1.2 Fractional arithmetic

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(Partial edit by Phil Ramsden: more to do. Idiomatic English, localisation of terminology and notation.)
 
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{{Vald flik|[[1.2 Bråkräkning|Teori]]}}
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{{Selected tab|[[1.2 Fractional arithmetic|Theory]]}}
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{{Ej vald flik|[[1.2 Övningar|Övningar]]}}
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{{Not selected tab|[[1.2 Exercises|Exercises]]}}
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{{Info|
{{Info|
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'''Innehåll:'''
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'''Contents:'''
-
* Addition och subtraktion av bråktal
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* Addition and subtraction of fractions
-
* Multiplikation och division av bråktal
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* Multiplication and division of fractions
}}
}}
{{Info|
{{Info|
-
'''Lärandemål:'''
+
'''Learning outcomes:'''
-
Efter detta avsnitt ska du ha lärt dig att:
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After this section you should have learned to:
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*Beräkna uttryck som innehåller bråktal, de fyra räknesätten och parenteser.
+
*Calculate the value of expressions containing fractions, the four arithmetic operations and parentheses.
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*Förkorta bråk så långt som möjligt.
+
*Cancel down fractions as far as possible (reduction).
-
*Bestämma minsta gemensamma nämnare (MGN).
+
*Determine the lowest common denominator (LCD).
}}
}}
-
==Förlängning och förkortning==
+
== Fraction modification ==
-
Ett rationellt tal kan skrivas på många sätt, beroende på vilken nämnare man väljer att använda. Exempelvis har vi att
+
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{osv.}</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>}}
-
Värdet av ett rationellt tal ändras inte när man multiplicerar eller dividerar täljare och nämnare med samma tal. Dessa operationer kallas ''förlängning'' respektive ''förkortning''.
+
The value of a rational number is not changed by multiplying or dividing the numerator and denominator by the same number. The division operation is called reduction or cancellation.
<div class="exempel">
<div class="exempel">
-
'''Exempel 1'''
+
''' Example 1'''
-
Förlängning:
+
Multiplying numerator and denominator by the same number:
<ol type="a">
<ol type="a">
-
<li><math>\frac{2}{3} = \frac{2\cdot 5}{3\cdot 5} = \frac{10}{15}</math></li>
+
<li><math>\frac{2}{3} = \frac{2\times 5}{3\times 5} = \frac{10}{15}</math></li>
-
<li><math>\frac{5}{7} = \frac{5\cdot 4}{7\cdot 4} = \frac{20}{28}</math></li>
+
<li><math>\frac{5}{7} = \frac{5\times 4}{7\times 4} = \frac{20}{28}</math></li>
</ol>
</ol>
-
Förkortning:
+
Dvinding numerator and denominator by the same number: (reducing or cancelling):
<ol type="a" start="3">
<ol type="a" start="3">
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</div>
</div>
-
Man bör alltid ange ett bråk förkortat så långt som möjligt. Detta kan vara arbetsamt när stora tal är inblandade, varför man redan under en pågående uträkning bör försöka hålla bråk i så förkortad form som möjligt.
+
We usually specify a fraction in a form where cancellation has been performed as far as possible: this is called expressing it in its lowest terms. This can be laborious when large numbers are involved which is why, in long calculations, you should usually cancel as you go along.
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== Addition och subtraktion av bråk ==
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==Addition and subtraction of fractions ==
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Vid addition och subtraktion av tal i bråkform måste bråken ha samma nämnare. Om så inte är fallet måste man först förlänga respektive bråk med lämpliga tal så att gemensam nämnare erhålles.
+
Fractions can only be added or subtracted if they have the same denominator. If they do not, they must each first be "top-and-bottom" multiplied in such a way that they do. This is called placing over a common denominator.
<div class="exempel">
<div class="exempel">
-
'''Exempel 2'''
+
''' Example 2'''
<ol type="a">
<ol type="a">
<li><math>\frac{3}{5}+\frac{2}{3}
<li><math>\frac{3}{5}+\frac{2}{3}
-
= \frac{3\cdot 3}{5\cdot 3} + \frac{2\cdot 5}{3\cdot 5}
+
= \frac{3\times 3}{5\times 3} + \frac{2\times 5}{3\times 5}
= \frac{9}{15} + \frac{10}{15} = \frac{9+10}{15}
= \frac{9}{15} + \frac{10}{15} = \frac{9+10}{15}
= \frac{19}{15}</math></li>
= \frac{19}{15}</math></li>
<li><math>\frac{5}{6}-\frac{2}{9}
<li><math>\frac{5}{6}-\frac{2}{9}
-
= \frac{5\cdot 3}{6\cdot 3} - \frac{2\cdot 2}{9\cdot 2}
+
= \frac{5\times 3}{6\times 3} - \frac{2\times 2}{9\times 2}
= \frac{15}{18} - \frac{4}{18} = \frac{15-4}{18}
= \frac{15}{18} - \frac{4}{18} = \frac{15-4}{18}
= \frac{11}{18}</math></li>
= \frac{11}{18}</math></li>
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</div>
</div>
-
Det viktiga är här att åstadkomma en gemensam nämnare, men man bör sträva efter att hitta en så låg gemensam nämnare som möjligt. Idealet är att hitta den minsta gemensamma nämnaren (MGN). Man kan alltid erhålla en gemensam nämnare genom att multiplicera de inblandade nämnarna med varandra. Detta är dock inte alltid nödvändigt.
+
A common denominator can always be found by simply multiplying the denominators of the two fractions together. Often, however, a smaller one can be found. The ideal is to find the lowest common denominator (LCD).
<div class="exempel">
<div class="exempel">
-
'''Exempel 3'''
+
'''Example 3'''
<ol type="a">
<ol type="a">
<li><math>\frac{7}{15}-\frac{1}{12}
<li><math>\frac{7}{15}-\frac{1}{12}
-
= \frac{7\cdot 12}{15\cdot 12}
+
= \frac{7\times 12}{15\times 12}
-
- \frac{1\cdot 15}{12\cdot 15}\vphantom{\Biggl(}</math><br>
+
- \frac{1\times 15}{12\times 15}\vphantom{\Biggl(}</math><br>
<math>\insteadof{\displaystyle\frac{7}{15}-\frac{1}{12}}{}{}
<math>\insteadof{\displaystyle\frac{7}{15}-\frac{1}{12}}{}{}
= \frac{84}{180}-\frac{15}{180} = \frac{69}{180} = \frac{69/3}{180/3}
= \frac{84}{180}-\frac{15}{180} = \frac{69}{180} = \frac{69/3}{180/3}
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<li><math>\frac{7}{15}-\frac{1}{12}
<li><math>\frac{7}{15}-\frac{1}{12}
-
= \frac{7\cdot 4}{15\cdot 4}- \frac{1\cdot 5}{12\cdot 5}
+
= \frac{7\times 4}{15\times 4}- \frac{1\times 5}{12\times 5}
= \frac{28}{60}-\frac{5}{60} = \frac{23}{60}</math></li>
= \frac{28}{60}-\frac{5}{60} = \frac{23}{60}</math></li>
<li><math>\frac{1}{8}+\frac{3}{4}-\frac{1}{6}
<li><math>\frac{1}{8}+\frac{3}{4}-\frac{1}{6}
-
= \frac{1\cdot 4\cdot 6}{8\cdot 4\cdot 6}
+
= \frac{1\times 4\times 6}{8\times 4\times 6}
-
+ \frac{3\cdot 8\cdot 6}{4\cdot 8\cdot 6}
+
+ \frac{3\times 8\times 6}{4\times 8\times 6}
-
- \frac{1\cdot 8\cdot 4}{6\cdot 8\cdot 4}\vphantom{\Biggl(}</math><br>
+
- \frac{1\times 8\times 4}{6\times 8\times 4}\vphantom{\Biggl(}</math><br>
<math>\insteadof{\frac{1}{8}+\frac{3}{4}-\frac{1}{6}}{}{}
<math>\insteadof{\frac{1}{8}+\frac{3}{4}-\frac{1}{6}}{}{}
= \frac{24}{192} + \frac{144}{192} - \frac{32}{192}
= \frac{24}{192} + \frac{144}{192} - \frac{32}{192}
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<li><math>\frac{1}{8}+\frac{3}{4}-\frac{1}{6}
<li><math>\frac{1}{8}+\frac{3}{4}-\frac{1}{6}
-
= \frac{1\cdot 3}{8\cdot 3} + \frac{3\cdot 6}{4\cdot 6}
+
= \frac{1\times 3}{8\times 3} + \frac{3\times 6}{4\times 6}
-
- \frac{1\cdot 4}{6\cdot 4}
+
- \frac{1\times 4}{6\times 4}
= \frac{3}{24} + \frac{18}{24} - \frac{4}{24}
= \frac{3}{24} + \frac{18}{24} - \frac{4}{24}
= \frac{17}{24}</math></li>
= \frac{17}{24}</math></li>
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</div>
</div>
-
Man bör vara så pass tränad i huvudräkning att man snabbt kan hitta MGN om nämnarna är av rimlig storlek. Att allmänt bestämma den minsta gemensamma nämnaren kräver att man studerar vilka primtal som ingår som faktorer i respektive nämnare.
+
If the denominators are of reasonable size, you can usually find the LCD by inspection. More generally, you can use prime factorisation.
-
 
