Mathematics: Fractions

A fraction is a number that is expressed as one integer written above another integer, with a dividing line between them . It represents the quotient of the two numbers 'x divided by y.' 
It can also be thought of as x out of y equal parts.
The top number of a fraction is called the numerator, and it represents the number of parts under consideration. The 1 in  means that 1 part out of the whole is being considered in the calculation. The bottom number of a fraction is called the denominator, and it represents the total number of equal parts. The 4 in  means that the whole consists of 4 equal parts. A fraction cannot have a denominator of zero; this is referred to as 'undefined.'
Fractions can be manipulated, without changing the value of the fraction, by multiplying or dividing (but not adding or subtracting) both the numerator and denominator by the same number. If you divide both numbers by a common factor, you are reducing or simplifying the fraction. Two fractions that have the same value but are expressed differently are known as equivalent fractions.

For example,  are all equivalent fractions. They can also all be reduced or simplified to .
When two fractions are manipulated so that they have the same denominator, this is known as finding a common denominator. The number chosen to be that common denominator should be the least common multiple of the two original denominators.

Example:
 the least common multiple of 4 and 6 is 12. Manipulating to achieve the common denominator: .

Proper Fractions and Mixed Numbers
A fraction whose denominator is greater than its numerator is known as a proper fraction, while a fraction whose numerator is greater than its denominator is known as an improper fraction. Proper fractions have values less than one and improper fractions have values greater than one.
A mixed number is a number that contains both an integer and a fraction. Any improper fraction can be rewritten as a mixed number.

Example:
.

Similarly, any mixed number can be rewritten as an improper fraction. Example:
.

Adding and Subtracting Fractions
If two fractions have a common denominator, they can be added or subtracted simply by adding or subtracting the two numerators and retaining the same denominator. If the two fractions do not already have the same denominator, one or both of them must be manipulated to achieve a common denominator before they can be added or subtracted. Example:
.

Multiplying Fractions
Two fractions can be multiplied by multiplying the two numerators to find the new numerator and the two denominators to find the new denominator. Example:
.

Dividing Fractions
Two fractions can be divided by flipping the numerator and denominator of the second fraction and then proceeding as though it were a multiplication. Example:
.

Multiplying a Mixed Number by a Whole Number or a Decimal
When multiplying a mixed number by something, it is usually best to convert it to an improper fraction first. Additionally, if the multiplicand is a decimal, it is most often simplest to convert it to a fraction. For instance, to multiply

by 3.5, begin by rewriting each quantity as a whole number plus a proper fraction. Remember, a mixed number is a fraction added to a whole number and a decimal is a representation of the sum of fractions, specifically tenths, hundredths, thousandths, and so on:

Next, the quantities being added need to be expressed with the same denominator. This is achieved by multiplying and dividing the whole number by the denominator of the fraction.
Recall that a whole number is equivalent to that number divided by 1:

When multiplying fractions, remember to multiply the numerators and denominators separately:



Now that the fractions have the same denominators, they can be added:

Finally, perform the last multiplication and then simplify: