Fractions
Fractions  A Definition
A Fraction is a number which can represent a part of a whole
each part represents \(\frac{1}{4}\), i.e. 1 of 4 pieces
 A Fraction can also be represented
as a division where the 'top' value is known as the Numerator and the
'bottom' value as the Denominator
Fraction = Rational Number = \(\frac{\mbox{Integer}}{\mbox{Integer}}\) i.e. a rational number is defined a number which can be represented as an integer divided by an integer. As shown later on, whole numbers can be represented as fractions, e.g. 3 can be represented as 3/1  so all whole numbers are also rational numbers.
As an example of an irrational number, there is the wellknown constant p. This cannot be accurately represented as a fraction (the figure of 22/7 which is often quoted is only an approximation).
Fractions  Common Terminology

A proper fraction is one in which the numerator (top) is
less than the denominator (bottom)
Result is always less than 1
\[ \mbox{e.g. }\frac{1}{4},\;\; \frac{1}{2},\;\; \frac{3}{4} \]
 An improper fraction is one in
which the numerator (top) is greater than the denominator
(bottom)
Result is always greater than 1
\[ \mbox{e.g. } \;\;\frac{4}{3},\;\; \frac{7}{4},\;\; \frac{5}{2}\]
 A mixed number is a number which explicitly
consists of a whole number and a 'fractional' part
(bottom)
e.g.\(\;\; 5\frac{1}{4},\; 9\frac{1}{2},\; 7\frac{3}{4} \)
 A reciprocal fraction is the
'Inverse' of another.
\[\frac{3}{4} \;\;\mbox{is the reciprocal of} \;\; \frac{4}{3} \] \[ \frac{4}{1}\;\; \mbox{is the reciprocal of}\;\; \frac{1}{4} \]
 A decimal fraction is the decimal
form of a fraction
\[\frac{1}{4} = 0.25 \] \[ \frac{3}{4} = 0.75\]  An equivalent fraction is
one which represents the same number as another fraction but is
expressed in different terms
\[ \frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{16}{32} \] \[ \frac{3}{7} = \frac{6}{14} = \frac{21}{49} \]  A fraction in its lowest terms is a fraction
which has been 'reduced to its simplest form', leaving the fraction in its
lowest possible terms. In other words, there is no whole number with which you can
simultaneously divide both the numerator and denominator.
\[ \frac{3}{4}, \;\;\; \frac{3}{7} \;\;\; \frac{1}{3} \;\;\; \frac{21}{43} \]
NOTE Any fraction
in which the Numerator and Denominator are the same, is always equal to
one
\[ \frac{1}{1} = 1; \;\;\; \frac{2}{2} = 1; \;\;\; \frac{3}{3} = 1; \;\;\; \frac{500}{500} = 1\] 
And remember : multiplying anything by 1 gives the same
result
\[8 \times 1 = 8; \;\;\; 8 \times \frac{1}{1} = 8; \;\;\; 8 \times \frac{2}{2} = 8\] 
Physical Analogy
Thinking of a cake divided into equal portions could be an aid to understanding some of the principles involved.
In order to add or subtract fractions, the denominator needs to be the same. Once this has been achieved, you just add / subtract across the top  the denominator is not added / subtracted.
If you have a cake cut into three equal parts (thirds), then two of these parts will equal 2/3. Stated mathematically
The denominator does not enter into the addition, and hopefully by thinking of the cake, you can see why this should be so.
Improper fractions like
are larger than 1, and can also be expressed as a mixed fraction thus
Mathematically, you would achieve this conversion by dividing the numerator (i.e. the top) by the denominator (i.e. the bottom). In general, this will produce a whole number plus a remainder.
 The whole number becomes the number in front of the 'fractional part'
 The remainder is placed over the original denominator to produce the 'fractional part'
If you could visualize several cakes, each cut into five equal portions, then you could visualize that 17 of these portions could be provided by three full cakes plus two extra portions from another cake. This would be in line with the figure of
derived above.
Equivalent fractions are fractions like \(\frac{1}{2}\) and \(\frac{3}{6}\) which, although displayed differently, actually represent the same number.
Using our cake analogy again, if you imagine one cake cut up into two equal pieces, and another cut up into six equal pieces, then you can see that 1 piece of the former cake equals three of the latter, i.e. \(\frac{1}{2} = \frac{3}{6}\).
Equivalent Fractions and Lowest Terms
You can convert a given fraction into another equivalent fraction by either multiplying or dividing top AND bottom by the same whole number, i.e. you produce a fraction which has both a different numerator and denominator but which represents the same number as the original fraction.
There are several reasons why you might want to produce equivalent fractions.
 Comparison of fractions The magnitude of fractions can be easily compared when they have the same denominator.
 Adding and subtracting fractions If the fractions to be added or subtracted have different denominators, then the first step is always to transform the fractions so that they are over the same denominator
 Answers to problems are usually stated in their lowest terms. This involves carrying out a division top and bottom if the numerator and denominator have a common factor.
Hints : If you are tested
 by asking you to choose the largest fraction from
a selection. In order to compare fractions, you will first have to
transform them so that they all have the same denominator.
 by asking whether a given fraction is in its lowest terms. Here you will have to consider whether there is an integer that divides both the numerator and denominator  if so, then the fraction is not in its lowest terms.
For example
\[ \frac{4}{8} \] can be reduced to\[ \frac{1}{2} \] by dividing the numerator and denominator by 4. The fraction is now in its lowest terms, because there is no number that will divide both the numerator and denominator, apart from 1, and dividing by 1 will obviously not change the figures in the numerator or denominator.
Multiplication of Fractions

