### Fractions

A fraction is a number written in the form , where 3 is called the numerator and 5 the denominator. There are two types of fractions, common and improper fractions. A common fraction is one whose numerator is less than the denominator; hence, all common fractions are less than 1. An improper fraction is one whose numerator is greater than the denominator; hence, all improper fractions are greater than 1.

Lowest Common Multiple (L.C.M.)

The lowest common multiple (L.C.M.) of a group of numbers is the lowest number that can be divided by each number in the group, without leaving a remainder.

Example

What is the L.C.M. of the numbers 3, 5 and 8?

Addition

If the denominators of two fractions to be added are the same, then add the numerators and place it over the denominator.

Example

Add the fractions:

If the denominators are different, then the L.C.M, the lowest common multiple, has to be found. On determining the L.C.M, divide it by each of the denominators, multiply the quotient by the respective numerators and then add the amounts, placing the result over the L.C.M.

Example

Add the fractions:

Subtraction

If the denominators of two fractions to be subtracted are the same, then add the numerators and place it over the denominator.

Example

Subtract the fractions:

If the denominators are different, then the L.C.M, the lowest common multiple, has to be found. On determining the L.C.M, divide it by each of the denominators, multiply the quotient by the respective numerators and then subtract the amounts, placing the result over the L.C.M.

Example

Subtract the fractions:

Multiplication

When multiplying, it is best to firstly ‘cancel’ the fractions into their lowest form. Fractions are cancelled by dividing denominators by numerators, or by dividing numerators by denominators.

Example

Multiply the fractions:

Division

When dividing: invert (inter change the numerator and the denominator: ) the fraction after the division sign, change the division sign to a multiplication sign, cancel the fractions, then multiply.

Example

Simplify: