1.2 Fractional arithmetic - Revision history

Phil: Partial edit by Phil Ramsden: more to do. Idiomatic English, localisation of terminology and notation.

Phil — Tue, 30 Dec 2008 13:17:51 GMT

Partial edit by Phil Ramsden: more to do. Idiomatic English, localisation of terminology and notation.

Jdinglis at 13:57, 26 October 2008

Jdinglis — Sun, 26 Oct 2008 13:57:24 GMT

←Older revision		Revision as of 13:57, 26 October 2008
Line 331:		Line 331:


-	'''Keep in mind that: '''	+	'''Keep in mind that...'''

-	~~Try always~~ to write an expression in the simplest possible terms. What is the "simplest" depends usually on the context.	+	You should try to write an expression in the simplest possible terms. What is the "simplest" depends usually on the context.

	It is important that you really master calculations with fractions. You should be able to find a common denominator and multiply or divide numerators and denominators by suitable numbers. These principles are basic when you have to calculate a rational expression and you will need them when you have to deal with other mathematical expressions and operations.		It is important that you really master calculations with fractions. You should be able to find a common denominator and multiply or divide numerators and denominators by suitable numbers. These principles are basic when you have to calculate a rational expression and you will need them when you have to deal with other mathematical expressions and operations.

Eakc07: Grammatical changes

Eakc07 — Tue, 21 Oct 2008 17:05:27 GMT

Grammatical changes

←Older revision		Revision as of 17:05, 21 October 2008
Line 28:		Line 28:
	A rational number can be written in many ways depending on the denominator one chooses to use. For example, we have that		A rational number can be written in many ways depending on the denominator one chooses to use. For example, we have that

-	{{Displayed math\|\|<math>0{,}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{etc.}</math>}}	+	{{Displayed math\|\|<math>0\text{.}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{etc.}</math>}}

	The value of a rational number is not changed by multiplying or dividing the numerator and denominator with the same number. The division operation is called cancellation.		The value of a rational number is not changed by multiplying or dividing the numerator and denominator with the same number. The division operation is called cancellation.
Line 51:		Line 51:
	</div>		</div>

-	One should always specify a fraction in a form where cancellation has been performed as far as possible (reduced fraction). This can be labourious when large numbers are involved, which is why, during an ongoing calculation one should try to keep all fractions maximally cancelled.	+	One should always specify a fraction in a form where cancellation has been performed as far as possible (reduced fraction). This can be labourious when large numbers are involved which is why, during an ongoing calculation, one should try to keep all fractions maximally cancelled.


	==Addition and subtraction of fractions ==		==Addition and subtraction of fractions ==

-	The addition and subtraction of fractions requires that the fractions have the same denominator. If this is not so, one must begin by multiplying the numerator and denominator of each fraction by a suitable number so that all the fractions then have a common denominator.	+	The addition and subtraction of fractions requires that the fractions have the same denominator. If this is not so one must begin by multiplying the numerator and denominator of each fraction by a suitable number so that all the fractions then have a common denominator.

	<div class="exempel">		<div class="exempel">
Line 72:		Line 72:
	</div>		</div>

-	The important point here is to obtain a common denominator, but we should try and find a common denominator which is as small as possible. The ideal ~~always~~ is to find the lowest common denominator (LCD). One can always obtain a common denominator by multiplying all the involved denominators with each other~~. However~~, this is not always necessary.	+	The important point here is to obtain a common denominator, but we should try and find a common denominator which is as small as possible. The ideal is to find the lowest common denominator (LCD). One can always obtain a common denominator by multiplying all the involved denominators with each other, however this is not always necessary.