+
<div class="exempel">
<div class="exempel">
-
'''Exempel 4'''
+
'''Example 4'''
<ol type="a">
<ol type="a">
-
<li>Beräkna <math>\ \frac{1}{60} + \frac{1}{42}</math>.<br/><br/>
+
<li>Simplify <math>\ \frac{1}{60} + \frac{1}{42}</math>.<br/><br/>
-
Delar vi upp 60 och 42 i så små heltalsfaktorer som möjligt, så kan vi bestämma det minsta heltal som är delbart med 60 och 42 genom att multiplicera ihop deras faktorer men undvika att ta med för många av faktorerna som talen har gemensamt
+
 
-
{{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{MGN} = 2\cdot 2\cdot 3\cdot 5\cdot 7 = 420\,\mbox{.}</math>}}
+
Our aim is to find the smallest number that both 60 and 42 go into. First, decompose 60 and 42 into their prime factors.
-
Vi kan då skriva
+
{{Displayed math||<math> \eqalign{60 &= 2\times 2\times 3\times 5\cr 42 &= 2\times 3\times 7}</math>}}
-
{{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>}}
+
 
 +
For each prime factor, choose the larger power:
 +
{{Displayed math||<math>\text{LCD} = 2\times 2\times 3\times 5\times 7 = 420\,\mbox{.}</math>}}
 +
We then can write
 +
{{Displayed math||<math>\frac{1}{60}+\frac{1}{42} = \frac{1\times 7}{60\times 7} + \frac{1\times 2\times 5}{42\times 2\times 5} = \frac{7}{420} + \frac{10}{420} =\frac{17}{420}\,\mbox{.}</math>}}
</li>
</li>
-
<li>Beräkna <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/>
-
Minsta gemensamma nämnare väljs så att den innehåller precis så
+
Here,
-
många primtalsfaktorer så att den blir delbar med 15, 6 och 18
+
{{Displayed math||<math>\left. \eqalign{15 &= 3\times 5\cr 6&=2\times 3\cr 18 &= 2\times 3\times 3} \right\} \quad\Rightarrow\quad \text{LCD} = 2\times 3\times 3\times5 = 90\,\mbox{.}</math>}}
-
{{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{MGN} = 2\cdot 3\cdot 3\cdot5 = 90\,\mbox{.}</math>}}
+
We then can write
-
Vi kan då skriva
+
{{Displayed math||<math> \frac{2}{15}+\frac{1}{6}-\frac{5}{18} = \frac{2\times 2\times 3}{15\times 2\times 3} + \frac{1\times 3\times 5}{6\times 3\times 5} - \frac{5\times 5}{18\times 5} = \frac{12}{90} + \frac{15}{90} - \frac{25}{90} = \frac{2}{90} = \frac{1}{45}\,\mbox{.}</math>}}
-
{{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>}}
+
</li>
</li>
</ol>
</ol>
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-
== Multiplikation ==
+
== Multiplication ==
-
 
