Fraction numbers - Addition, Subtraction, Multiplication and Division

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What is a Fraction?

In mathematics, a fraction is a number of the form a/b, where b ≠ 0. It represents some part of a whole.
The part ‘a’ above the dividing line is called the numerator of a fraction, and the part below ‘b’ is called a denominator.
In most cases, the numerator and denominator are both integers. The denominator denotes the total number of parts which comprise the whole, and the numerator denotes the number of parts of the total.

Consider the example below of a pizza cut into 4 slices.

For example, the fraction 3/5 indicates that there are 5 parts in total which make up the whole, and 3 is the number of parts out of the 5. Here, 3 is the numerator and 5 is the denominator.

Fraction examples -

When all four slices of the pizza are present, the fraction is written as 4/4. The 4 in the denominator indicates that 4 slices make up the whole pizza.
When one slice of the pizza has been eaten, 3 slices remain which is indicated by the fraction ¾. The numerator 3 indicates the number of parts of the total 4 slices.
When 2 slices are eaten, 2 slices remain, indicated by 2/4. Finally, when only one slice remains, it is represented by ¼.

Properties of fractions

Given below are some important properties of fractions.

Numerator – The number which is above the dividing line of a fraction.

Denominator – The number which is below the dividing line of a fraction.

Commutative property – The addition and multiplication of fractions is commutative. This implies that, the order in which these fractions are written does not matter: the final answer is the same. $$ {a\over b}+{c\over d} = {c\over d} + {a\over b} \; and \; {a\over b}*{c\over d} = {c\over d} * {a\over b} $$

Associative property – The grouping of fractions for addition and multiplication does not affect the final answer. Hence, fraction addition and multiplication are associative. $$ ({a\over b}+{c\over d})+{e\over f} = {a\over b} + ({c\over d} + {e\over f}) \; and \; ({a\over b} * {c\over d}) * {e\over f} = {a\over b} * ({c\over d} * {e\over f}) $$

Distributive property – The multiplication of a fraction over the addition of two or more fractions gets distributed. This means the following. $$ {a\over b}*({c\over d} + {e\over f}) = ({a\over b}*{c\over d}) + ({a\over b}*{e\over f}) $$

Identity property – The additive identity of any fraction is 0. This means that, when 0 is added to any fraction, we get back the fraction itself. The multiplicative identity of a fraction is 1. Therefore, 1 multiplied by any fraction gives back the original fraction. $$ {a\over b}+0 = 0+{a\over b} = {a\over b} \; and \; {a\over b} *1 \; = \; 1*{a\over b} = {a\over b} $$

Inverse property – The inverse of any fraction is its reciprocal. The reciprocal of a fraction is another fraction where the positions of the numerator and denominator are interchanged. When a fraction is multiplied by its reciprocal, the result is 1. $$ {a\over b}*{b\over a} = 1 $$

The fraction b/a is known as the reciprocal (or multiplicative inverse) of the fraction a/b.

Fraction formulas

The formulas for different arithmetic operations on fractions are as follows.

Formula Example
Addition: For fraction addition, a least common denominator needs to be found so that all the fractions being added have the same denominator.$$ {a\over b} + {c\over d} = {{a*d}\over{b*d}}+{{c*b}\over{d*b}}={{ad+bc}\over bd} $$ For example, $$ {2\over5}+{3\over4} = {{2*4}\over{5*4}} + {{3*5}\over{4*5}} = {{8+15}\over 20} = {23\over 20} $$
Subtraction: For subtraction also, a least common denominator needs to be found.$$ {a\over b} - {c\over d} = {{a*d}\over{b*d}}-{{c*b}\over{d*b}}={{ad-bc}\over bd} $$ For example, $$ {3\over4}-{2\over5} = {{3*5}\over{4*5}} - {{2*4}\over{5*4}} = {{15-8}\over 20} = {7\over 20} $$
Multiplication: Unlike addition or subtraction, for multiplying fractions, we do not need to find the least common denominator.$$ {a\over b} * {c\over d} = {{a*c}\over{b*d}} $$ For example, $$ {3\over4}*{2\over5} = {6\over20} = {3\over10} $$
Division: When dividing one fraction by the other, the first fraction is multiplied by the reciprocal of the second fraction.$$ {{a\over b} \over {c\over d}} = {a\over b}*{d\over c} = {{a*d}\over {b*c}} $$ For example, $$ {{3\over4}\over{2\over5}} = {3\over4}*{5\over2} = {{3*5}\over{4*2}} = {15\over8} $$

Steps for arithmetic operations on fractions

The steps for different arithmetic operations on fractions are as follows:

1) Addition and subtraction:

    a) For adding or subtracting two or more fractions, we first need to find the least common denominator of all the fractions.

    b) For each fraction, multiply the denominator with the number so that the least common denominator is obtained. Multiply the numerator also with the same number.

    c) Repeat this step for all the fractions.

    d) For addition, add the numbers in the numerator, and for subtraction, subtract them.

    e) Reduce the fraction to its lowest term.

2) Multiplication:

    a) Unlike addition and subtraction, we do need to find the least common denominator for multiplication and division of fractions. In this case, first multiply all the numerators together, and the denominators together.

    b) Finally, reduce the fraction to its lowest term.

3) Division:

    a) For division, we first need to know which fraction is being divided by the other.

    b) Keep the first fraction as it is.

    c) Replace the second fraction with its reciprocal.

    d) Multiply the two fractions, and reduce the result to its lowest term.

Areas of application

Some of the real-world applications of fractions are listed below.

1) Dividing a pizza or pie into equal slices involves cutting it up into a number of fractions which make up the whole pizza (or pie).

2) In examinations, grades are assigned according to the marks obtained out of the total marks for the exam. For example, a score of 14/20 indicates that the person scored 14 marks out of the total 20.

3) When we go shopping and find something on discount, there is a deduction on the original price which is calculated using fractions. For example, if a dress worth INR 1000 is available on 50% discount, the new price of the dress is 1000 – 50% of 1000 = INR 500.

4) In recipes, exact measurements of the ingredients are indicated using fractions. For example, ½ a teaspoon of sugar or 1 ½ cups of flour.

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