Line 106:		Line 106:
	</div>		</div>

-	One should be sufficiently proficient in ~~doing~~ mental arithmetic that one can quickly find the LCD if the denominators are of reasonable size. ~~To generally~~ determine the lowest common denominator requires investigating which prime numbers make up the denominator.	+	One should be sufficiently proficient in mental arithmetic so that one can quickly find the LCD if the denominators are of reasonable size. In general, to determine the lowest common denominator requires investigating which prime numbers make up the denominator.
	<div class="exempel">		<div class="exempel">

Line 113:		Line 113:
	<li>Simplify <math>\ \frac{1}{60} + \frac{1}{42}</math>.<br/><br/>		<li>Simplify <math>\ \frac{1}{60} + \frac{1}{42}</math>.<br/><br/>

-	Decompose 60 and 42 into their smallest integer factors. This way we can determine the minimum number that is divisible by 60 and 42. This is achieved by multiplying together the factors ~~but~~ avoid ~~the inclusion of~~ too many of the factors ~~that the numbers have in common~~.	+	Decompose 60 and 42 into their smallest integer factors. This way we can determine the minimum number that is divisible by 60 and 42. This is achieved by multiplying together the factors. However we need to avoid including too many of the factors.
	{{Displayed math\|\|<math>\left.\eqalign{60 &= 2\cdot 2\cdot 3\cdot 5\cr 42 &= 2\cdot 3\cdot 7}\right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 2\cdot 3\cdot 5\cdot 7 = 420\,\mbox{.}</math>}}		{{Displayed math\|\|<math>\left.\eqalign{60 &= 2\cdot 2\cdot 3\cdot 5\cr 42 &= 2\cdot 3\cdot 7}\right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 2\cdot 3\cdot 5\cdot 7 = 420\,\mbox{.}</math>}}
	We then can write		We then can write
Line 121:		Line 121:
	<li> Simplify <math>\ \frac{2}{15}+\frac{1}{6}-\frac{5}{18}</math>.<br/><br/>		<li> Simplify <math>\ \frac{2}{15}+\frac{1}{6}-\frac{5}{18}</math>.<br/><br/>

-	The lowest common denominator is chosen so that it contains just enough primes in order to be divisible by 15, 6 and 18	+	The lowest common denominator is chosen so that it contains just enough primes in order to be divisible by 15, 6 and 18
	{{Displayed math\|\|<math>\left. \eqalign{15 &= 3\cdot 5\cr 6&=2\cdot 3\cr 18 &= 2\cdot 3\cdot 3} \right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 3\cdot 3\cdot5 = 90\,\mbox{.}</math>}}		{{Displayed math\|\|<math>\left. \eqalign{15 &= 3\cdot 5\cr 6&=2\cdot 3\cr 18 &= 2\cdot 3\cdot 3} \right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 3\cdot 3\cdot5 = 90\,\mbox{.}</math>}}
	We then can write		We then can write
Line 132:		Line 132:
	== Multiplication ==		== Multiplication ==

-	When a fraction is multiplied by an integer, only the numerator is multiplied by the integer. It is obvious that~~, for example,~~ <math>\tfrac{1}{3}</math> multiplied by 2 gives <math>\tfrac{2}{3}</math>, ~~that is,~~	+	When a fraction is multiplied by an integer, only the numerator is multiplied by the integer. For example, it is obvious that <math>\tfrac{1}{3}</math> multiplied by 2 gives <math>\tfrac{2}{3}</math>,

	{{Displayed math\|\|<math>\frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}</math>}}		{{Displayed math\|\|<math>\frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}</math>}}
Line 145:		Line 145:
	</div>		</div>

-	Before doing a multiplication, one should always check whether it is possible to perform a cancellation. This is done by deleting any common factors in the numerator and denominator.	+	Before doing a multiplication one should always check whether it is possible to perform a cancellation. This is done by deleting any common factors in the numerator and denominator.
	<div class="exempel">		<div class="exempel">
	''' Example 6'''		''' Example 6'''
Line 157:		Line 157:
	</div>		</div>

-	In 6b ~~one has cancelled~~ the 3 at an earlier stage than in 6a.	+	In 6b the 3 has been cancelled at an earlier stage than in 6a.

	<div class="exempel">		<div class="exempel">
Line 180:		Line 180:
	== Division ==		== Division ==

-	If <math>\tfrac{1}{4}</math> is divided ~~into~~ 2 one gets the answer <math>\tfrac{1}{8}</math>. If <math>\tfrac{1}{2}</math> is divided ~~into~~ 5 one gets the result <math>\tfrac{1}{10}</math>. We have that	+	If <math>\tfrac{1}{4}</math> is divided by 2 one gets the answer <math>\tfrac{1}{8}</math>. If <math>\tfrac{1}{2}</math> is divided by 5 one gets the result <math>\tfrac{1}{10}</math>. We have that

	{{Displayed math\|\|<math>\frac{\displaystyle \frac{1}{4}}{2} = \frac{1}{4\cdot 2} = \frac{1}{8} \qquad \mbox{ and } \qquad \frac{\displaystyle \frac{1}{2}}{5} = \frac{1}{2\cdot 5} = \frac{1}{10}\,\mbox{.}</math>}}		{{Displayed math\|\|<math>\frac{\displaystyle \frac{1}{4}}{2} = \frac{1}{4\cdot 2} = \frac{1}{8} \qquad \mbox{ and } \qquad \frac{\displaystyle \frac{1}{2}}{5} = \frac{1}{2\cdot 5} = \frac{1}{10}\,\mbox{.}</math>}}

-	When a fraction is divided by an integer, the denominator is multiplied by the integer.	+	When a fraction is divided by an integer the denominator is multiplied by the integer.

	<div class="exempel">		<div class="exempel">
Line 194:		Line 194:
	</div>		</div>

-	When a number is divided by a fraction, the number is multiplied by the inverted ("up-side-down") fraction . For example, dividing by <math>\frac{1}{2}</math> is the same as multiplying by<math>\frac{2}{1}</math> that is 2.	+	When a number is divided by a fraction the number is multiplied by the inverted ("up-side-down") fraction . For example, dividing by <math>\frac{1}{2}</math> is the same as multiplying by<math>\frac{2}{1}</math> that is 2.

	<div class="exempel">		<div class="exempel">
Line 218:		Line 218:
	</div>		</div>

-	How can division with a fraction turn into fraction multiplication? The explanation is that if a fraction is multiplied by its inverted fraction, the product is always 1~~, for~~ example,	+	How can division with a fraction turn into fraction multiplication? The explanation is that if a fraction is multiplied by its inverted fraction the product is always 1. For example,

	{{Displayed math\|\|<math>\frac{2}{3}\cdot\frac{3}{2} = \frac{\not{2}}{\not{3}}\cdot\frac{\not{3}}{\not{2}} = 1 \qquad \mbox{ or } \qquad \frac{9}{17}\cdot\frac{17}{9} = \frac{\not{9}}{\not{17}}\cdot\frac{\not{17}}{\not{9}} = 1\mbox{.}</math>}}		{{Displayed math\|\|<math>\frac{2}{3}\cdot\frac{3}{2} = \frac{\not{2}}{\not{3}}\cdot\frac{\not{3}}{\not{2}} = 1 \qquad \mbox{ or } \qquad \frac{9}{17}\cdot\frac{17}{9} = \frac{\not{9}}{\not{17}}\cdot\frac{\not{17}}{\not{9}} = 1\mbox{.}</math>}}

-	If in a division of fractions one multiplies the numerator and denominator with the inverse of the denominator then the resulting fraction will have denominator 1~~, and thus~~ the result is the numerator multiplied by the inverse of the original denominator.	+	If in a division of fractions one multiplies the numerator and denominator with the inverse of the denominator then the resulting fraction will have denominator 1. Thus the result is the numerator multiplied by the inverse of the original denominator.

	<div class="exempel">		<div class="exempel">
Line 237:		Line 237:
	== Fractions as a proportion of a whole ==		== Fractions as a proportion of a whole ==

-	Rational numbers are numbers that we can ~~write~~ as fractions, ~~convert~~ to decimal form, or ~~mark~~ on a real-number axis. In our everyday language they are also used to describe the proportion of something. Below are given some examples. Note how we use the word "of", which can lead to a multiplication or a division.	+	Rational numbers are numbers that can be writen as fractions, they can subsequently be converted to decimal form or be marked on a real-number axis. In our everyday language they are also used to describe the proportion of something. Below are given some examples. Note how we use the word "of", which can lead to a multiplication or a division.