+
-
När ett bråk multipliceras med ett heltal, multipliceras endast täljaren med heltalet. Det är uppenbart att om t.ex. <math>\tfrac{1}{3}</math> multipliceras med 2 så blir resultatet <math>\tfrac{2}{3}</math>, dvs.
+
-
 
+
-
{{Fristående formel||<math>\frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}</math>}}
+
-
Om två bråk multipliceras med varandra, multipliceras täljarna med varandra och nämnarna med varandra.
+
When a fraction is multiplied by an integer, only the numerator is multiplied; for example, it is obvious that <math>\tfrac{1}{3}</math> multiplied by 2 is equal to <math>\tfrac{2}{3}</math>.
 +
If two fractions are multiplied together, then the numerators are multiplied together and the denominators are multiplied together.
<div class="exempel">
<div class="exempel">
-
'''Exempel 5'''
+
''' Example 5'''
<ol type="a">
<ol type="a">
-
<li><math>8\cdot\frac{3}{7} = \frac{8\cdot 3}{7} = \frac{24}{7}</math></li>
+
<li><math>8\times\frac{3}{7} = \frac{8\times 3}{7} = \frac{24}{7}</math></li>
-
<li><math>\frac{2}{3}\cdot \frac{1}{5} = \frac{2\cdot 1}{3\cdot 5} = \frac{2}{15}</math></li>
+
<li><math>\frac{2}{3}\times \frac{1}{5} = \frac{2\times 1}{3\times 5} = \frac{2}{15}</math></li>
</ol>
</ol>
</div>
</div>
-
Innan man genomför multiplikationen bör man alltid kontrollera om det är möjligt att förkorta bråket. Detta utförs genom att ''stryka'' eventuella gemensamma faktorer i täljare och nämnare.
+
Before doing a multiplication you should always check whether you can cancel first. Note that you can cancel on both sides of the multiplication sign.
-
 
+
<div class="exempel">
<div class="exempel">
-
'''Exempel 6'''
+
''' Example 6'''
-
Jämför uträkningarna:
+
Compare the calculations:
<ol type="a">
<ol type="a">
-
<li><math>\frac{3}{5}\cdot\frac{2}{3} = \frac{3\cdot 2}{5\cdot 3} = \frac{6}{15} = \frac{6/3}{15/3} = \frac{2}{5}</math></li>
+
<li><math>\frac{3}{5}\times\frac{2}{3} = \frac{3\times 2}{5\times 3} = \frac{6}{15} = \frac{6/3}{15/3} = \frac{2}{5}</math></li>
-
<li><math>\frac{3}{5}\cdot\frac{2}{3} = \frac{\not{3}\cdot 2}{5\cdot \not{3}} = \frac{2}{5}</math></li>
+
<li><math>\frac{3}{5}\times\frac{2}{3} = \frac{\not{3}\times 2}{5\times \not{3}} = \frac{2}{5}</math></li>
</ol>
</ol>
</div>
</div>
-
Att stryka treorna i 6b innebär ju bara att man förkortar bråket med 3 i ett tidigare skede.
+
In 6b the 3 has been cancelled at an earlier stage than in 6a.
-
+
 
