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Fractions


Fractions - A Definition


  • A Fraction is a number which can represent a part of a whole

    each part represents \(\frac{1}{4}\), i.e. 1 of 4 pieces

  • A Fraction can also be represented as a division where the 'top' value is known as the Numerator and the 'bottom' value as the Denominator


  • Fraction = Rational Number = \(\frac{\mbox{Integer}}{\mbox{Integer}}\)

    i.e. a rational number is defined a number which can be represented as an integer divided by an integer. As shown later on, whole numbers can be represented as fractions, e.g. 3 can be represented as 3/1 - so all whole numbers are also rational numbers.

    As an example of an irrational number, there is the well-known constant p. This cannot be accurately represented as a fraction (the figure of 22/7 which is often quoted is only an approximation).

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Fractions - Common Terminology


  • A proper fraction is one in which the numerator (top) is less than the denominator (bottom)

    Result is always less than 1

    \[ \mbox{e.g. }\frac{1}{4},\;\; \frac{1}{2},\;\; \frac{3}{4} \]

  • An improper fraction is one in which the numerator (top) is greater than the denominator (bottom)

    Result is always greater than 1

    \[ \mbox{e.g. } \;\;\frac{4}{3},\;\; \frac{7}{4},\;\; \frac{5}{2}\]

  • A mixed number is a number which explicitly consists of a whole number and a 'fractional' part (bottom)


    e.g.\(\;\; 5\frac{1}{4},\; 9\frac{1}{2},\; 7\frac{3}{4} \)

  • A reciprocal fraction is the 'Inverse' of another.

    \[\frac{3}{4} \;\;\mbox{is the reciprocal of} \;\; \frac{4}{3} \]

    \[ \frac{4}{1}\;\; \mbox{is the reciprocal of}\;\; \frac{1}{4} \]

  • A decimal fraction is the decimal form of a fraction

    \[\frac{1}{4} = 0.25 \] \[ \frac{3}{4} = 0.75\]

  • An equivalent fraction is one which represents the same number as another fraction but is expressed in different terms

    \[ \frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{16}{32} \] \[ \frac{3}{7} = \frac{6}{14} = \frac{21}{49} \]

  • A fraction in its lowest terms is a fraction which has been 'reduced to its simplest form', leaving the fraction in its lowest possible terms. In other words, there is no whole number with which you can simultaneously divide both the numerator and denominator.

    \[ \frac{3}{4}, \;\;\; \frac{3}{7} \;\;\; \frac{1}{3} \;\;\; \frac{21}{43} \]
    Note that it is more usual to state answers to calculations in their lowest terms, e.g. 15/20 and 3/4 represent the same number, but it is more usual to quote the number as 3/4.


NOTE Any fraction in which the Numerator and Denominator are the same, is always equal to one

\[ \frac{1}{1} = 1; \;\;\; \frac{2}{2} = 1; \;\;\; \frac{3}{3} = 1; \;\;\; \frac{500}{500} = 1\]


And remember : multiplying anything by 1 gives the same result

\[8 \times 1 = 8; \;\;\; 8 \times \frac{1}{1} = 8; \;\;\; 8 \times \frac{2}{2} = 8\]

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Physical Analogy


Thinking of a cake divided into equal portions could be an aid to understanding some of the principles involved.


  In order to add or subtract fractions, the denominator needs to be the same. Once this has been achieved, you just add / subtract across the top - the denominator is not added / subtracted.

If you have a cake cut into three equal parts (thirds), then two of these parts will equal 2/3. Stated mathematically

\[ \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \]

The denominator does not enter into the addition, and hopefully by thinking of the cake, you can see why this should be so.

Improper fractions like

\[ \frac{17}{5} \]

are larger than 1, and can also be expressed as a mixed fraction thus

\[ 3 \frac{2}{5} \]

Mathematically, you would achieve this conversion by dividing the numerator (i.e. the top) by the denominator (i.e. the bottom). In general, this will produce a whole number plus a remainder.

  • The whole number becomes the number in front of the 'fractional part'
  • The remainder is placed over the original denominator to produce the 'fractional part'

If you could visualize several cakes, each cut into five equal portions, then you could visualize that 17 of these portions could be provided by three full cakes plus two extra portions from another cake. This would be in line with the figure of

\[ 3 \frac{2}{5} \]

derived above.