	<div class="exempel">		<div class="exempel">
Line 266:		Line 266:
	== Mixed expressions ==		== Mixed expressions ==

-	When fractions appear in calculations one~~, of course,~~ must follow the usual methods for arithmetic operations and their priority (multiplication / division before addition / subtraction). Remember also that the numerator and denominator in a division are calculated separately before the division is performed ( "invisible parentheses").	+	When fractions appear in calculations one must follow the usual methods for arithmetic operations and their priority (multiplication / division before addition / subtraction). Remember also that the numerator and denominator involved in a division are calculated separately before the division is performed ( "invisible parentheses").

	<div class="exempel">		<div class="exempel">
Line 335:		Line 335:
	Try always to write an expression in the simplest possible terms. What is the "simplest" depends usually on the context.		Try always to write an expression in the simplest possible terms. What is the "simplest" depends usually on the context.

-	It is important that you really master calculations with fractions. You should be able to find a common denominator, multiply or divide numerators and denominators by suitable numbers ~~etc~~. These principles are basic when you have to calculate a rational expression ~~that includes variables~~ and you will need them when you have to deal with other mathematical expressions and operations.	+	It is important that you really master calculations with fractions. You should be able to find a common denominator and multiply or divide numerators and denominators by suitable numbers. These principles are basic when you have to calculate a rational expression and you will need them when you have to deal with other mathematical expressions and operations.

	Rational expressions that contain variables (x, y, ...) and include fractions are very common when studying functions, especially increment ratios, limits and derivatives.		Rational expressions that contain variables (x, y, ...) and include fractions are very common when studying functions, especially increment ratios, limits and derivatives.

Eakc07 at 16:37, 21 October 2008

Eakc07 — Tue, 21 Oct 2008 16:37:49 GMT

←Older revision		Revision as of 16:37, 21 October 2008
Line 17:		Line 17:
	'''Learning outcomes:'''		'''Learning outcomes:'''

-	After this section, you should have learned to:	+	After this section you should have learned to:
	*Calculate expressions containing fractions, the four arithmetic operations and parentheses.		*Calculate expressions containing fractions, the four arithmetic operations and parentheses.
-	*Cancel down fractions as far as possible (reduction).	+	*Cancel down fractions as far as possible (reduction).
	*Determine the lowest common denominator (LCD).		*Determine the lowest common denominator (LCD).
	}}		}}
Line 26:		Line 26:
	== Fraction modification ==		== Fraction modification ==

-	A rational number can be written in many ways, depending on the denominator one chooses to use. For example, we have that	+	A rational number can be written in many ways depending on the denominator one chooses to use. For example, we have that

	{{Displayed math\|\|<math>0{,}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{etc.}</math>}}		{{Displayed math\|\|<math>0{,}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{etc.}</math>}}

Tekbot: Robot: Automated text replacement (-{{Fristående formel +{{Displayed math)

Tekbot — Wed, 10 Sep 2008 13:53:59 GMT

Robot: Automated text replacement (-{{Fristående formel +{{Displayed math)

Tekbot: Robot: Automated text replacement (-{{Vald flik +{{Selected tab)

Tekbot — Wed, 10 Sep 2008 13:48:55 GMT

Robot: Automated text replacement (-{{Vald flik +{{Selected tab)

←Older revision		Revision as of 13:48, 10 September 2008
Line 2:		Line 2:
	{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"		{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
-	{{~~Vald flik~~\|[[1.2 Fractional arithmetic\|Theory]]}}	+	{{Selected tab\|[[1.2 Fractional arithmetic\|Theory]]}}
	{{Not selected tab\|[[1.2 Exercises\|Exercises]]}}		{{Not selected tab\|[[1.2 Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|

Tekbot: Robot: Automated text replacement (-{{Ej vald flik +{{Not selected tab)

Tekbot — Wed, 10 Sep 2008 13:43:32 GMT

Robot: Automated text replacement (-{{Ej vald flik +{{Not selected tab)

