Introduction To Fractions

Have you ever broken a bar of chocolate and shared it with a friend? How many friends did you share it with? One, two, may be more. So you did not have the whole chocolate, you had part of it and gave part of it your friend or friends. Let us say you have chocolate bar of similar shape below:

Count the number of equal sized tablets that are there in the chocolate. There are a total of $18$ equal sized portion in it. Let us say you cousin has comes to visit you, and you want to share the chocolate with him. You may break your chocolate through the middle across the dotted line like shown below, and give one part to him and keep the rest for yourself.

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You would normally say, $\unicode{x201C}\text{Here, you can have half of my chocolate}\unicode{x201D}$. And, you would be perfectly right. What you gave him was $1$ of $2$ equal parts. That is a fraction of your chocolate and is written as $\dfrac{1}{2}$

Instead, if you had broken your chocolate into $18$ smaller tablets and given your friend $9$ tablets, you would have given him exactly the same quantity as before. But this time you write it as $\dfrac{9}{18}$ , but you know that this is exactly same as what you gave him before. How? We will see in just a little while.

Now, let us consider one more case. Let us say you have two chocolate and you gave one whole chocolate and half of the other one to your friend like shown below.

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What you have given your friend is called $\unicode{x201C}\text{One and a half}\unicode{x201D}$ chocolates and this is written as $1\dfrac{1}{2}$. As you can see this is exactly same as $1 + \dfrac{1}{2}$.

Now let us look at different types of fraction:

Fractions where the numerator is less than the denominator, like $\dfrac{1}{2}$ , $\dfrac{3}{4}$ , $\dfrac{2}{3}$ , etc are called:

$Proper\ Fractions$

Fractions where the numerator is greater than the denominator, like $\dfrac{5}{3}$ , $\dfrac{7}{4}$ , $\dfrac{10}{7}$ are called:

$Improper\ Fraction$

Fractions that contain an integer value and a fraction part, like $1\dfrac{1}{2}$ , $4\dfrac{2}{3}$, $7\dfrac{3}{5}$ are called:

$Mixed\ Fraction$

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Fraction Reduction:

Now, let's say instead of making $2$ equal sized parts, you make $3$ equal sized parts of the chocolate like shown below and give $2$ of them to your friend, you would have given him two third of your chocolate, which is written as $\dfrac{2}{3}$

Like this your friend's share contains $12$ small tablets. But you could have given him exactly the same quantity by breaking the chocolate into $18$ small tablets and giving him $12$ out of $18$ small tablets. This time the fraction would be $\dfrac{12}{18}$

So, we see that $\dfrac{12}{18} = \dfrac{2}{3}$. How does this happen?

One very important property of a fraction is as follows:

$\unicode{x201C}\text{When you multiply the numerator and denominator by the same number, other than zero, the value of the fraction does not change}\unicode{x201D}$

So, let us go back to our fraction $\dfrac{12}{18}$

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We can see that both the numerator and denominator are divisible by $2$ and $3$. So, we divide both the numerator and the denominator first by $2$ and then by $3$.

Dividing by $2$, we get:

$\dfrac{\cancel{12}\raise{5pt}{6}}{\cancel{18}\lower{5pt}{9}} = \dfrac{6}{9}$

then dividing again by $3$ we get,

$\dfrac{\cancel{6} \raise{5pt}{2}}{\cancel{9} \lower{5pt}{3}} = \dfrac{2}{3}$

This is how we get

As you will see when we divide we are striking out the original numerator and denominator and writing the new ones.

This is how we get $\dfrac{12}{18} = \dfrac{2}{3}$

We will take a look at one more example of reduction. Remember our chocolate sharing problem, where we saw that 9 out of 18 tablets was same as half of the chocolate. This is how it goes:

First we cancel out 3 from the numerator and denominator, to get:

$\dfrac{\cancel{9} \raise{5pt}{3}}{\cancel{18} \lower{5pt}{6}} = \dfrac{3}{6}$

Then we cancel out one more $3$ from numerator and denominator, to get:

$\dfrac{\cancel{3} \raise{5pt}{1}}{\cancel{6} \lower{5pt}{2}} = \dfrac{1}{2}$

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Thus:

$\dfrac{9}{18} = \dfrac{1}{2}$

Converting Improper Fraction To Mixed:

Remember that in improper fraction the numerator is greater than the denominator? We will see, how to convert this into mixed fraction.

This is done by a simple long division. All you need to do is divide the numerator by the denominator, and write the mixed fraction as:

$quotient \dfrac{remainder}{divisor}$

We will try an example, and we will convert the improper fraction $\dfrac{19}{7}$ to a mixed fraction. So let us divide $19$ by the divisor $7$:

$\begin{array}{rl} \phantom{00}2 & \Leftarrow & \hbox{The quotient is 2}\\[-5pt]7 \enclose{longdiv} {19} \\ \underline{14} \\ 5 & \Leftarrow & \hbox{The remainder is 5}\end{array}$

Now we write the values of the quotient, remainder and divisor as we have shown before and get:

$2\dfrac{5}{7}$

Thus:

$\dfrac{19}{7} = 2\dfrac{5}{7}$

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Converting Mixed Fraction to Improper Fraction:

This is exactly the reverse step of Improper to Mixed conversion. Here we have a mixed fraction of in the form:

$\text{Integer}\dfrac{\text{numerator}}{\text{denominator}}$

We convert this to improper fraction by using:

$\text{New numerator = Integer} \times \text{denominator + numerator}$ , and our improper fraction would be:

$\dfrac{New\ numerator}{denominator}$

Remember when you convert, either a reduced mixed fraction to improper fraction or the other way, the denominator does not change. We will take and example by converting $3\dfrac{7}{11}$ to improper fraction.

Our new numerator = $3 \times 11 + 7 = 33 + 7 = 40$. Hence our improper fraction is:

$\dfrac{40}{11}$

Thus:

$3\dfrac{7}{11} = \dfrac{40}{11}$