Fundamentals of Fractions

In Mathematics, fractions are defined as the parts of a whole. The whole can be an object or a group of objects. In real life, when we cut a piece of cake from the whole of it, then the portion is the fraction of the cake. A fraction is a word that is originated from Latin. In Latin, “Fractus” means “broken”.  In ancient times, the fraction was represented using words. Later, it was introduced in numerical form
The fraction is also termed as a portion or section of any quantity. It is denoted by using ‘/’ symbol, such as a/b. For example, in 2/4 is a fraction where the upper part denotes the numerator and the lower part is the denominator. 
Fraction : A fraction represents a numerical value, which defines the parts of a whole.
Generally, the fraction can be a portion of any quantity out of the whole thing and the whole can be any specific things or value
The basics of fractions explain the top and bottom numbers of a fraction. The top number represents the number of selected or shaded parts of a whole whereas the bottom number represents the total number of parts
Suppose a number has to be divided into four parts, then it is represented as x/4. So the fraction here, x/4, defines 1/4th of the number x. Hence, 1/4 is the fraction here.  It means one in four equal parts. It can be read as one-fourth or 1/4. This is known as fraction
Types of Fractions
Based on the properties of numerator and denominator, fractions are sub-divided into different types. They are
1.Proper Fractions : The proper fractions are those where the numerator is less than the denominator. For example, 8/9 will be a proper fraction since
“numerator < denominator”.
2. Improper Fraction: The improper fraction is a fraction where the numerator happens to be greater than the denominator. For example, 9/8 will be an improper fraction since “numerator > denominator”.
3.  Mixed Fraction: A mixed fraction is a combination of the integer part and a proper fraction. These are also called mixed numbers or mixed numerals. 
    For example  \(3\frac{2}{3}=\frac{\left[ (3\times 3)+2 \right]}{3}=\frac{11}{3}\)
4.Like fractions :- Fractions that have same denominators are called like fractions.
    Ex:- 1) \(4\over 5\) and \(2\over 3\) are called like fractions
2)\({2 \over 7},{3 \over 7},{11 \over 7}\) etc are called like fractions
5.Unlike fractions:- Fractions that have different denominators are called unlike fractions.
    Ex:- \(1 \over 4\) and \(2 \over 3\) are unlike fractions
6. Equivalent Fractions : Two fractions are equivalent to each other if after simplification either of two fractions is equal to the other one.
    For example, 2/3 and 4/6 are equivalent fractions.
    Since, 4/6 = (2×2)/(2×3) = 2/3
Unit Fraction : A fraction is known as a unit fraction when the numerator is equal to 1
One half of whole = ½
    One-third of whole = 1/3
    One-fourth of whole = ¼
    One-fifth of whole = 1/5
8.    Decimal fraction :- A fraction whose denominator is 10, 100, 1000,.....etc is called decimal fraction.
    Ex:-  \({3 \over 10},{9 \over 100}\)etc
9.    Vulgar fraction : A fraction whose denominator is a whole number other than      10,100,1000........etc are called vulgar fractions.
    Ex:- \({2 \over 3},{4 \over 5},{7 \over 8},{13 \over 9},....\) etc
10.    Complex fraction :- A fraction whose numerator and denominator are fractions those are called complex fractions.  
    Ex:- \(\frac{1/2}{3/4},\frac{7/8}{11/9},\frac{9/8}{13/14}......\) etc
Fraction on a Number Line
We have already learned to represent the integers, such as 0, 1, 2, -1, -2, on a number line. In the same way, we can represent fractions on a number line
For example, if we have to represent 1/5 and 3/5 parts of a whole, then it can be represented as shown in the below figure.