<div class="exempel">
<div class="exempel">
-
'''Exempel 7'''
+
'''Example 7'''
<ol type="a">
<ol type="a">
-
<li><math>\frac{7}{10}\cdot \frac{2}{7}
+
<li><math>\frac{7}{10}\times \frac{2}{7}
-
= \frac{\not{7}}{10}\cdot \frac{2}{\not{7}}
+
= \frac{\not{7}}{10}\times \frac{2}{\not{7}}
-
= \frac{1}{10}\cdot \frac{2}{1}
+
= \frac{1}{10}\times \frac{2}{1}
-
= \frac{1}{\not{2} \cdot 5}\cdot \frac{\not{2}}{1}
+
= \frac{1}{\not{2} \times 5}\times \frac{\not{2}}{1}
-
= \frac{1}{5}\cdot \frac{1}{1} =\frac{1}{5}</math></li>
+
= \frac{1}{5}\times \frac{1}{1} =\frac{1}{5}</math></li>
-
<li><math>\frac{14}{15}\cdot \frac{20}{21}
+
<li><math>\frac{14}{15}\times \frac{20}{21}
-
= \frac{2 \cdot 7}{3 \cdot 5}\cdot \frac{4 \cdot 5}{3 \cdot 7}
+
= \frac{2 \times 7}{3 \times 5}\times \frac{4 \times 5}{3 \times 7}
-
= \frac{2 \cdot \not{7}}{3 \cdot 5}\cdot \frac{4 \cdot 5}{3 \cdot \not{7}}
+
= \frac{2 \times \not{7}}{3 \times 5}\times \frac{4 \times 5}{3 \times \not{7}}
-
= \frac{2}{3 \cdot \not{5}}\cdot \frac{4 \cdot \not{5}}{3}
+
= \frac{2}{3 \times \not{5}}\times \frac{4 \times \not{5}}{3}
-
= \frac{2}{3}\cdot\frac{4}{3}
+
= \frac{2}{3}\times\frac{4}{3}
-
= \frac{2\cdot 4}{3\cdot 3}
+
= \frac{2\times 4}{3\times 3}
= \frac{8}{9}</math></li>
= \frac{8}{9}</math></li>
</ol>
</ol>
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== Division ==
== Division ==
-
Om <math>\tfrac{1}{4}</math> delas i 2 så blir svaret <math>\tfrac{1}{8}</math>. Om <math>\tfrac{1}{2}</math> delas i 5 så blir resultatet <math>\tfrac{1}{10}</math>. Vi har alltså att
+
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
-
{{Fristående formel||<math>\frac{\displaystyle \frac{1}{4}}{2} = \frac{1}{4\cdot 2} = \frac{1}{8} \qquad \mbox{ och } \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\times 2} = \frac{1}{8} \qquad \mbox{ and } \qquad \frac{\displaystyle \frac{1}{2}}{5} = \frac{1}{2\times 5} = \frac{1}{10}\,\mbox{.}</math>}}
-
När ett bråk divideras med ett heltal, multipliceras alltså nämnaren med heltalet.
+
When a fraction is divided by an integer the denominator is multiplied by the integer.
<div class="exempel">
<div class="exempel">
-
'''Exempel 8'''
+
'''Example 8'''
<ol type="a">
<ol type="a">
-
<li><math>\frac{3}{5}\Big/4 = \frac{3}{5\cdot 4} = \frac{3}{20}</math></li>
+
<li><math>\frac{3}{5}\Big/4 = \frac{3}{5\times 4} = \frac{3}{20}</math></li>
-
<li><math>\frac{6}{7}\Big/3 = \frac{6}{7\cdot 3} = \frac{2\cdot\not{3}}{7\cdot \not{3}} = \frac{2}{7}</math></li>
+
<li><math>\frac{6}{7}\Big/3 = \frac{6}{7\times 3} = \frac{2\times\not{3}}{7\times \not{3}} = \frac{2}{7}</math></li>
</ol>
</ol>
</div>
</div>
-
När ett tal divideras med ett bråk, multipliceras talet med bråket inverterat ("uppochnervänt"). Att t.ex. dividera med <math>\frac{1}{2}</math> är ju samma sak som att multiplicera med <math>\frac{2}{1}</math> dvs. 2.
+
More generally, when a number is divided by a fraction the number is multiplied by the same fraction, inverted (that is, upside down). For example, dividing by <math>\frac{1}{2}</math> is the same as multiplying by<math>\frac{2}{1}</math>; that is, by 2.
<div class="exempel">
<div class="exempel">
-
'''Exempel 9'''
+
'''Example 9'''
<ol type="a">
<ol type="a">
-
<li><math>\frac{3}{\displaystyle \frac{1}{2}}
+
<li><math>3\div\frac{1}{2}
-
= 3\cdot \frac{2}{1} = \frac{3\cdot 2}{1} = 6</math></li>
+
= 3\times \frac{2}{1} = \frac{3\times 2}{1} = 6</math></li>
-
<li><math>\frac{5}{\displaystyle
+
<li><math>5\div\frac{3}{7} = 5\times\frac{7}{3}
-
\frac{3}{7}} = 5\cdot\frac{7}{3}
+
= \frac{5\times 7}{3} = \frac{35}{3}</math></li>
-
= \frac{5\cdot 7}{3} = \frac{35}{3}</math></li>
+
<li><math>
<li><math>
-
\frac{\displaystyle \frac{2}{3}}{\displaystyle \frac{5}{8}}
+
\frac{2}{3}\div\frac{5}{8}
-
= \frac{2}{3}\cdot \frac{8}{5} = \frac{2\cdot 8}{3\cdot 5}
+
= \frac{2}{3}\times \frac{8}{5} = \frac{2\times 8}{3\times 5}
= \frac{16}{15}</math></li>
= \frac{16}{15}</math></li>
-
<li><math>\frac{\displaystyle \frac{3}{4}}{\displaystyle \frac{9}{10}}
+
<li><math>\frac{3}{4}\div\frac{9}{10}
-
= \frac{3}{4}\cdot \frac{10}{9}
+
= \frac{3}{4}\times \frac{10}{9}
-
= \frac{\not{3}}{2\cdot\not{2}}
+
= \frac{\not{3}}{2\times\not{2}}
-
\cdot\frac{\not{2} \cdot 5}{\not{3} \cdot 3}
+
\times\frac{\not{2} \times 5}{\not{3} \times 3}
-
= \frac{5}{2\cdot 3}
+
= \frac{5}{2\times 3}
= \frac{5}{6}</math></li>
= \frac{5}{6}</math></li>
</ol>
</ol>
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</div>
</div>
-
Hur kan bråkdivision förvandlas till multiplikation? Förklaringen är att om ett bråk multipliceras med sitt inverterade bråk blir produkten alltid 1, t.ex.
+
Why is it that division by a fraction is the same as multiplication by the same fraction, upside down? The explanation is that if a fraction is multiplied by "itself upside down", 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{eller} \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>}}
-
Om man i en bråkdivision förlänger täljare och nämnare med nämnarens inverterade bråk, får man alltid 1 i nämnaren och resultatet blir täljaren multiplicerad med den ursprungliga nämnarens inverterade bråk.
+
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">
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'''Exempel 10'''
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''' Example 10'''
<math>\frac{\displaystyle \frac{2}{3}}{\displaystyle \frac{5}{7}}
<math>\frac{\displaystyle \frac{2}{3}}{\displaystyle \frac{5}{7}}
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== Bråk som andelar ==
+
== Fractions as a proportion of a whole ==
-
Rationella tal är alltså tal som kan skrivas i bråkform, omvandlas till decimalform, eller markeras på en tallinje. I vårt vardagliga språkbruk används också bråk när man beskriver andelar av något. Här nedan ges några exempel. Lägg märke till hur vi använder ordet "''av''", vilket kan betyda såväl multiplikation som 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">
-
'''Exempel 11'''
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'''Example 11'''
<ol type="a">
<ol type="a">
-
<li>Olle satsade 20 kr och Stina 50 kr.<br><br>
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<li>Jack invested 20 EUR and Jill 50 EUR. <br><br>
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Olles andel är &nbsp;<math>\frac{20}{50 + 20} = \frac{20}{70} = \frac{2}{7}</math>&nbsp; och han bör alltså få &nbsp;<math>\frac{2}{7}</math> av vinsten.</li><br><br>
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Jack´s share is &nbsp;<math>\frac{20}{50 + 20} = \frac{20}{70} = \frac{2}{7}</math>&nbsp; and he must be given &nbsp;<math>\frac{2}{7}</math> of the profits. .</li><br><br>
-
<li>Hur stor del utgör 45 kr av 100 kr? <br><br>
+
<li> What proportion is 45 EUR of 100 EUR? <br><br>
 +
'''Answer:''' 45 EUR is &nbsp;<math>\frac{45}{100} = \frac{9}{20}</math>&nbsp;of 100 EUR. .</li><br><br>
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'''Svar:''' 45 kr är &nbsp;<math>\frac{45}{100} = \frac{9}{20}</math>&nbsp; av 100 kr.</li><br><br>
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<li> What proportion is <math>\frac{1}{3}</math>litres of <math>\frac{1}{2}</math> litre? <br><br>
-
<li>Hur stor del utgör <math>\frac{1}{3}</math> liter av <math>\frac{1}{2}</math> liter? <br><br>
+
'''Answer:''' <math>\frac{1}{3}</math> litres is <math>\frac{\displaystyle \frac{1}{3}}{\displaystyle \frac{1}{2}} = \frac{1}{3}\cdot\frac{2}{1} = \frac{2}{3} </math>&nbsp; of &nbsp;<math>\frac{1}{2}</math> litres.</li><br><br>
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'''Svar:''' <math>\frac{1}{3}</math> liter är <math>\frac{\displaystyle \frac{1}{3}}{\displaystyle \frac{1}{2}} = \frac{1}{3}\cdot\frac{2}{1} = \frac{2}{3} </math>&nbsp; av &nbsp;<math>\frac{1}{2}</math> liter.</li><br><br>
+
<li>How much is &nbsp;<math>\frac{5}{8} </math>&nbsp; of 1000?<br><br>
-
<li>Hur mycket är &nbsp;<math>\frac{5}{8} </math>&nbsp; av 1000?<br><br>
+
'''Answer:''' <math>\frac{5}{8}\cdot 1000 = \frac{5000}{8} = 625</math></li><br><br>
-
'''Svar:''' <math>\frac{5}{8}\cdot 1000 = \frac{5000}{8} = 625</math></li><br><br>
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<li> How much is &nbsp;<math>\frac{2}{3}</math>&nbsp; of &nbsp;<math>\frac{6}{7}</math> ?<br><br>
-
<li>Hur mycket är &nbsp;<math>\frac{2}{3}</math>&nbsp; av &nbsp;<math>\frac{6}{7}</math> ?<br><br>
+
'''Answer:''' <math>\frac{2}{3}\cdot\frac{6}{7} = \frac{2}{\not{3}} \cdot \frac{2 \cdot \not{3}}{7} = \frac{2 \cdot 2}{7} = \frac{4}{7}</math></li>
-
 
