# Differences

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articles:fractions [2009/07/14 14:38] 127.0.0.1 external edit |
articles:fractions [2009/11/19 22:48] (current) daddydoos |
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- | ====== Fractions - The Basics ====== | + | ====== Fractions - The Basics ====== |

- | + | | |

- | ===== What is a fraction? ===== | + | |

- | + | ||

- | The best way to consider a fraction is as part of a pie... | + | ===== What is a fraction? ===== |

- | + | | |

- | {{ :pie.gif |Picture of a pie}}
| + | |

- | + | ||

- | ...where the pie represents the number 1. | + | The best way to consider a fraction is as part of a pie... |

- | + | | |

- | {{ :pie_above.gif |Picture of a pie - looking down from above}} = 1 | + | |

- | + | ||

- | + | {{ :pie.gif |Picture of a pie}} | |

- | For example, half of the pie would be represented in maths as... | + | |

- | + | ||

- | <math>\frac{1}{2}</math>
| + | |

- | + | ...where the pie represents the number 1. | |

- | i.e. | + | |

- | + | ||

- | {{ :pie_above_half.gif |Picture of exactly half of a pie}} | + | |

- | + | {{ :pie_above.gif |Picture of a pie - looking down from above}} = 1 | |

- | <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.
| + | |

- | + | For example, half of the pie would be represented in maths as... | |

- | ===== 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 | + | |

- | + | <latex>\frac{1}{2}</latex> | |

- | <math>\frac{2}{1}</math>
| + | |

- | + | ||

- | To write a whole number as a fraction simply write that whole number over 1. | + | |

- | + | i.e. | |

- | <math>n \equiv \frac{n}{1}</math>
| + | |

- | + | ||

- | ...putting it mathematically :-) | + | |

- | + | {{ :pie_above_half.gif |Picture of exactly half of a pie}} | |

- | ===== 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... | + | |

- | + | <latex>\small \frac{1}{2}</latex> literally means one divided by two. | |

- | {{ :fractions_table.gif |A fractions table}} | + | |

- | + | ||

- | 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>.
| + | |

- | + | 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**. | |

- | In other words... | + | |

- | <math>\frac{1}{2} = \frac{2}{4}</math>
| + | |

- | + | ||

- | 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 amount. They are equivalent. See if you can spot other equivalent fractions in the table above. | + | 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 (<latex>\small \pi</latex> or 3.1415927...) can't be represented as a fraction. Neither can <latex>\small \sqrt{2}</latex>. Both of these numbers are irrational. |

- | + | | |

- | 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 ===== | + | ===== Whole Numbers As Fractions ===== |

- | + | | |

- | If we are adding fractions that have the same denominator then this is easy... | + | |

- | <math>\frac{3}{14} + \frac{5}{14} = ?</math>
| + | |

- | + | 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 | |

- | Because we have 3 lots of 14<sup>th</sup>s and 5 lots of 14<sup>th</sup>s then the sum of these two fractions will be 8 (5 + 3) lots of 14<sup>th</sup>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:-
| + | |

- | + | ||

- | <math>\frac{1}{4} + \frac{2}{5} = ?</math>
| + | |

- | + | <latex>\frac{2}{1}</latex> | |

- | 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>...
| + | |

- | + | To write a whole number as a fraction simply write that whole number over 1. | |

- | <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>.
| + | |

- | + | <latex>n \equiv \frac{n}{1}</latex> | |

- | So our question can be rewritten as
| + | |

- | + | ||

- | <math>\frac{5}{20} + \frac{8}{20} = \frac{13}{20}</math>
| + | |

- | + | ...putting it mathematically :-) | |

- | ===== Subtracting Fractions ===== | + | |

- | + | ||

- | The trick is the same when subtracting fractions: we need to find a common denominator. For example... | + | |

- | + | ===== Equivalent Fractions ===== | |

- | <math>\frac{7}{9} - \frac{1}{4} = ?</math>
| + | |

- | + | ||

- | Convert <math>\small \frac{7}{9}</math> and <math>\small \frac{1}{4}</math> into equivalent fractions where the denominators are both the same...
| + | |

- | + | 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... | |

- | <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>...
| + | |

- | + | {{ :fractions_table.gif |A fractions table}} | |

- | If we choose equivalent fractions from the lists above with the same denominators then our question becomes... | + | |

