Introduction to Fractions

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Fractions and the Number Line

Imagine a container which can be either empty or full or indeed can contain any amount of liquid to mane it any level between empty and full. It is useful to have a means of noting just how full the container is.

Imagine that there are $5$ children each with one container and enough liquid (lemonade) to fill three such containers. If each child were to receive an equal amount of lemonade, how full would each container be ? Clearly each container would have some lemonade in it i.e. it would not be empty but it would not be full. It would be somewhere between empty and full. If empty were to be represented by zero and if full were to be represented by one, what kind of a number would the partly-full container represent ?

There is a need for numbers which are greater than zero and less than one. More generally, there is a need for numbers which are not equal to whole numbers (integers). Such numbers are called fractions.

Fractions are said to be Rational Numbers, symbol \( \mathbb{Q} \)


Activity / Demonstration :- Identify fractions from pie-charts or number-lines.

Fractions and the Number Line

Consider the number-line between \(0\) and \(1\) (see the figure below).

Whole.jpg


All numbers between \(0\) and \(1\) lie between the corresponding points on the line. Now consider a number which lies midway between \(0\) and \(1\) as shown on the number-line (below).

Half.jpg

This point which is equally spaced between \(0\) and \(1\) is denoted by the number \(\displaystyle \frac{1}{2}\) also written as \(1/2\) and pronounced as one half or sometimes a half.

Now consider the two points which divide the number-line between \(0\) and \(1\) into three equal portions (below).

Third.jpg


The two points are denoted by \(\displaystyle \frac{1}{3}\) (or \(1/3\)) and \(\displaystyle \frac{2}{3}\) (or \(2/3\)) and are referred to as one third ( or a third ) and two thirds.

Similarly (see below), the three points which divide the part of the number-line between \(0\) and \(1\) into four parts are called \(\displaystyle \frac{1}{4}\), \(\displaystyle \frac{2}{4}\) and \(\displaystyle \frac{3}{4}\), called one fourth, two fourths, three fourths or, more commonly one quarter, two quarters and three quarters. Note that two quarters is the same number as one half.

Quarter.jpg


Furthermore, the numbers one fifth, two fifths, three fifths and four fifths are equally spaced between zero and one (below). The number three-fifths i.e. \(\displaystyle \frac{3}{5}\) denotes a number that lies three of five equal steps between zero and one. The number \(\displaystyle \frac 35\) i.e. three fifths represents the fraction to which each of the containers is filled in the example above involving the five children.


Fifth.jpg

In general, the number \(\frac {m}{n}\) represents the number that lies \(m\) of \(n\) equal steps along the way from zero to one. For example, the fraction \(\frac{23}{29}\) twenty three twenty ninths lies at a point \(23\) of \(29\) equal intervals along the way from zero to one.

23.jpg


The number on the top of the fraction is called the numerator while the number on the bottom is called the denominator. For many cases, the word used is the same as that denoting order e.g. one third, one fifth, two sevenths, one twentieth, one twenty-seventh one hundredth. There are some cases such as halves and quarters (and possibly twenty-oneths, thirty-twoiths etc.) where alternative terminology is used.

Fractions could also be represented by amounts of circles filled in e.g. as below.

Pie.jpg

The shaded areas of the two circles represent the fractions one quarter (\(\displaystyle \frac 14\)) and three fifths (\(\displaystyle \frac 35\)).

In general, the number \(\frac {m}{n}\) represents the number that lies \(m\) of \(n\) equal steps along the way from zero to one.

Quick Quiz

Ask students to identify some fractions from pie-chart diagrams.

Fractions and Division

Consider the fraction \(\frac 35\). This fraction lies three of five equally-spaced intervals along the number-line from zero to one (see below).

Fifth.jpg

However, looking at the fraction in the context of the number-line between zero and three (Figure \(\ref{nl5b}<math>), [[File:Threefifths.jpg]] the fraction <math>\frac 35<math> lies one of '''five''' equal spacings between zero and '''three'''. Hence the fraction <math>\frac 35\) is the number \(3\) divided by the number \(5\).