+
-
'''Svar:''' <math>\frac{2}{3}\cdot\frac{6}{7} = \frac{2}{\not{3}} \cdot \frac{2 \cdot \not{3}}{7} = \frac{2 \cdot 2}{7} = \frac{4}{7}</math></li>
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</ol>
</ol>
</div>
</div>
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== Blandade uttryck ==
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== Mixed expressions ==
-
När bråk förekommer i räkneuttryck gäller naturligtvis metoderna för de fyra räknesätten som vanligt, samt prioriteringsreglerna (multiplikation/division före addition/subtraktion). Kom också ihåg att täljare och nämnare i ett divisionsuttryck beräknas var för sig innan divisionen utförs ("osynliga parenteser").
+
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">
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'''Exempel 12'''
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''' Example 12'''
<ol type="a">
<ol type="a">
<li><math>\frac{1}{\displaystyle \frac{2}{3}+\frac{3}{4}}
<li><math>\frac{1}{\displaystyle \frac{2}{3}+\frac{3}{4}}
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</div>
</div>
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[[1.2 Övningar|Övningar]]
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[[1.2 Exercises|Exercises]]
<div class="inforuta" style="width: 580px">
<div class="inforuta" style="width: 580px">
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'''Råd för inläsning'''
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'''Study advice'''
-
 
+
-
'''Grund- och slutprov'''
+
-
Efter att du har läst texten och arbetat med övningarna ska du göra
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'''Basic and final tests'''
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grund- och slutprovet för att bli godkänd på detta avsnitt. Du hittar
+
-
länken till proven i din student lounge.
+
 +
After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
-
'''Tänk på att:'''
 
