1.2 Fractional arithmetic

From Förberedande kurs i matematik 1

(Difference between revisions)
Jump to: navigation, search
m (Robot: Automated text replacement (-{{Fristående formel +{{Displayed math))
Current revision (13:17, 30 December 2008) (edit) (undo)
(Partial edit by Phil Ramsden: more to do. Idiomatic English, localisation of terminology and notation.)
 
(3 intermediate revisions not shown.)
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 the value of 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\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 by the same number. The division operation is called reduction or cancellation.
<div class="exempel">
<div class="exempel">
''' Example 1'''
''' Example 1'''
-
Same number multiplication:
+
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>
-
Same number division (Cancellation down):
+
Dvinding numerator and denominator by the same number: (reducing or cancelling):
<ol type="a" start="3">
<ol type="a" start="3">
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.
+
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 ==
==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.
+
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">
Line 62: Line 62:
<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>
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.
+
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).
Line 79: Line 79:
<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}
Line 86: Line 86:
<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}
Line 99: Line 99:
<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>
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.
+
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">
Line 113: Line 114:
<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.
+
 
-
{{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>}}
+
Our aim is to find the smallest number that both 60 and 42 go into. First, decompose 60 and 42 into their prime factors.
 +
{{Displayed math||<math> \eqalign{60 &= 2\times 2\times 3\times 5\cr 42 &= 2\times 3\times 7}</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
We then can write
-
{{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>}}
+
{{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> 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
+
Here,
-
{{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\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>}}
We then can write
We then can write
-
{{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>}}
+
{{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>}}
</li>
</li>
</ol>
</ol>
Line 132: Line 137:
== 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; for example, it is obvious that <math>\tfrac{1}{3}</math> multiplied by 2 is equal to <math>\tfrac{2}{3}</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 together, then the numerators are multiplied together and the denominators are multiplied together.
<div class="exempel">
<div class="exempel">
''' Example 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>
-
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 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">
''' Example 6'''
''' Example 6'''
Line 151: Line 154:
Compare the calculations:
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>
-
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">
'''Example 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>
Line 180: Line 183:
== 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\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>}}
-
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">
'''Example 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>
-
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.
+
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">
'''Example 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>
Line 218: Line 220:
</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,
+
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,
{{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 239:
== 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 268:
== 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 331: Line 333:
-
'''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, 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.

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"