Examples
\[ \frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{ 2 \times 4} = \frac{3}{8} \;\;\mbox{which is already in its simplest form} \] 
\[ \frac{2}{5} \times \frac{2}{4} = \frac{2 \times 2} {5 \times 4} = \frac{4}{20} \;\;\mbox{and then simplify : } \frac{4}{20} = \frac{1}{5} \] 
In the
last example above, the answer is simplified by dividing the numerator and
denominator by 4. This maintains the ratio of the fraction (see Equivalent Fractions under
"FractionsTerminology"). Go to
here for more information on this technique. (link inactive at the moment)

This link gives some background information on common misconceptions in multiplying fractions.(link not active at the moment)
Multiplication of Fractions, using cancelling
When multiplying fractions in the previous section, you multiplied right across the top and right across the bottom, and then reduced the resulting fraction to its lowest terms, if that was necessary. We now introduce a 'shortcut' called cancelling  once you have mastered it, the figures involved will be smaller and the resulting calculation easier.
In the previous section, you reduced a fraction to its lowest terms by dividing a fraction top and bottom by the same whole number.
In a multiplication, you can divide the top of one fraction by a whole number and the bottom of an another fraction by the same whole number.By analogy with reducing a fraction to its lowest terms, make sure that you divide the top of a fraction and the bottom of another by the same whole number.
Example
Here I can 'cancel' the 16 and 12.
Both 16 and 12 can be divided by 4, so I
 divide the top of the lefthand fraction by 4
 and the bottom of the righthand fraction by 4
Multiplying right across the top, and right across the bottom now produces
which is already in its lowest terms.
So you can see the advantages of cancelling
 the numbers to be multiplied are smaller and easier to handle
 the answer is usually already in its lowest terms  it will be in its lowest terms if all the possible cancellations have been carried out
Example
Cancelling top left and bottom right by 7 produces
Cancelling bottom left and top right by 9 produces
The whole calculation was much easier than if you had tried it without cancellation.

Division of Fractions
Method


In the last example, above, the answer is simplified by dividing the numerator and denominator by 4. This maintains the ratio of the fraction.(see Equivalent Fractions under "FractionsTermiology").
You might find it interesting to visit this link which is concerned with misconceptions arising from dividing a whole number by a fraction. Remember that a whole number can be represented as a fraction by putting it over 1, e.g. 8 = 8/1.
And this link continues these ideas into more general situations.

Addition of Fractions
Method
If the denominators are equal, then you add the fractions straightforwardly

If the denominators are different, then you are unable to add straightawayYou have to follow these steps 

Examples
\[ \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1 \] 


Subtraction of Fractions
This procedure is directly analogous to addition of fractions.
Method
The steps are exactly the same as for Addition of Fractions, except for Step 3
where we subtract rather then add.

Examples



Percentage / Decimal to Fractional Form

asks you to convert a percentage to a fraction, and present the answer in its lowest terms.
The technique is
 Place the percentage over 100.
For example
If you are given 56%, you would produce
\[ \frac{56}{100} \]  and then reduce this fraction to its lowest terms, if possible, using the methods mentioned above.
 Place the percentage over 100.

asks you to convert a decimal to a fraction, and present the answer in its lowest terms.
The technique is
 Consider how many places of decimals you have.
 If x represents the number of decimal places, then
form a fraction with
 the numerator being the "decimal figures"
 the denominator being 10 to the power x.
For example
If you are given 0.67, you have two places of decimals, and would therefore produce
\[ \frac{67}{100} \] Other examples
0.543 becomes \(\frac{543}{1000}\) 0.4 becomes \(\frac{4}{10}\) 0.45 becomes \(\frac{45}{100}\) 0.786 becomes \(\frac{786}{1000}\) 0.985678 becomes \(\frac{985678}{1000\ 000}\)
 and then reduce this fraction to its lowest terms, if possible, using the methods mentioned above.
Links to Other Sites
 All About Fractions : links to about two dozen seperate areas.
 Fractions from WebMath (Discovery Channel)
 Fractions from George Mason University
 Fractions from learn.co.uk (from the Guardian newspaper)
 Fraction Help and Tutorials from about.com
 'Football' game using Fractions
Questions from Past Papers
The table below shows the heights of 40 children measured to the nearest 0.1 cm.
1. How many children are less than 160 cm. tall ?
 A 8
 B 18
 C 30
 D 40
2. 20% of the children are ?
 A less than 150 cm tall
 B between 150 cm and 170 cm tall
 C at least 170 cm tall
 D at least 180 cm tall
3. What fraction of the children are over 160 cm tall ?
 \[ A \ \ \frac{3}{10} \]
 \[ B \ \ \frac{9}{20} \]
 \[ C \ \ \frac{1}{2} \]
 \[ D \ \ \frac{11}{20} \]