-
Sträva alltid efter att skriva ett uttryck i enklast möjliga form. Vad
+
'''Keep in mind that...'''
-
som är "enklast" beror dock oftast på sammanhanget.
+
-
Det är viktigt att du verkligen behärskar bråkräkning. Att du kan
+
You should try to write an expression in the simplest possible terms. What is the "simplest" depends usually on the context.
-
hitta en gemensam nämnare, förkorta och förlänga etc. Principerna är
+
-
nämligen grundläggande när man ska räkna med rationella uttryck som
+
-
innehåller variabler och för att du ska kunna hantera andra
+
-
matematiska uttryck och operationer.
+
-
Rationella uttryck med bråk som innehåller variabler
+
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.
-
(''x'', ''y'', ...) är mycket vanliga när man studerar funktioner,
+
-
speciellt ändringskvoter, gränsvärden och derivata.
+
 +
Rational expressions that contain variables (x, y, ...) and include fractions are very common when studying functions, especially increment ratios, limits and derivatives.
-
'''Lästips'''
 
-
För dig som vill fördjupa dig ytterligare eller behöver en längre
+
'''Reviews'''
-
förklaring.
+
-
[http://en.wikipedia.org/wiki/Fraction_(mathematics) Läs mer om bråk och bråkräkning i engelska Wikipedia ]
+
For those of you who want to deepen your studies or need more detailed explanations consider the following references
-
[http://www.fritext.se/matte/brak/brak.html Bråkräkning - Fri text ]
+
[http://en.wikipedia.org/wiki/Fraction_(mathematics)Learn more about the fractions and calculating with fractions in the English Wikipedia ]
-
'''Länktips'''
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'''Useful web sites'''
-
[http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html Experimentera interaktivt med bråk ]
+
[http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html Experimenting interactively with fractions ]
-
[http://www.math.kth.se/~gunnarj/BIENNALEN/fall4.html Spela primtalskanonen]
+
[http://www.math.kth.se/~gunnarj/BIENNALEN/fall4.html Play the prime number canon]
-
[http://www.theducation.se/kurser/experiment/gyma/applets/ex13_brakaddition/Ex13Applet.html Här kan du få en bild av hur det går till när man lägger ihop bråk. ]
 
</div>
</div>

Current revision

       Theory          Exercises      

Contents:

  • Addition and subtraction of fractions
  • Multiplication and division of fractions

Learning outcomes:

After this section you should have learned to:

  • Calculate the value of expressions containing fractions, the four arithmetic operations and parentheses.
  • Cancel down fractions as far as possible (reduction).
  • Determine the lowest common denominator (LCD).


Fraction modification

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

\displaystyle 0\text{.}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{etc.}

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

Example 1

Multiplying numerator and denominator by the same number:

  1. \displaystyle \frac{2}{3} = \frac{2\times 5}{3\times 5} = \frac{10}{15}
  2. \displaystyle \frac{5}{7} = \frac{5\times 4}{7\times 4} = \frac{20}{28}

Dvinding numerator and denominator by the same number: (reducing or cancelling):

  1. \displaystyle \frac{9}{12} = \frac{9/3}{12/3} = \frac{3}{4}
  2. \displaystyle \frac{72}{108} = \frac{72/2}{108/2} = \frac{36}{54} = \frac{36/6}{54/6} = \frac{6}{9} = \frac{6/3}{9/3} = \frac{2}{3}

We usually specify a fraction in a form where cancellation has been performed as far as possible: this is called expressing it in its lowest terms. This can be laborious when large numbers are involved which is why, in long calculations, you should usually cancel as you go along.


Addition and subtraction of fractions

Fractions can only be added or subtracted if they have the same denominator. If they do not, they must each first be "top-and-bottom" multiplied in such a way that they do. This is called placing over a common denominator.

Example 2

  1. \displaystyle \frac{3}{5}+\frac{2}{3} = \frac{3\times 3}{5\times 3} + \frac{2\times 5}{3\times 5} = \frac{9}{15} + \frac{10}{15} = \frac{9+10}{15} = \frac{19}{15}
  2. \displaystyle \frac{5}{6}-\frac{2}{9} = \frac{5\times 3}{6\times 3} - \frac{2\times 2}{9\times 2} = \frac{15}{18} - \frac{4}{18} = \frac{15-4}{18} = \frac{11}{18}

A common denominator can always be found by simply multiplying the denominators of the two fractions together. Often, however, a smaller one can be found. The ideal is to find the lowest common denominator (LCD).


Example 3

  1. \displaystyle \frac{7}{15}-\frac{1}{12} = \frac{7\times 12}{15\times 12} - \frac{1\times 15}{12\times 15}\vphantom{\Biggl(}
    \displaystyle \insteadof{\displaystyle\frac{7}{15}-\frac{1}{12}}{}{} = \frac{84}{180}-\frac{15}{180} = \frac{69}{180} = \frac{69/3}{180/3} = \frac{23}{60}
  2. \displaystyle \frac{7}{15}-\frac{1}{12} = \frac{7\times 4}{15\times 4}- \frac{1\times 5}{12\times 5} = \frac{28}{60}-\frac{5}{60} = \frac{23}{60}
  3. \displaystyle \frac{1}{8}+\frac{3}{4}-\frac{1}{6} = \frac{1\times 4\times 6}{8\times 4\times 6} + \frac{3\times 8\times 6}{4\times 8\times 6} - \frac{1\times 8\times 4}{6\times 8\times 4}\vphantom{\Biggl(}
    \displaystyle \insteadof{\frac{1}{8}+\frac{3}{4}-\frac{1}{6}}{}{} = \frac{24}{192} + \frac{144}{192} - \frac{32}{192} = \frac{136}{192} = \frac{136/8}{192/8} = \frac{17}{24}
  4. \displaystyle \frac{1}{8}+\frac{3}{4}-\frac{1}{6} = \frac{1\times 3}{8\times 3} + \frac{3\times 6}{4\times 6} - \frac{1\times 4}{6\times 4} = \frac{3}{24} + \frac{18}{24} - \frac{4}{24} = \frac{17}{24}

If the denominators are of reasonable size, you can usually find the LCD by inspection. More generally, you can use prime factorisation.