←Older revision		Revision as of 13:43, 10 September 2008
Line 3:		Line 3:
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
	{{Vald flik\|[[1.2 Fractional arithmetic\|Theory]]}}		{{Vald flik\|[[1.2 Fractional arithmetic\|Theory]]}}
-	{{~~Ej vald flik~~\|[[1.2 Exercises\|Exercises]]}}	+	{{Not selected tab\|[[1.2 Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|
	\|}		\|}

Tekbot: Robot: Automated text replacement (-1.2 Bråkräkning +1.2 Fractional arithmetic)

Tekbot — Wed, 10 Sep 2008 13:11:11 GMT

Robot: Automated text replacement (-1.2 Bråkräkning +1.2 Fractional arithmetic)

←Older revision		Revision as of 13:11, 10 September 2008
Line 2:		Line 2:
	{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"		{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
-	{{Vald flik\|[[1.2 ~~Bråkräkning~~\|Theory]]}}	+	{{Vald flik\|[[1.2 Fractional arithmetic\|Theory]]}}
	{{Ej vald flik\|[[1.2 Exercises\|Exercises]]}}		{{Ej vald flik\|[[1.2 Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|

Tekbot: Robot: Automated text replacement (-Övningar +Exercises)

Tekbot — Wed, 10 Sep 2008 10:37:21 GMT

Robot: Automated text replacement (-Övningar +Exercises)

←Older revision		Revision as of 10:37, 10 September 2008
Line 3:		Line 3:
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
	{{Vald flik\|[[1.2 Bråkräkning\|Theory]]}}		{{Vald flik\|[[1.2 Bråkräkning\|Theory]]}}
-	{{Ej vald flik\|[[1.2 ~~Övningar~~\|Exercises]]}}	+	{{Ej vald flik\|[[1.2 Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|
	\|}		\|}
Line 320:		Line 320:
	</div>		</div>

-	[[1.2 ~~Övningar~~\|Exercises]]	+	[[1.2 Exercises\|Exercises]]

Tekbot: 1.2 Bråkräkning moved to 1.2 Fractional arithmetic: Robot: moved page

Tekbot — Tue, 09 Sep 2008 12:18:43 GMT

1.2 Bråkräkning moved to 1.2 Fractional arithmetic: Robot: moved page

←Older revision

Revision as of 12:18, 9 September 2008

←Older revision		Revision as of 13:48, 10 September 2008
Line 2:		Line 2:
	{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"		{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
-	{{~~Vald flik~~\|[[1.2 Fractional arithmetic\|Theory]]}}	+	{{Selected tab\|[[1.2 Fractional arithmetic\|Theory]]}}
	{{Not selected tab\|[[1.2 Exercises\|Exercises]]}}		{{Not selected tab\|[[1.2 Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|

←Older revision		Revision as of 13:43, 10 September 2008
Line 3:		Line 3:
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
	{{Vald flik\|[[1.2 Fractional arithmetic\|Theory]]}}		{{Vald flik\|[[1.2 Fractional arithmetic\|Theory]]}}
-	{{~~Ej vald flik~~\|[[1.2 Exercises\|Exercises]]}}	+	{{Not selected tab\|[[1.2 Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|
	\|}		\|}

←Older revision		Revision as of 13:11, 10 September 2008
Line 2:		Line 2:
	{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"		{\| border="0" cellspacing="0" cellpadding="0" height="30" width="100%"
	\| style="border-bottom:1px solid #797979" width="5px" \|		\| style="border-bottom:1px solid #797979" width="5px" \|
-	{{Vald flik\|[[1.2 ~~Bråkräkning~~\|Theory]]}}	+	{{Vald flik\|[[1.2 Fractional arithmetic\|Theory]]}}
	{{Ej vald flik\|[[1.2 Exercises\|Exercises]]}}		{{Ej vald flik\|[[1.2 Exercises\|Exercises]]}}
	\| style="border-bottom:1px solid #797979" width="100%"\|		\| style="border-bottom:1px solid #797979" width="100%"\|

