1.2 Fractional arithmetic

From Förberedande kurs i matematik 1

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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|
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'''Innehåll:'''
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'''Contents:'''
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* Addition och subtraktion av bråktal
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* Addition and subtraction of fractions
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* Multiplikation och division av bråktal
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* Multiplication and division of fractions
}}
}}
{{Info|
{{Info|
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'''Lärandemål:'''
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'''Learning outcomes:'''
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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.
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*Calculate expressions containing fractions, the four arithmetic operations and parentheses.
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*Förkorta bråk så långt som möjligt.
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*Cancel down fractions as far as possible (reduction).
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*Bestämma minsta gemensamma nämnare (MGN).
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*Determine the lowest common denominator (LCD).
}}
}}
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==Förlängning och förkortning==
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== Fraction modification ==
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Ett rationellt tal kan skrivas på många sätt, beroende på vilken nämnare man väljer att använda. Exempelvis har vi att
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A rational number can be written in many ways depending on the denominator one chooses to use. For example, we have that
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{{Fristående formel||<math>0{,}25 = \frac{25}{100} = \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16}\quad\textrm{osv.}</math>}}
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{{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>}}
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Värdet av ett rationellt tal ändras inte när man multiplicerar eller dividerar täljare och nämnare med samma tal. Dessa operationer kallas ''förlängning'' respektive ''förkortning''.
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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.
<div class="exempel">
<div class="exempel">
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'''Exempel 1'''
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''' Example 1'''
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Förlängning:
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Same number multiplication:
<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\cdot 5}{3\cdot 5} = \frac{10}{15}</math></li>
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</ol>
</ol>
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Förkortning:
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Same number division (Cancellation down):
<ol type="a" start="3">
<ol type="a" start="3">
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</div>
</div>
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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.
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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.
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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.
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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">
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'''Exempel 2'''
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''' Example 2'''
<ol type="a">
<ol type="a">
<li><math>\frac{3}{5}+\frac{2}{3}
<li><math>\frac{3}{5}+\frac{2}{3}
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</div>
</div>
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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.
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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.
<div class="exempel">
<div class="exempel">
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'''Exempel 3'''
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'''Example 3'''
<ol type="a">
<ol type="a">
<li><math>\frac{7}{15}-\frac{1}{12}
<li><math>\frac{7}{15}-\frac{1}{12}
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</div>
</div>
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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.
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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.
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<div class="exempel">
<div class="exempel">
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'''Exempel 4'''
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'''Example 4'''
<ol type="a">
<ol type="a">
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<li>Beräkna <math>\ \frac{1}{60} + \frac{1}{42}</math>.<br/><br/>
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<li>Simplify <math>\ \frac{1}{60} + \frac{1}{42}</math>.<br/><br/>
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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
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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.
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{{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>}}
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{{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>}}
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Vi kan då skriva
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We then can write
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{{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>}}
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{{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>
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<li>Beräkna <math>\ \frac{2}{15}+\frac{1}{6}-\frac{5}{18}</math>.<br/><br/>
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<li> Simplify <math>\ \frac{2}{15}+\frac{1}{6}-\frac{5}{18}</math>.<br/><br/>
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Minsta gemensamma nämnare väljs så att den innehåller precis så
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The lowest common denominator is chosen so that it contains just enough primes in order to be divisible by 15, 6 and 18
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många primtalsfaktorer så att den blir delbar med 15, 6 och 18
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{{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>}}
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{{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>}}
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We then can write
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Vi kan då skriva
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{{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>}}
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{{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>}}
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</li>
</li>
</ol>
</ol>