Example 4

  1. Simplify \displaystyle \ \frac{1}{60} + \frac{1}{42}.

    Our aim is to find the smallest number that both 60 and 42 go into. First, decompose 60 and 42 into their prime factors.
    \displaystyle \eqalign{60 &= 2\times 2\times 3\times 5\cr 42 &= 2\times 3\times 7}

    For each prime factor, choose the larger power:

    \displaystyle \text{LCD} = 2\times 2\times 3\times 5\times 7 = 420\,\mbox{.}

    We then can write

    \displaystyle \frac{1}{60}+\frac{1}{42} = \frac{1\times 7}{60\times 7} + \frac{1\times 2\times 5}{42\times 2\times 5} = \frac{7}{420} + \frac{10}{420} =\frac{17}{420}\,\mbox{.}
  2. Simplify \displaystyle \ \frac{2}{15}+\frac{1}{6}-\frac{5}{18}.

    Here,
    \displaystyle \left. \eqalign{15 &= 3\times 5\cr 6&=2\times 3\cr 18 &= 2\times 3\times 3} \right\} \quad\Rightarrow\quad \text{LCD} = 2\times 3\times 3\times5 = 90\,\mbox{.}

    We then can write

    \displaystyle \frac{2}{15}+\frac{1}{6}-\frac{5}{18} = \frac{2\times 2\times 3}{15\times 2\times 3} + \frac{1\times 3\times 5}{6\times 3\times 5} - \frac{5\times 5}{18\times 5} = \frac{12}{90} + \frac{15}{90} - \frac{25}{90} = \frac{2}{90} = \frac{1}{45}\,\mbox{.}


Multiplication

When a fraction is multiplied by an integer, only the numerator is multiplied; for example, it is obvious that \displaystyle \tfrac{1}{3} multiplied by 2 is equal to \displaystyle \tfrac{2}{3}.

If two fractions are multiplied together, then the numerators are multiplied together and the denominators are multiplied together.

Example 5

  1. \displaystyle 8\times\frac{3}{7} = \frac{8\times 3}{7} = \frac{24}{7}
  2. \displaystyle \frac{2}{3}\times \frac{1}{5} = \frac{2\times 1}{3\times 5} = \frac{2}{15}

Before doing a multiplication you should always check whether you can cancel first. Note that you can cancel on both sides of the multiplication sign.

Example 6

Compare the calculations:

  1. \displaystyle \frac{3}{5}\times\frac{2}{3} = \frac{3\times 2}{5\times 3} = \frac{6}{15} = \frac{6/3}{15/3} = \frac{2}{5}
  2. \displaystyle \frac{3}{5}\times\frac{2}{3} = \frac{\not{3}\times 2}{5\times \not{3}} = \frac{2}{5}

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

Example 7

  1. \displaystyle \frac{7}{10}\times \frac{2}{7} = \frac{\not{7}}{10}\times \frac{2}{\not{7}} = \frac{1}{10}\times \frac{2}{1} = \frac{1}{\not{2} \times 5}\times \frac{\not{2}}{1} = \frac{1}{5}\times \frac{1}{1} =\frac{1}{5}
  2. \displaystyle \frac{14}{15}\times \frac{20}{21} = \frac{2 \times 7}{3 \times 5}\times \frac{4 \times 5}{3 \times 7} = \frac{2 \times \not{7}}{3 \times 5}\times \frac{4 \times 5}{3 \times \not{7}} = \frac{2}{3 \times \not{5}}\times \frac{4 \times \not{5}}{3} = \frac{2}{3}\times\frac{4}{3} = \frac{2\times 4}{3\times 3} = \frac{8}{9}


Division

If \displaystyle \tfrac{1}{4} is divided by 2 one gets the answer \displaystyle \tfrac{1}{8}. If \displaystyle \tfrac{1}{2} is divided by 5 one gets the result \displaystyle \tfrac{1}{10}. We have that

\displaystyle \frac{\displaystyle \frac{1}{4}}{2} = \frac{1}{4\times 2} = \frac{1}{8} \qquad \mbox{ and } \qquad \frac{\displaystyle \frac{1}{2}}{5} = \frac{1}{2\times 5} = \frac{1}{10}\,\mbox{.}

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

Example 8

  1. \displaystyle \frac{3}{5}\Big/4 = \frac{3}{5\times 4} = \frac{3}{20}
  2. \displaystyle \frac{6}{7}\Big/3 = \frac{6}{7\times 3} = \frac{2\times\not{3}}{7\times \not{3}} = \frac{2}{7}

More generally, when a number is divided by a fraction the number is multiplied by the same fraction, inverted (that is, upside down). For example, dividing by \displaystyle \frac{1}{2} is the same as multiplying by\displaystyle \frac{2}{1}; that is, by 2.

Example 9

  1. \displaystyle 3\div\frac{1}{2} = 3\times \frac{2}{1} = \frac{3\times 2}{1} = 6
  2. \displaystyle 5\div\frac{3}{7} = 5\times\frac{7}{3} = \frac{5\times 7}{3} = \frac{35}{3}
  3. \displaystyle \frac{2}{3}\div\frac{5}{8} = \frac{2}{3}\times \frac{8}{5} = \frac{2\times 8}{3\times 5} = \frac{16}{15}
  4. \displaystyle \frac{3}{4}\div\frac{9}{10} = \frac{3}{4}\times \frac{10}{9} = \frac{\not{3}}{2\times\not{2}} \times\frac{\not{2} \times 5}{\not{3} \times 3} = \frac{5}{2\times 3} = \frac{5}{6}

Why is it that division by a fraction is the same as multiplication by the same fraction, upside down? The explanation is that if a fraction is multiplied by "itself upside down", the product is always 1. For example,

\displaystyle \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{.}

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.