- | + | ||

- | <math>\frac{28}{36} - \frac{9}{36} = \frac{19}{36}</math>
| + | |

- | + | See that 2 lots of quarters <latex>\small \left(\frac{1}{4}+\frac{1}{4}\right)</latex> represents the same amount as one half <latex>\small \left(\frac{1}{2}\right)</latex>. | |

- | ===== Multiplying Fractions ===== | + | |

- | + | ||

- | This is much easier! To find the answer to | + | |

- | + | In other words... | |

- | <math>\frac{2}{5} \times \frac{3}{7}</math>
| + | |

- | + | <latex>\frac{1}{2} = \frac{2}{4}</latex> | |

- | 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. | + | |

- | + | ||

- | <math>\frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35}</math>
| + | |

- | + | 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 amount. They are equivalent. See if you can spot other equivalent fractions in the table above. | |

- | ===== Dividing Fractions By Fractions ===== | + | |

- | + | ||

- | Again, this is pretty straightforward. To divide one fraction by another you just have to remember... | + | |

- | + | 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. | |

- | //**The number you are dividing by turn upside down and multiply**// | + | |

- | + | ||

- | For example, | + | |

- | + | ===== Adding Fractions ===== | |

- | <math>\frac{2}{5} \div \frac{3}{7} = ?</math>
| + | |

- | + | ||

- | 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...
| + | |

- | + | If we are adding fractions that have the same denominator then this is easy... | |

- | <math>\frac{2}{5} \times \frac{7}{3} = \frac{2 \times 7}{5 \times 3} = \frac{14}{15}</math>
| + | |

- | + | <latex>\frac{3}{14} + \frac{5}{14} = ?</latex> | |

- | For further fractions help [[[[articles:more_fractions|click here!]] | + | |

+ | |||

+ | |||

+ | Because we have 3 lots of 14<sup>th</sup>s and 5 lots of 14<sup>th</sup>s then the sum of these two fractions will be 8 (5 + 3) lots of 14<sup>th</sup>s, or <latex>\small \frac{8}{14}</latex>. The sum is easy because the denominators are the same. Maths with fractions becomes complicated when the denominators are different, for example:- | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{1}{4} + \frac{2}{5} = ?</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | 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! | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16} = \frac{5}{20} = \frac{6}{24}</latex>... | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{2}{5} = \frac{4}{10} = \frac{6}{15} = \frac{8}{20} = \frac{10}{25} = \frac{12}{30}</latex>... | ||

+ | | ||

+ | |||

+ | |||

+ | Can you spot the two fractions from each list that have the same denominators? That's right: - <latex>\small \frac{5}{20}</latex> and <latex>\small \frac{8}{20}</latex>. | ||

+ | | ||

+ | |||

+ | |||

+ | So our question can be rewritten as | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{5}{20} + \frac{8}{20} = \frac{13}{20}</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | ===== Subtracting Fractions ===== | ||

+ | | ||

+ | |||

+ | |||

+ | The trick is the same when subtracting fractions: we need to find a common denominator. For example... | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{7}{9} - \frac{1}{4} = ?</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | Convert <latex>\small \frac{7}{9}</latex> and <latex>\small \frac{1}{4}</latex> into equivalent fractions where the denominators are both the same... | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{7}{9} = \frac{14}{18} = \frac{21}{27} = \frac{28}{36} = \frac{35}{45} = \frac{42}{54}</latex>... | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\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}</latex>... | ||

+ | | ||

+ | |||

+ | |||

+ | If we choose equivalent fractions from the lists above with the same denominators then our question becomes... | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{28}{36} - \frac{9}{36} = \frac{19}{36}</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | ===== Multiplying Fractions ===== | ||

+ | | ||

+ | |||

+ | |||

+ | This is much easier! To find the answer to | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{2}{5} \times \frac{3}{7}</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | 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. | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35}</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | ===== 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, | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{2}{5} \div \frac{3}{7} = ?</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | The number we are dividing by is <latex>\small \frac{3}{7}</latex> so let's turn this upside down to give <latex>\small \frac{7}{3}</latex> and then multiply... | ||

+ | | ||

+ | |||

+ | |||

+ | <latex>\frac{2}{5} \times \frac{7}{3} = \frac{2 \times 7}{5 \times 3} = \frac{14}{15}</latex> | ||

+ | | ||

+ | |||

+ | |||

+ | For further fractions help [[[[articles:more_fractions|click here!]] | ||