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== Multiplikation ==
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== Multiplication ==
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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.
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{{Fristående formel||<math>\frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}</math>}}
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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>,
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Om två bråk multipliceras med varandra, multipliceras täljarna med varandra och nämnarna med varandra.
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{{Displayed math||<math>\frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}</math>}}
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If two fractions are multiplied with each other, then the numerators are multiplied together and the denominators are multiplied together.
<div class="exempel">
<div class="exempel">
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'''Exempel 5'''
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''' 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\cdot\frac{3}{7} = \frac{8\cdot 3}{7} = \frac{24}{7}</math></li>
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</div>
</div>
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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.
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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.
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<div class="exempel">
<div class="exempel">
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'''Exempel 6'''
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''' Example 6'''
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Jämför uträkningarna:
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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}\cdot\frac{2}{3} = \frac{3\cdot 2}{5\cdot 3} = \frac{6}{15} = \frac{6/3}{15/3} = \frac{2}{5}</math></li>
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</div>
</div>
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Att stryka treorna i 6b innebär ju bara att man förkortar bråket med 3 i ett tidigare skede.
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In 6b the 3 has been cancelled at an earlier stage than in 6a.
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<div class="exempel">
<div class="exempel">
-
'''Exempel 7'''
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'''Example 7'''
<ol type="a">
<ol type="a">
<li><math>\frac{7}{10}\cdot \frac{2}{7}
<li><math>\frac{7}{10}\cdot \frac{2}{7}
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== Division ==
== Division ==
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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
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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
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{{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>}}
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{{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>}}
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När ett bråk divideras med ett heltal, multipliceras alltså nämnaren med heltalet.
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When a fraction is divided by an integer the denominator is multiplied by the integer.
<div class="exempel">
<div class="exempel">
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'''Exempel 8'''
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'''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\cdot 4} = \frac{3}{20}</math></li>
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</div>
</div>
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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.
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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">
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'''Exempel 9'''
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'''Example 9'''
<ol type="a">
<ol type="a">
<li><math>\frac{3}{\displaystyle \frac{1}{2}}
<li><math>\frac{3}{\displaystyle \frac{1}{2}}
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</div>
</div>
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Hur kan bråkdivision förvandlas till multiplikation? Förklaringen är att om ett bråk multipliceras med sitt inverterade bråk blir produkten alltid 1, t.ex.
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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,
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{{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>}}
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{{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>}}
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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.
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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 ==
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== Fractions as a proportion of a whole ==
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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.
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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">
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'''Exempel 11'''
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'''Example 11'''
<ol type="a">
<ol type="a">
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<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>
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<li>Hur stor del utgör 45 kr av 100 kr? <br><br>
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<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>
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<li>Hur stor del utgör <math>\frac{1}{3}</math> liter av <math>\frac{1}{2}</math> liter? <br><br>
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'''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>
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<li>How much is &nbsp;<math>\frac{5}{8} </math>&nbsp; of 1000?<br><br>
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<li>Hur mycket är &nbsp;<math>\frac{5}{8} </math>&nbsp; av 1000?<br><br>
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'''Answer:''' <math>\frac{5}{8}\cdot 1000 = \frac{5000}{8} = 625</math></li><br><br>
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'''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>
+
</ol>
</ol>
</div>
</div>
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== Blandade uttryck ==
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== Mixed expressions ==
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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").
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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">
-
'''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'''
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'''Grund- och slutprov'''
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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
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länken till proven i din student lounge.
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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.
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'''Tänk på att:'''
 
