1.2 Brüche
Aus Online Mathematik Brückenkurs 1
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Version vom 13:43, 10. Sep. 2008
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
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:
- \displaystyle \frac{2}{3} = \frac{2\cdot 5}{3\cdot 5} = \frac{10}{15}
- \displaystyle \frac{5}{7} = \frac{5\cdot 4}{7\cdot 4} = \frac{20}{28}
Same number division (Cancellation down):
- \displaystyle \frac{9}{12} = \frac{9/3}{12/3} = \frac{3}{4}
- \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
- \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}
- \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 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.
Example 3
- \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} - \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}
- \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} - \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 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.
Example 4
- 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 but avoid the inclusion of too many of the factors that the numbers have in common. Vorlage:Fristående formel We then can write Vorlage:Fristående formel - 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 Vorlage:Fristående formel We then can write Vorlage:Fristående formel
Multiplication
When a fraction is multiplied by an integer, only the numerator is multiplied by the integer. It is obvious that, for example, \displaystyle \tfrac{1}{3} multiplied by 2 gives \displaystyle \tfrac{2}{3}, that is,
If two fractions are multiplied with each other, then the numerators are multiplied together and the denominators are multiplied together.
Example 5
- \displaystyle 8\cdot\frac{3}{7} = \frac{8\cdot 3}{7} = \frac{24}{7}
- \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:
- \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}
- \displaystyle \frac{3}{5}\cdot\frac{2}{3} = \frac{\not{3}\cdot 2}{5\cdot \not{3}} = \frac{2}{5}
In 6b one has cancelled the 3 at an earlier stage than in 6a.
Example 7
- \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}
- \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 into 2 one gets the answer \displaystyle \tfrac{1}{8}. If \displaystyle \tfrac{1}{2} is divided into 5 one gets the result \displaystyle \tfrac{1}{10}. We have that
When a fraction is divided by an integer, the denominator is multiplied by the integer.
Example 8
- \displaystyle \frac{3}{5}\Big/4 = \frac{3}{5\cdot 4} = \frac{3}{20}
- \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
- \displaystyle \frac{3}{\displaystyle \frac{1}{2}} = 3\cdot \frac{2}{1} = \frac{3\cdot 2}{1} = 6
- \displaystyle \frac{5}{\displaystyle \frac{3}{7}} = 5\cdot\frac{7}{3} = \frac{5\cdot 7}{3} = \frac{35}{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}
- \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,
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.
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 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.
Example 11
- 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. . - What proportion is 45 EUR of 100 EUR?
Answer: 45 EUR is \displaystyle \frac{45}{100} = \frac{9}{20} of 100 EUR. . - 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. - How much is \displaystyle \frac{5}{8} of 1000?
Answer: \displaystyle \frac{5}{8}\cdot 1000 = \frac{5000}{8} = 625 - 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, 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").
Example 12
- \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}
- \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}
- \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}
- \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}
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:
Try always to write an expression in the simplest possible terms. What is the "simplest" depends usually on the context.
It is important that you really master calculations with fractions. You should be able to find a common denominator, multiply or divide numerators and denominators by suitable numbers etc. These principles are basic when you have to calculate a rational expression that includes variables and you will need them when you have to deal with other mathematical expressions and operations.
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