Equivalent fractions are fractions like \(\frac{1}{2}\) and \(\frac{3}{6}\) which, although displayed differently, actually represent the same number.

Using our cake analogy again, if you imagine one cake cut up into two equal pieces, and another cut up into six equal pieces, then you can see that 1 piece of the former cake equals three of the latter, i.e. \(\frac{1}{2} = \frac{3}{6}\).

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Equivalent Fractions and Lowest Terms



You can convert a given fraction into another equivalent fraction by either multiplying or dividing top AND bottom by the same whole number, i.e. you produce a fraction which has both a different numerator and denominator but which represents the same number as the original fraction.

There are several reasons why you might want to produce equivalent fractions.

  • Comparison of fractions The magnitude of fractions can be easily compared when they have the same denominator.
  • Adding and subtracting fractions If the fractions to be added or subtracted have different denominators, then the first step is always to transform the fractions so that they are over the same denominator
  • Answers to problems are usually stated in their lowest terms. This involves carrying out a division top and bottom if the numerator and denominator have a common factor.

Hints : If you are tested

  • by asking you to choose the largest fraction from a selection. In order to compare fractions, you will first have to transform them so that they all have the same denominator.

  • by asking whether a given fraction is in its lowest terms. Here you will have to consider whether there is an integer that divides both the numerator and denominator - if so, then the fraction is not in its lowest terms.

    For example

    \[ \frac{4}{8} \]
    can be reduced to
    \[ \frac{1}{2} \]

    by dividing the numerator and denominator by 4. The fraction is now in its lowest terms, because there is no number that will divide both the numerator and denominator, apart from 1, and dividing by 1 will obviously not change the figures in the numerator or denominator.

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Multiplication of Fractions



  1. 1. Multiply the numerators together across the top, and the denominators together across the bottom
  2. 2. Reduce to simplest form


Examples

\[ \frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{ 2 \times 4} = \frac{3}{8} \;\;\mbox{which is already in its simplest form} \]

\[ \frac{2}{5} \times \frac{2}{4} = \frac{2 \times 2} {5 \times 4} = \frac{4}{20} \;\;\mbox{and then simplify : } \frac{4}{20} = \frac{1}{5} \]



In the last example above, the answer is simplified by dividing the numerator and denominator by 4. This maintains the ratio of the fraction (see Equivalent Fractions under "Fractions-Terminology"). Go to here for more information on this technique. (link inactive at the moment)

Quick Quiz     Carry out the following calculations (giving the answers in their lowest terms)
  1. \(\frac{3}{4} \times \frac{6}{7}\)
  2. \(\frac{16}{3} \times \frac{8}{12}\)
  3. \(\frac{4}{3} \times \frac{2}{7}\)
  4. \(\frac{1}{2} \times \frac{4}{5}\)
  5. \(\frac{7}{5} \times \frac{11}{3}\)
  1. \(\frac{9}{5} \times \frac{6}{5}\)
  2. \(\frac{63}{45} \times \frac{9}{7}\)
  3. \(\frac{5}{13} \times \frac{5}{6}\)
  4. \(\frac{34}{42} \times \frac{2}{7}\)



This link gives some background information on common misconceptions in multiplying fractions.(link not active at the moment)

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Multiplication of Fractions, using cancelling

When multiplying fractions in the previous section, you multiplied right across the top and right across the bottom, and then reduced the resulting fraction to its lowest terms, if that was necessary. We now introduce a 'short-cut' called cancelling - once you have mastered it, the figures involved will be smaller and the resulting calculation easier.

In the previous section, you reduced a fraction to its lowest terms by dividing a fraction top and bottom by the same whole number.

In a multiplication, you can divide the top of one fraction by a whole number and the bottom of an another fraction by the same whole number.
By analogy with reducing a fraction to its lowest terms, make sure that you divide the top of a fraction and the bottom of another by the same whole number.

Example

  • \[\frac{16}{3} \times \frac{5}{12}\]

    • Here I can 'cancel' the 16 and 12.

    Both 16 and 12 can be divided by 4, so I

    • divide the top of the left-hand fraction by 4
    • and the bottom of the right-hand fraction by 4
    to produce

  • \[\frac{4}{3} \times \frac{5}{3}\]

    • Multiplying right across the top, and right across the bottom now produces

    \[ \frac{20}{9} \]

    which is already in its lowest terms.