Similarly, the fraction \(\frac{23}{29}\) represents the result of the division \(23 \div 29\) as this number lies one of \(29\) equal intervals between \(0\) and \(23\).

More generally, the fraction \(\frac{m}{n}\) is the result of the division \(m \div n\).

Furthermore, a division can be expressed as a fraction e.g. the division \(5 \div 8\) has the result \(\displaystyle \frac 58\).

The fraction \( \displaystyle \frac{m}{n} \) is equal to \(m\) divided by \(n\)

Worked Example 1

If Alice is aged \(9\) and Bob is aged \(7\), what fraction of Alice's age is Bob ?

Solution The fraction is \(7 \div 9\) i.e. \(\displaystyle \frac 79\).

Activity :- Applet inviting student to enter two numbers and applet will show the fraction m/n on number line and give markers to show that the relevant fraction (m/n) lies m of n equal intervals between 0 and 1 as well as 1 of n equal intervals between 0 and m.

Proper Fractions, Improper Fractions and Mixed Numbers

Consider the number line with the fractions \(\displaystyle \frac 13\) and \(\displaystyle \frac 23\) included.

Integerthirds.jpg

Fractions that lie between zero and one are referred to as proper fractions ; the numerator and denominator are both positive and the numerator is less than the denominator. However, factions can lie at the ends of, or indeed outside, the interval zero to one.

The fraction \(\displaystyle \frac 03\) is equal to zero and the fraction \(\displaystyle \frac 33\) is equal to one; these two fractions lie at the ends of the interval zero to one.

The fractions \(\displaystyle \frac 43\) (four thirds) and \(\displaystyle \frac 53\) (five thirds) lie at points dividing the interval from one to two into three equal parts. These are referred to as improper fractions i.e. the numerator is greater than the denominator.

The fractions \(\displaystyle \frac{-2}{3}\) and \(\displaystyle \frac{-1}{3}\) lie between \(-1\) and \(0\) and divide the interval between \(-1\) and \(0\) into three equal parts. See the figure below

Improperthirds.jpg

So, fractions can exist that are negative and fractions can exist that are greater than one.

Note that the fraction \(\displaystyle \frac n1\) is equal to \(n\) for any numerator \(n\) e.g. \(\displaystyle \frac 71 = 7\), \(\displaystyle \frac {238}{1} = 238\).

Improper fractions can also be expressed in a different form i.e. that of a mixed number. Consider the fraction \(\displaystyle \frac 43\); this fraction lies between \(1\) and \(2\) and is greater than one by the amount \(\displaystyle \frac 13\). So, the improper fraction \(\displaystyle \frac 43\) can be written as \(\displaystyle 1 + \frac{1}{3}\) and hence as \(\displaystyle 1 \frac{1}{3}\). A mixed number consists of an integer part and a fractional part consisting of a numerator and a denominator. For example the mixed number \(\displaystyle 4 \frac 27\) has an integer part of \(4\) and a fractional part of \(\displaystyle \frac 27\) with numerator \(2\) and denominator \(7\).

Numbers such as \(\displaystyle 1 \frac 13\) or \(\displaystyle 2 \frac 25\) are called mixed numbers. They are equal to the improper fractions \(\displaystyle \frac 43\) and \(\displaystyle \frac{12}{5}\) respectively.

Improper fractions and mixed numbers are useful in different circumstances; improper fractions are useful when fractions are multiplied and divided whereas mixed numbers can be more useful for addition and subtraction and also to get a 'feel' for the size of a fraction.

\( \displaystyle \frac{m}{n} \) with \(m < n\) and \(m , n > 0 \) is known as a proper fraction.

\( \displaystyle \frac{m}{n} \) with \(m > n\) and \(m , n > 0 \) is known as an improper fraction.

