/10 = (four integers and 2 tenths).

Proper Fraction

The fraction in which numerator is less than the Denominator or The fraction in which the degree of the numerator is less than the degree of the denominator is called Proper fraction.

Examples of Proper Fraction

• 1/2
• 3/4
• 5/9