←Older revision		Revision as of 13:53, 10 September 2008
Line 28:		Line 28:
	A rational number can be written in many ways, depending on the denominator one chooses to use. For example, we have that		A rational number can be written in many ways, depending on the denominator one chooses to use. For example, we have that

-	{{~~Fristående formel~~\|\|<math>0{,}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{etc.}</math>}}	+	{{Displayed math\|\|<math>0{,}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{etc.}</math>}}

	The value of a rational number is not changed by multiplying or dividing the numerator and denominator with the same number. The division operation is called cancellation.		The value of a rational number is not changed by multiplying or dividing the numerator and denominator with the same number. The division operation is called cancellation.
Line 114:		Line 114:

	Decompose 60 and 42 into their smallest integer factors. This way we can determine the minimum number that is divisible by 60 and 42. This is achieved by multiplying together the factors but avoid the inclusion of too many of the factors that the numbers have in common.		Decompose 60 and 42 into their smallest integer factors. This way we can determine the minimum number that is divisible by 60 and 42. This is achieved by multiplying together the factors but avoid the inclusion of too many of the factors that the numbers have in common.
-	{{~~Fristående formel~~\|\|<math>\left.\eqalign{60 &= 2\cdot 2\cdot 3\cdot 5\cr 42 &= 2\cdot 3\cdot 7}\right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 2\cdot 3\cdot 5\cdot 7 = 420\,\mbox{.}</math>}}	+	{{Displayed math\|\|<math>\left.\eqalign{60 &= 2\cdot 2\cdot 3\cdot 5\cr 42 &= 2\cdot 3\cdot 7}\right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 2\cdot 3\cdot 5\cdot 7 = 420\,\mbox{.}</math>}}
	We then can write		We then can write
-	{{~~Fristående formel~~\|\|<math>\frac{1}{60}+\frac{1}{42} = \frac{1\cdot 7}{60\cdot 7} + \frac{1\cdot 2\cdot 5}{42\cdot 2\cdot 5} = \frac{7}{420} + \frac{10}{420} =\frac{17}{420}\,\mbox{.}</math>}}	+	{{Displayed math\|\|<math>\frac{1}{60}+\frac{1}{42} = \frac{1\cdot 7}{60\cdot 7} + \frac{1\cdot 2\cdot 5}{42\cdot 2\cdot 5} = \frac{7}{420} + \frac{10}{420} =\frac{17}{420}\,\mbox{.}</math>}}
	</li>		</li>

Line 122:		Line 122:

	The lowest common denominator is chosen so that it contains just enough primes in order to be divisible by 15, 6 and 18		The lowest common denominator is chosen so that it contains just enough primes in order to be divisible by 15, 6 and 18
-	{{~~Fristående formel~~\|\|<math>\left. \eqalign{15 &= 3\cdot 5\cr 6&=2\cdot 3\cr 18 &= 2\cdot 3\cdot 3} \right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 3\cdot 3\cdot5 = 90\,\mbox{.}</math>}}	+	{{Displayed math\|\|<math>\left. \eqalign{15 &= 3\cdot 5\cr 6&=2\cdot 3\cr 18 &= 2\cdot 3\cdot 3} \right\} \quad\Rightarrow\quad \text{LCD} = 2\cdot 3\cdot 3\cdot5 = 90\,\mbox{.}</math>}}
	We then can write		We then can write
-	{{~~Fristående formel~~\|\|<math> \frac{2}{15}+\frac{1}{6}-\frac{5}{18} = \frac{2\cdot 2\cdot 3}{15\cdot 2\cdot 3} + \frac{1\cdot 3\cdot 5}{6\cdot 3\cdot 5} - \frac{5\cdot 5}{18\cdot 5} = \frac{12}{90} + \frac{15}{90} - \frac{25}{90} = \frac{2}{90} = \frac{1}{45}\,\mbox{.}</math>}}	+	{{Displayed math\|\|<math> \frac{2}{15}+\frac{1}{6}-\frac{5}{18} = \frac{2\cdot 2\cdot 3}{15\cdot 2\cdot 3} + \frac{1\cdot 3\cdot 5}{6\cdot 3\cdot 5} - \frac{5\cdot 5}{18\cdot 5} = \frac{12}{90} + \frac{15}{90} - \frac{25}{90} = \frac{2}{90} = \frac{1}{45}\,\mbox{.}</math>}}
	</li>		</li>
	</ol>		</ol>
Line 134:		Line 134:
	When a fraction is multiplied by an integer, only the numerator is multiplied by the integer. It is obvious that, for example, <math>\tfrac{1}{3}</math> multiplied by 2 gives <math>\tfrac{2}{3}</math>, that is,		When a fraction is multiplied by an integer, only the numerator is multiplied by the integer. It is obvious that, for example, <math>\tfrac{1}{3}</math> multiplied by 2 gives <math>\tfrac{2}{3}</math>, that is,