    So you can see the advantages of cancelling

    • the numbers to be multiplied are smaller and easier to handle
    • the answer is usually already in its lowest terms - it will be in its lowest terms if all the possible cancellations have been carried out

    Example

    \[ \frac{63}{45} \times \frac{9}{7} \]

    Cancelling top left and bottom right by 7 produces

    \[ \frac{9}{45} \times \frac{9}{1} \]

    Cancelling bottom left and top right by 9 produces

    \[ \frac{9}{5} \times \frac{1}{1} \]
    This obviously just becomes
    \[ \frac{9}{5} \]

    The whole calculation was much easier than if you had tried it without cancellation.

    Quick Quiz     Carry out the following calculations
    1. 4/3 × 6/7
    2. 6/32 × 8/12
    3. 5/36 × 24/7
    4. 3/11 × 44/21
    5. 2/25 × 5/8
    1. 9/15 × 5/18
    2. 3/45 × 5/6
    3. 5/13 × 4/5
    4. 35/42 × 2/7

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    Division of Fractions

    Method

    1. Replace the division sign by a multiplication sign, and invert the second term.
    2. Treat as a multiplication, as outlined in the previous section.
    3. Reduce to simplest form

    \[ \frac{3}{4} \div \frac{4}{5} = \frac{3}{4} \times \frac{5}{4} = \frac{(3\times 5)}{(4 × 4)} = \frac{15}{16} \]

    already in its simplest form

    \[ \frac{2}{4} \div \frac{3}{4} = \frac{2}{4} \times \frac{4}{3} = \frac{(2\times 4)}{(4\times 3)} = \frac{8}{12}\]

    (when simplified)\[ = \frac{2}{3} \]   

    In the last example, above, the answer is simplified by dividing the numerator and denominator by 4. This maintains the ratio of the fraction.(see Equivalent Fractions under "Fractions-Termiology").

    You might find it interesting to visit this link which is concerned with misconceptions arising from dividing a whole number by a fraction. Remember that a whole number can be represented as a fraction by putting it over 1, e.g. 8 = 8/1.

    And this link continues these ideas into more general situations.

    Quick Quiz     Carry out the following calculations, giving your answers in lowest terms
    1. 1. 3/4 ÷ 6/7
    2. 2. 16/3 ÷ 2/3
    3. 3. 4/3 ÷ 2/7
    4. 4. 1/2 ÷ 4/5
    5. 5. 7/5 ÷ 11/3
    1. 6. 9/5 ÷ 6/5
    2. 7. 63/45 ÷ 9/7
    3. 8. 5/13 ÷ 5/6
    4. 9. 34/42 ÷ 2/7

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    Addition of Fractions

    Method



    If the denominators are equal, then you add the fractions straightforwardly

    • Addition involves adding just the numerators

      Example

      \[ \frac{3}{7} + \frac{2}{7} = \frac{5}{7} \]

    If the denominators are different, then you are unable to add straightaway


    You have to follow these steps -

    1. Find a common denominator

    2. Convert the fractions, using the idea of equivalent fractions, such that all fractions now have this common denominator (reducing to lowest terms involves dividing top and bottom by the same number. Here we multiply top and bottom by the same number).

    3. Add the fractions

    4. Reduce to simplest form

    Examples

    \[ \frac{1}{2} + \frac{1}{2} = \frac{2}{2} = 1 \]

    Denominators the same, therefore 'straightforward'. The denominator remains unchanged and the two numerators are added.
    \[ \frac{2}{3} + \frac{1}{2} = \left(\frac{2}{3} \times \frac{2}{2}\right) + \left(\frac{1}{2} \times \frac{3}{3}\right) = \frac{4}{6} + \frac{3}{6} = \frac{7}{6} \]

    1. Lowest Common Denominator (LCD) of 2 and 3 is 6.
    2. Convert fractions - multiply 2/3 by 2 top and bottom, and multiply 1/2 by 3 top and bottom.
    3. Both fractions are now over the same denominator - add as before.

    Both this link and this link give some background information on common misconceptions in adding fractions.

    Quick Quiz     Carry out the following calculations, giving your answers in their lowest terms
    1. \[1.\ \frac{3}{8} + \frac{2}{8} \]
    2. \[2.\ \frac{1}{5} + \frac{3}{5} \]
    3. \[3.\ \frac{3}{7} + \frac{2}{7} \]
    4. \[4.\ \frac{4}{3} + \frac{2}{5} \]
    5. \[5.\ \frac{1}{2} + \frac{4}{5} \]
    1. \[6.\ \frac{2}{5} + \frac{1}{3} \]
    2. \[7.\ \frac{9}{5} + \frac{6}{5} \]
    3. \[8.\ \frac{5}{12} + \frac{1}{6} \]
    4. \[9.\ \frac{11}{42} + \frac{2}{7} \]

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    Subtraction of Fractions

    This procedure is directly analogous to addition of fractions.

    Method

    The steps are exactly the same as for Addition of Fractions, except for Step 3 where we subtract rather then add.
    1. Find a common denominator
    2. Convert the fractions
    3. Subtract the fractions
    4. Reduce to simplest form

    Examples


    \[ \frac{2}{5} - \frac{1}{5} = \frac{1}{5} \]

    Denominators the same, therefore subtract 'straightforwardly'. The denominator remains unchanged and it is only the two numerators that are involved in the subtraction process.
    \[ \frac{2}{3} - \frac{1}{4} \] \[ = \left(\frac{2}{3} \times \frac{4}{4}\right) - \left(\frac{1}{4} \times \frac{3}{3}\right) \] \[ = \frac{8}{12} - \frac{3}{12} \] \[ = \frac{5}{12} \]

    1. Lowest Common Denominator (LCD) of 3 and 4 is 12.
    2. Convert fractions
    3. Add their new values.

    Quick Quiz     Carry out the following calculations, giving your answers in their lowest terms
    1. \[1.\ \ \frac{3}{8} - \frac{2}{8} \]
    2. \[2.\ \ \frac{3}{5} - \frac{1}{5} \]
    3. \[3.\ \ \frac{3}{7} - \frac{2}{7} \]
    4. \[4.\ \ \frac{4}{3} - \frac{2}{5} \]
    5. \[5.\ \ \frac{4}{5} - \frac{1}{2} \]
    1. \[6.\ \ \frac{2}{5} - \frac{1}{3} \]
    2. \[7.\ \ \frac{9}{5} - \frac{6}{5} \]
    3. \[8.\ \ \frac{5}{12} - \frac{1}{6} \]
    4. \[9.\ \ \frac{17}{42} - \frac{2}{7} \]

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    Percentage / Decimal to Fractional Form

    • asks you to convert a percentage to a fraction, and present the answer in its lowest terms.

      The technique is

      1. Place the percentage over 100.

        For example

        If you are given 56%, you would produce

        \[ \frac{56}{100} \]

      2. and then reduce this fraction to its lowest terms, if possible, using the methods mentioned above.

    • asks you to convert a decimal to a fraction, and present the answer in its lowest terms.

      The technique is

      1. Consider how many places of decimals you have.
      2. If x represents the number of decimal places, then form a fraction with

        • the numerator being the "decimal figures"
        • the denominator being 10 to the power x.

          For example

          If you are given 0.67, you have two places of decimals, and would therefore produce

          \[ \frac{67}{100} \]

          Other examples

          0.543becomes \(\frac{543}{1000}\)
          0.4becomes\(\frac{4}{10}\)
          0.45becomes\(\frac{45}{100}\)
          0.786becomes\(\frac{786}{1000}\)
          0.985678becomes\(\frac{985678}{1000\ 000}\)

      3. and then reduce this fraction to its lowest terms, if possible, using the methods mentioned above.

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    Questions from Past Papers

    The table below shows the heights of 40 children measured to the nearest 0.1 cm.


    1.   How many children are less than 160 cm. tall ?

    • A    8
    • B    18
    • C    30
    • D    40

    2.   20% of the children are ?

    • A    less than 150 cm tall
    • B    between 150 cm and 170 cm tall
    • C    at least 170 cm tall
    • D    at least 180 cm tall

    3.   What fraction of the children are over 160 cm tall ?

    • \[ A \ \ \frac{3}{10} \]
    • \[ B \ \ \frac{9}{20} \]
    • \[ C \ \ \frac{1}{2} \]
    • \[ D \ \ \frac{11}{20} \]

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