\( \displaystyle p\frac{m}{n} \) is known as a mixed number. It is the sum of the integer \( p \) and the proper fraction \( \displaystyle \frac{m}{n} \)

Conversions between improper fractions and mixed numbers

Improper fractions and be converted to mixed numbers and vice-versa.

To convert a mixed number to an improper fraction, write the numerator as the original numerator plus the product of the denominator and the integer part; the denominator stays the same.

Worked example 2

Write the mixed number \(\displaystyle 4 \frac 27\) as an improper fraction.

Solution In \(\displaystyle 4 \frac 27\), the integer part is \(4\), the numerator \(2\) and the denominator \(7\). The fraction becomes an improper fraction with numerator equal to \(2 + 4 \times 7 = 30\) and denominator equal to \(7\). Hence \(\displaystyle 4 \frac 27 = \frac{30}{7}\).

To convert an improper fraction to a mixed number, divide the denominator into the numerator. The quotient will become the integer part and the remainder will be the numerator in the fractional part. The denominator in the mixed number will be the same as the denominator in the improper fraction.

Worked example 3

Write the improper fraction \(\displaystyle \frac{19}{6}\) as a mixed number.

Solution :- The numerator is \(19\) and the denominator is \(6\). Dividing \(19\) by \(6\) gives a quotient of \(3\) and a remainder of \(1\). Hence the integer part is \(3\) and the numerator in the fractional part is \(1\). Hence \(\displaystyle \frac{19}{6} = 3 \frac 16\).

Equivalent Fractions

It was noted that the fractions \(\displaystyle \frac 12\) and \(\displaystyle \frac 24\) are really the same fraction. The fraction \(\displaystyle \frac 24\) can be written as \[ \frac 24 = \frac {1 \times 2}{2 \times 2} = \frac {1}{2} \times \frac{2}{2} = \frac{1}{2} \times 1 = \frac{1}{2} \]

Similarly, the following fractions are also equal to \(\displaystyle \frac 12\) , \(\displaystyle \frac 36\) , \(\displaystyle \frac 48\) , \(\displaystyle \frac 7{14}\) , \(\displaystyle \frac {10}{20}\) , \(\displaystyle \frac {26}{52}\) , \(\displaystyle \frac {1000}{2000}\).

The following fractions are equal to \(\displaystyle \frac 13\) , \(\displaystyle \frac 26\) , \(\displaystyle \frac 4{12}\) , \(\displaystyle \frac 5{15}\) , \(\displaystyle \frac {10}{30}\) , \(\displaystyle \frac {32}{96}\) , \(\displaystyle \frac {1000}{3000}\)

The following fractions are equal to \(\displaystyle \frac 25\) , \(\displaystyle \frac 4{10}\) , \(\displaystyle \frac {10}{25}\) , \(\displaystyle \frac {12}{30}\) , \(\displaystyle \frac {32}{80}\) , \(\displaystyle \frac {1000}{2500}\)


In general, multiplying the top and bottom (numerator and denominator) of a fraction by the same number leaves the fraction unchanged.

Worked Example 4

Consider the fraction \(\displaystyle \frac 37\). Write this fraction with a denominator of \(35\)..

Solution The denominator of the original fraction is \(7\) but it is desired to write the fraction with a denominator of \(35\) i.e. \(5\) times the original denominator. This is achieved by multiplying both the numerator and denominator by \(5\) i.e.

\[ \frac 37 = \frac{5 \times 3}{5 \times 7} = \frac{15}{35} \]

The fractions \(\displaystyle \frac 37\) and \(\displaystyle \frac{15}{35}\) (and indeed \(\displaystyle \frac{21}{49}\) etc.) are the same fraction but the form \(\displaystyle \frac 37\) has special significance as it is the form with the lowest numbers involved. Similarly, although \(\displaystyle \frac 45\) and \(\displaystyle \frac {8}{10}\) are equal, the form \(\displaystyle \frac 45\) has special significance as this is the form with the lowest terms involved.