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Sträva alltid efter att skriva ett uttryck i enklast möjliga form. Vad
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'''Keep in mind that...'''
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som är "enklast" beror dock oftast på sammanhanget.
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Det är viktigt att du verkligen behärskar bråkräkning. Att du kan
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You should try to write an expression in the simplest possible terms. What is the "simplest" depends usually on the context.
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hitta en gemensam nämnare, förkorta och förlänga etc. Principerna är
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nämligen grundläggande när man ska räkna med rationella uttryck som
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innehåller variabler och för att du ska kunna hantera andra
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matematiska uttryck och operationer.
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Rationella uttryck med bråk som innehåller variabler
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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.
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(''x'', ''y'', ...) är mycket vanliga när man studerar funktioner,
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speciellt ändringskvoter, gränsvärden och derivata.
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 +
Rational expressions that contain variables (x, y, ...) and include fractions are very common when studying functions, especially increment ratios, limits and derivatives.
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'''Lästips'''
 
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För dig som vill fördjupa dig ytterligare eller behöver en längre
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'''Reviews'''
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förklaring.
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[http://en.wikipedia.org/wiki/Fraction_(mathematics) Läs mer om bråk och bråkräkning i engelska Wikipedia ]
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For those of you who want to deepen your studies or need more detailed explanations consider the following references
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[http://www.fritext.se/matte/brak/brak.html Bråkräkning - Fri text ]
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[http://en.wikipedia.org/wiki/Fraction_(mathematics)Learn more about the fractions and calculating with fractions in the English Wikipedia ]
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'''Länktips'''
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'''Useful web sites'''
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[http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html Experimentera interaktivt med bråk ]
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[http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html Experimenting interactively with fractions ]
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[http://www.math.kth.se/~gunnarj/BIENNALEN/fall4.html Spela primtalskanonen]
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[http://www.math.kth.se/~gunnarj/BIENNALEN/fall4.html Play the prime number canon]
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[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. ]
 
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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 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 with the same number. The division operation is called cancellation.

Example 1

Same number multiplication:

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

Same number division (Cancellation down):

  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}

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

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.

Example 2

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

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.


Example 3

  1. \displaystyle \frac{7}{15}-\frac{1}{12} = \frac{7\cdot 12}{15\cdot 12} - \frac{1\cdot 15}{12\cdot 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\cdot 4}{15\cdot 4}- \frac{1\cdot 5}{12\cdot 5} = \frac{28}{60}-\frac{5}{60} = \frac{23}{60}
  3. \displaystyle \frac{1}{8}+\frac{3}{4}-\frac{1}{6} = \frac{1\cdot 4\cdot 6}{8\cdot 4\cdot 6} + \frac{3\cdot 8\cdot 6}{4\cdot 8\cdot 6} - \frac{1\cdot 8\cdot 4}{6\cdot 8\cdot 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\cdot 3}{8\cdot 3} + \frac{3\cdot 6}{4\cdot 6} - \frac{1\cdot 4}{6\cdot 4} = \frac{3}{24} + \frac{18}{24} - \frac{4}{24} = \frac{17}{24}

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.

Example 4

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

    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.
    \displaystyle \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{.}

    We then can write

    \displaystyle \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{.}
  2. Simplify \displaystyle \ \frac{2}{15}+\frac{1}{6}-\frac{5}{18}.

    The lowest common denominator is chosen so that it contains just enough primes in order to be divisible by 15, 6 and 18
    \displaystyle \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{.}

    We then can write

    \displaystyle \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{.}


Multiplication

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

\displaystyle \frac{1}{3}\cdot 2 = \frac{1\cdot 2}{3} = \frac{2}{3}\,\mbox{.}

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

Example 5

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

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.

Example 6

Compare the calculations:

  1. \displaystyle \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}
  2. \displaystyle \frac{3}{5}\cdot\frac{2}{3} = \frac{\not{3}\cdot 2}{5\cdot \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}\cdot \frac{2}{7} = \frac{\not{7}}{10}\cdot \frac{2}{\not{7}} = \frac{1}{10}\cdot \frac{2}{1} = \frac{1}{\not{2} \cdot 5}\cdot \frac{\not{2}}{1} = \frac{1}{5}\cdot \frac{1}{1} =\frac{1}{5}
  2. \displaystyle \frac{14}{15}\cdot \frac{20}{21} = \frac{2 \cdot 7}{3 \cdot 5}\cdot \frac{4 \cdot 5}{3 \cdot 7} = \frac{2 \cdot \not{7}}{3 \cdot 5}\cdot \frac{4 \cdot 5}{3 \cdot \not{7}} = \frac{2}{3 \cdot \not{5}}\cdot \frac{4 \cdot \not{5}}{3} = \frac{2}{3}\cdot\frac{4}{3} = \frac{2\cdot 4}{3\cdot 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\cdot 2} = \frac{1}{8} \qquad \mbox{ and } \qquad \frac{\displaystyle \frac{1}{2}}{5} = \frac{1}{2\cdot 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\cdot 4} = \frac{3}{20}
  2. \displaystyle \frac{6}{7}\Big/3 = \frac{6}{7\cdot 3} = \frac{2\cdot\not{3}}{7\cdot \not{3}} = \frac{2}{7}

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

Example 9

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

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,

\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

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