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fractions [2006/09/08 19:48]
fractions [2009/07/14 14:34] (current)
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 +====== Fractions - The Basics ======
 +
 +===== What is a fraction? =====
 +
 +The best way to consider a fraction is as part of a pie...
 +
 +<html><center></html>
 +{{pie.gif|}}
 +<html></center></html>
 +
 +...where the pie represents the number 1.
 +
 +<html><center></html>{{pie_above.gif|}} = 1
 +<html></center></html>
 +
 +
 +For example, half of the pie would be represented in maths as...
 +
 +<html><center></html>
 +<math>\frac{1}{2}</math>
 +<html></center></html>
 +
 +i.e.
 +
 +<html><center></html>{{pie_above_half.gif|}}<html></center></html>
 +
 +<math>\small \frac{1}{2}</math> literally means one divided by two.
 +
 +The two numbers contained in the fraction each have a name: the number on the top is called the **numerator** and the number on the bottom is called the **denominator**.
 +
 +It is important to note here that not all numbers can be represented as a fraction - a fact that is not immediately obvious and indeed this is an instance where the great Pythagoras got his maths wrong! Numbers that can be represented by fractions are called **rational numbers** and those that can't are **irrational numbers**. For example, pi (<math>\small \pi</math> or 3.1415927...) can't be represented as a fraction. Neither can <math>\small \sqrt{2}</math>. Both of these numbers are irrational.
 +
 +===== Whole Numbers As Fractions =====
 +
 +Whole numbers can be represented as a fraction because, for example, two divided by one is just 2. So the number '2' as a fraction is written as
 +
 +<html><center></html>
 +<math>\frac{2}{1}</math>
 +<html></center></html>
 +
 +To write a whole number as a fraction simply write that whole number over 1.
 +
 +<html><center></html>
 +<math>n \equiv \frac{n}{1}</math>
 +<html></center></html>
 +
 +...putting it mathematically :-)
 +
 +===== Equivalent Fractions =====
 +
 +Below is a table showing how different fractions fit into the number 1. It is worthwhile seeing if you can understand this table before you read on...
 +
 +<html><center></html>
 +{{fractions_table.gif|}}
 +<html></center></html>
 +
 +See that 2 lots of quarters <math>\small \left(\frac{1}{4}+\frac{1}{4}\right)</math> represents the same amount as one half <math>\small \left(\frac{1}{2}\right)</math>.
 +
 +In other words...
 +<html><center></html>
 +<math>\frac{1}{2} = \frac{2}{4}</math>
 +<html></center></html>
 +
 +One divided by two leaves me with a half of one. Two divided by four leaves me with a half of two. The amount represented by both fractions is one half. Technically speaking, the ratio of the numerator and demoninator in both fractions is the same. Although the numerators and denominators are different in both fractions both of these fractions represent the same number. They are equivalent. See if you can spot other equivalent fractions in the table above.
 +
 +This idea that we can convert a fraction into an equivalent fraction - same amount but different numbers top and bottom - becomes very important when we start working with them, as you will now see.
 +
 +===== Adding Fractions =====
 +
 +If we are adding fractions that have the same denominator then this is easy...
 +<html><center></html>
 +<math>\frac{3}{14} + \frac{5}{14} = ?</math>
 +<html></center></html>
 +
 +Because we have 3 lots of 14<html><sup></html>th<html></sup></html>s and 5 lots of 14<html><sup></html>th<html></sup></html>s then the sum of these two fractions will be 8 (5 + 3) lots of 14<html><sup></html>th<html></sup></html>s, or <math>\small \frac{8}{14}</math>. The sum is easy because the denominators are the same. Maths with fractions becomes complicated when the denominators are different, for example:-
 +
 +<html><center></html>
 +<math>\frac{1}{4} + \frac{2}{5} = ?</math>
 +<html></center></html>
 +
 +The trick to adding these two fractions together is to convert one or both of the fractions into equivalent fractions so that the denominators become the same number. In other words, if we find a **common denominator** then they will be easy to add together!
 +
 +<math>\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16} = \frac{5}{20} = \frac{6}{24}</math>...
 +
 +<math>\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \frac{8}{20} = \frac{10}{25} = \frac{12}{30}</math>...
 +
 +Can you spot the two fractions from each list that have the same denominators? That's right: - <math>\small \frac{5}{20}</math> and <math>\small \frac{8}{20}</math>.
 +
 +So our question can be rewritten as
 +
 +<html><center></html>
 +<math>\frac{5}{20} + \frac{8}{20} = \frac{13}{20}</math>
 +<html></center></html>
 +
 +===== Subtracting Fractions =====
 +
 +The trick is the same when subtracting fractions: we need to find a common denominator. For example...
 +
 +<html><center></html>
 +<math>\frac{7}{9} - \frac{1}{4} = ?</math>
 +<html></center></html>
 +
 +Convert <math>\small \frac{7}{9}</math> and <math>\small \frac{1}{4}</math> into equivalent fractions where the denominators are both the same...
 +
 +<math>\frac{7}{9} = \frac{14}{18} = \frac{21}{27} = \frac{28}{36} = \frac{35}{45} = \frac{42}{54}</math>...
 +
 +<math>\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16} = \frac{5}{20} = \frac{6}{24} = \frac{7}{28} = \frac{8}{32} = \frac{9}{36} = \frac{10}{40}</math>...
 +
 +If we choose equivalent fractions from the lists above with the same denominators then our question becomes...
 +
 +
 +<html><center></html>
 +<math>\frac{28}{36} - \frac{9}{36} = \frac{19}{36}</math>
 +<html></center></html>
 +
 +===== Multiplying Fractions =====
 +
 +This is much easier! To find the answer to
 +
 +<html><center></html>
 +<math>\frac{2}{5} \times \frac{3}{7}</math>
 +<html></center></html>
 +
 +we need to find the new numerator by simply multiplying the two numerators together. Do the same with the denominators to find the new denominator.
 +
 +<html><center></html>
 +<math>\frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35}</math>
 +<html></center></html>
 +
 +===== Dividing Fractions By Fractions =====
 +
 +Again, this is pretty straightforward. To divide one fraction by another you just have to remember...
 +
 +//**The number you are dividing by turn upside down and multiply**//
 +
 +For example,
 +
 +<html><center></html>
 +<math>\frac{2}{5} \div \frac{3}{7} = ?</math>
 +<html></center></html>
 +
 +The number we are dividing by is <math>\small \frac{3}{7}</math> so let's turn this upside down to give <math>\small \frac{7}{3}</math> and then multiply...
 +
 +<html><center></html>
 +<math>\frac{2}{5} \times \frac{7}{3} = \frac{2 \times 7}{5 \times 3} = \frac{14}{15}</math>
 +<html></center></html>
 +
 +




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