The form with the lowest numbers involved is said to be a fraction in its lowest terms. The convention is that, in the absence of any reason to the contrary, any fraction is presented in its lowest terms. Sometimes, when a fraction is presented in its lowest terms, the numerator and denominator will be prime numbers; sometimes one of them will be a prime number and the other one not so while on other occasions neither the numerator or the denominator will be a prime number although they will have no factor in common.

To convert a fraction to its lowest terms, identify any common factors in the numerator and the denominator and cancel them. Sometimes there may be several different factors to cancel. Sometimes it can help to write the numerator and denominator as products of factors.

Worked Example 5

Convert the following fractions to their lowest terms or state that they are in the lowest terms already.

\(\displaystyle \frac{7}{11}\) , \(\displaystyle \frac{5}{21}\) , \(\displaystyle \frac{22}{36}\) , \(\displaystyle \frac{10}{21}\) , \(\displaystyle \frac{12}{42}\) , \(\displaystyle \frac{312}{468}\).

Solution :-

\(\displaystyle \frac{7}{11}\) : As both the numerator and denominator are prime there is no simplification that can take place. The fraction is already in its lowest terms.

\(\displaystyle \frac{5}{21}\) : The numerator is a prime number so no further simplification can be carried out even though the denominator is not a prime. This fraction is already in its lowest terms.

\(\displaystyle \frac{22}{36}\) : Here, both the numerator and the denominator contain a factor of two so \[ \frac{22}{36} = \frac{2 \times 11}{2 \times 18} = \frac{11}{18} \nonumber \] As the new numerator is a prime number, no further simplification can take place. So \(\displaystyle \frac{22}{36} = \frac{11}{18}\).

\(\displaystyle \frac{10}{21}\) : This fraction can be written as \(\displaystyle \frac{2 \times 5}{3 \times 7}\). Although neither the numerator or the denominator is a prime, none of their factors are common. So, \(\displaystyle \frac{10}{21}\) is already in its lowest terms.

\(\displaystyle \frac{12}{42}\) : Both numerator and denominator contain a factor of two so, canceling this factor of two, \(\displaystyle \frac{12}{42} = \frac{6}{21}\). However, both numerator and denominator now contain a factor of three. Canceling this factor of three, \(\displaystyle \frac{12}{42} = \frac{6}{21} = \frac 27\). As the numerator and denominator are now primes, no further simplification can take place so \(\displaystyle \frac{12}{42} = \frac 27\). Of course, it could also be noted from the beginning that both numerator and denominator contain a factor of \(6\) and have this factor cancel.

\(\displaystyle \frac{312}{468}\). This looks a trickier one from the outset but some cancelation can be noted from the beginning. Both numerator and denominator contain a factor of two so \(\displaystyle \frac{312}{468} = \frac{2 \times 156}{2 \times 234} = \frac{156}{234}\). Once again, top and bottom contain a factor of two so the fraction reduces to \(\displaystyle \frac{156}{234} = \frac{2 \times 78}{2 \times 117} = \frac{78}{117}\). It is now possible to spot that a factor of three exists in both terms so \(\displaystyle \frac{78}{117} = \frac{3 \times 26}{3 \times 39} = \frac{26}{39}\). Now it can be spotted that there is a factor of \(13\) in both numerator and denominator so \(\displaystyle \frac{26}{39} = \frac{13 \times 2}{13 \times 3} = \frac 23\). So, after several rounds of calculation, it can be seen that \(\displaystyle \frac{312}{468} = \frac 23\).

An alternative way to approach this final part would be to write both the numerator and denominator as a product of all factors and cancel all the common factors i.e. \[ \frac{312}{468} = \frac{2 \times 2 \times 2 \times 3 \times 13}{2 \times 2 \times 3 \times 3 \times 13} = \frac{2}{3} \]

Demonstration / Activity :- get students to identify two equivalent fracfrom a selection.


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