Since the denominator is equal to 5, thus 1 is divided into 5 equal parts, on the number line. Now the first section is 1/5 and the third section is 3/5.
Comparison of fraction by cross-multiplication method
    If two fractions \(a\over b\) and \(c \over d\)  are to be compared, we cross multiply
     i) If  \(a\times d>b\times c\), then \({a \over b}>{c \over d}\)          ii) If \(a\times d<b\times c\) , then \({a\over b}<{c\over d}\)
    iii) If \(a\times d=b\times c\) , then \({a\over b}={c\over d}\)
Ex:- Compare the \(2 \over3\)  and \(5\over 6\)
Solution:-On cross mutiplication  we get \(2\times6=12\) and \(3\times 5=15\) clearly, then 12 < 15
    \({2\over 3}<{5\over6}\) 
 Note :
    1) Every fraction is an irrational number, every irrational number may not be a fraction
    2) Fraction is always positive, where as rational number may be positive or negative.
Reducing a fraction: Reducing is what we do when we want to make a smaller version of a fraction that still has the same mathematical value as the original.
Ex: \({4 \over 8}={4\div4 \over 8\div 4}={1 \over 2}\)
    In this numerator and denominator both are divided by 4, we get 1 over 2
Comparison of fractions :
1)    Converting fractions to decimals
Ex: \(\frac{3}{8}or\frac{5}{12}\) which is greater?
    Sol: 
     \(\begin{align} & \frac{3}{8}=0.375(or)\frac{5}{12}=0.4166.. \\ & \therefore \frac{5}{12}>\frac{3}{8}\Rightarrow \frac{5}{12}\text{is greater}. \\ \end{align} \)
2)Making fractions as like fractions.
    Ex: i)\({2 \over 3}(or){4 \over 15}\)  which is smaller?
    Sol: Given \({2 \over 3} and {4 \over 15}\), Here LCM of 3 & 15 is 15.
   Make the denominators = LCM of denominators
    \(\begin{align} & \frac{2}{3}=\frac{2\times 5}{3\times 5}=\frac{10}{15}\text{and}\frac{4}{15} \\ & \therefore \frac{4}{15}<\frac{10}{15}\Rightarrow \frac{4}{15}<\frac{2}{3} \\ & \therefore \frac{4}{15}\text{is smaller} \\ \end{align}\)
\(\begin{align} & ii)\frac{4}{9}\text{}\frac{5}{9}\left( \because 4<5 \right)\text{denominator same} \\ & \text{iii)}\frac{3}{8}(or)\frac{5}{12}\text{which is greater?} \\ \end{align} \)  

Sol: 8 x 3 = 24, 12 x 2 = 24
         Make the denominator equal to LCM of denominators.
\(\begin{align} & \therefore \frac{3\times 3}{8\times 3}=\frac{9}{24}and\frac{5\times 2}{12\times 2}=\frac{10}{24} \\ & \therefore \frac{9}{24}<\frac{10}{24}\left( \because 9<10 \right) \\ & \text{So }\frac{5}{12}\text{ is greater} \\ \end{align} \)    
i.e., to compare fractions we convert unlike fractions into like fractions by making denominator equal to LCM of denominators.
How to arrange fractions in ascending and descending order:
1.    Check the denominators of all the given fractions. If all the denomiantors are equal then simply compare the numerators and arrange fractions according to it.
2.    If the denominators are not same then take the LCM of all the denominators and make all the denominators equal to LCM, i.e., we convert fractions into like fractions. Then check the numerators arrange them in ascending order (from smaller to bigger) or descending order (from bigger to smaller fractions).
Ex: Arrange the fractions \({2 \over 3},{1 \over 2},{5 \over 6}\) in ascending oder
    Sol: Given fractions \({2 \over 3},{1 \over 2},{5 \over 6}\) denominators are not equal.
        LCM of 3, 2, 6 = 6
        \(\begin{align} & \therefore \frac{2\times 2}{3\times 2},\frac{1\times 3}{2\times 3},\frac{5\times 1}{6\times 1}, \\ & \Rightarrow \frac{4}{6},\frac{3}{6},\frac{5}{6} \\ \end{align} \)
    Ascending oder is \(\frac{1}{2}<\frac{2}{3}<\frac{5}{6}\)
    or Ascending order is \(\frac{1}{2}<\frac{2}{3}<\frac{5}{6}\)