-	{{~~Fristående formel~~\|\|<math>\frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}</math>}}	+	{{Displayed math\|\|<math>\frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}</math>}}

	If two fractions are multiplied with each other, then the numerators are multiplied together and the denominators are multiplied together.		If two fractions are multiplied with each other, then the numerators are multiplied together and the denominators are multiplied together.
Line 182:		Line 182:
	If <math>\tfrac{1}{4}</math> is divided into 2 one gets the answer <math>\tfrac{1}{8}</math>. If <math>\tfrac{1}{2}</math> is divided into 5 one gets the result <math>\tfrac{1}{10}</math>. We have that		If <math>\tfrac{1}{4}</math> is divided into 2 one gets the answer <math>\tfrac{1}{8}</math>. If <math>\tfrac{1}{2}</math> is divided into 5 one gets the result <math>\tfrac{1}{10}</math>. We have that

-	{{~~Fristående formel~~\|\|<math>\frac{\displaystyle \frac{1}{4}}{2} = \frac{1}{4\cdot 2} = \frac{1}{8} \qquad \mbox{ and } \qquad \frac{\displaystyle \frac{1}{2}}{5} = \frac{1}{2\cdot 5} = \frac{1}{10}\,\mbox{.}</math>}}	+	{{Displayed math\|\|<math>\frac{\displaystyle \frac{1}{4}}{2} = \frac{1}{4\cdot 2} = \frac{1}{8} \qquad \mbox{ and } \qquad \frac{\displaystyle \frac{1}{2}}{5} = \frac{1}{2\cdot 5} = \frac{1}{10}\,\mbox{.}</math>}}

	When a fraction is divided by an integer, the denominator is multiplied by the integer.		When a fraction is divided by an integer, the denominator is multiplied by the integer.
Line 220:		Line 220:
	How can division with a fraction turn into fraction multiplication? The explanation is that if a fraction is multiplied by its inverted fraction, the product is always 1, for example,		How can division with a fraction turn into fraction multiplication? The explanation is that if a fraction is multiplied by its inverted fraction, the product is always 1, for example,

-	{{~~Fristående formel~~\|\|<math>\frac{2}{3}\cdot\frac{3}{2} = \frac{\not{2}}{\not{3}}\cdot\frac{\not{3}}{\not{2}} = 1 \qquad \mbox{ or } \qquad \frac{9}{17}\cdot\frac{17}{9} = \frac{\not{9}}{\not{17}}\cdot\frac{\not{17}}{\not{9}} = 1\mbox{.}</math>}}	+	{{Displayed math\|\|<math>\frac{2}{3}\cdot\frac{3}{2} = \frac{\not{2}}{\not{3}}\cdot\frac{\not{3}}{\not{2}} = 1 \qquad \mbox{ or } \qquad \frac{9}{17}\cdot\frac{17}{9} = \frac{\not{9}}{\not{17}}\cdot\frac{\not{17}}{\not{9}} = 1\mbox{.}</math>}}

	If in a division of fractions one multiplies the numerator and denominator with the inverse of the denominator then the resulting fraction will have denominator 1, and thus the result is the numerator multiplied by the inverse of the original denominator.		If in a division of fractions one multiplies the numerator and denominator with the inverse of the denominator then the resulting fraction will have denominator 1, and thus the result is the numerator multiplied by the inverse of the original denominator.