Example 10

\displaystyle \frac{\displaystyle \frac{2}{3}}{\displaystyle \frac{5}{7}} = \frac{\displaystyle \frac{2}{3}\cdot\displaystyle \frac{7}{5}}{\displaystyle \frac{5}{7}\cdot\displaystyle \frac{7}{5}} = \frac{\displaystyle \frac{2}{3}\cdot\displaystyle \frac{7}{5}}{1} = \frac{2}{3}\cdot\frac{7}{5}


Fractions as a proportion of a whole

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.

Example 11

  1. Jack invested 20 EUR and Jill 50 EUR.

    Jack´s share is  \displaystyle \frac{20}{50 + 20} = \frac{20}{70} = \frac{2}{7}  and he must be given  \displaystyle \frac{2}{7} of the profits. .


  2. What proportion is 45 EUR of 100 EUR?

    Answer: 45 EUR is  \displaystyle \frac{45}{100} = \frac{9}{20} of 100 EUR. .


  3. What proportion is \displaystyle \frac{1}{3}litres of \displaystyle \frac{1}{2} litre?

    Answer: \displaystyle \frac{1}{3} litres is \displaystyle \frac{\displaystyle \frac{1}{3}}{\displaystyle \frac{1}{2}} = \frac{1}{3}\cdot\frac{2}{1} = \frac{2}{3}   of  \displaystyle \frac{1}{2} litres.


  4. How much is  \displaystyle \frac{5}{8}   of 1000?

    Answer: \displaystyle \frac{5}{8}\cdot 1000 = \frac{5000}{8} = 625


  5. How much is  \displaystyle \frac{2}{3}  of  \displaystyle \frac{6}{7} ?

    Answer: \displaystyle \frac{2}{3}\cdot\frac{6}{7} = \frac{2}{\not{3}} \cdot \frac{2 \cdot \not{3}}{7} = \frac{2 \cdot 2}{7} = \frac{4}{7}


Mixed expressions

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").

Example 12

  1. \displaystyle \frac{1}{\displaystyle \frac{2}{3}+\frac{3}{4}} = \frac{1}{\displaystyle \frac{2\cdot 4}{3\cdot 4} + \frac{3\cdot 3}{4\cdot 3}} = \frac{1}{\displaystyle \frac{8}{12} + \frac{9}{12}} = \frac{1}{\displaystyle \frac{17}{12}} = 1\cdot\frac{12}{17} = \frac{12}{17}


  2. \displaystyle \frac{\displaystyle \frac{4}{3} - \frac{1}{6}}{\displaystyle \frac{4}{3}+\frac{1}{6}} = \frac{\displaystyle \frac{4 \cdot 2}{3 \cdot 2} - \frac{1}{6}}{\displaystyle \frac{4 \cdot 2}{3 \cdot 2} + \frac{1}{6}} = \frac{\displaystyle \frac{8}{6} - \frac{1}{6}}{\displaystyle \frac{8}{6} + \frac{1}{6}} = \frac{\displaystyle \frac{7}{6}}{\displaystyle \frac{9}{6}} = \frac{7}{\not{6}}\cdot\frac{\not{6}}{9} = \frac{7}{9}


  3. \displaystyle \frac{3-\displaystyle \frac{3}{5}}{\displaystyle \frac{2}{3}-2} = \frac{\displaystyle \frac{3 \cdot 5}{5}- \frac{3}{5}}{\displaystyle \frac{2}{3} - \frac{2 \cdot 3}{3}} = \frac{\displaystyle \frac{15}{5} - \frac{3}{5}}{\displaystyle \frac{2}{3} - \frac{6}{3}} = \frac{\displaystyle \frac{12}{5}}{-\displaystyle \frac{4}{3}} = \frac{12}{5}\cdot\left(-\frac{3}{4}\right) = -\frac{3\cdot \not{4} }{5} \cdot \frac{3}{\not{4}} = -\frac{3\cdot 3}{5} = -\frac{9}{5}


  4. \displaystyle \frac{\displaystyle\frac{1}{\frac{1}{2}+\frac{1}{3}}-\frac{3}{5} \cdot\frac{1}{3}}{\displaystyle\frac{2}{3}\big/\frac{1}{5} -\frac{\frac{1}{4}-\frac{1}{3}}{2}} = \frac{\displaystyle\frac{1}{\frac{3}{6}+\frac{2}{6}} -\frac{3\cdot1}{5\cdot3}}{\displaystyle\frac{2}{3}\cdot\frac{5}{1} -\frac{\frac{3}{12}-\frac{4}{12}}{2}} = \frac{\displaystyle \frac{1}{\displaystyle \frac{5}{6}} - \frac{1}{5}}{\displaystyle \frac{10}{3} - \frac{-\displaystyle \frac{1}{12}}{2}} \displaystyle \qquad\quad{}= \frac{\displaystyle \frac{6}{5} - \frac{1}{5}}{\displaystyle \frac{10}{3} + \frac{1}{24}} = \frac{1}{\displaystyle \frac{80}{24}+\frac{1}{24}} = \frac{1}{\displaystyle \frac{81}{24}} = \frac{24}{81} = \frac{8}{27}

Exercises


Study advice

Basic and final tests

After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.


Keep in mind that...

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.

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


Reviews

For those of you who want to deepen your studies or need more detailed explanations consider the following references

more about the fractions and calculating with fractions in the English Wikipedia


Useful web sites

Experimenting interactively with fractions

Play the prime number canon

Retrieved from "http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-ImperialCollege/index.php/1.2_Fractional_arithmetic"