1.2 Fractional arithmetic

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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

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