Fractions & Divisions

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Introduction to Fractions & Divisions

In order to master the skill of dividing fractions, you need just two things – to understand the concept of reciprocity and to know how to multiply fractions. Simply put, the core concept that governs the division of fractions is to find out how many parts of the denominator makes up the numerator.

The Big Idea: Fractions & Divisions

A Simple Idea: The Reciprocal of a fraction

A fraction is simply one integer divided by another. The number which gets divided is called the numerator, while the number which it is divided by is referred to as the denominator. If either the numerator or denominator is a negative number, then the fraction itself becomes negative. Conversely, if both numerator and denominator have the same sign (meaning both are positive or both are negative) then we can think of both signs cancelling each other out, and the fraction stays positive.

When we need to find the reciprocal of a fraction, all that needs to be done is to invert the numbers. So, the numerator of the fraction becomes the denominator of the reciprocal, while the denominator of the fraction becomes the numerator of the reciprocal. The sign of a reciprocal stays unchanged, though. So, the reciprocal of a negative or positive fraction is also negative or positive respectively.

Here are two simple examples:

  • The reciprocal of \({3 \over 4}\) is \({4 \over 3}\)

  • The reciprocal of \(-{5 \over 7}\) is \(-{7 \over 5}\)

Division of a whole number by a fraction

Okay, so we have spoken about the reciprocal and how division involving fractions essentially boils down to the number of parts of the denominator that makes up the numerator. Let’s visualise this statement with an example:

2 divided by \({1 \over 3}\);

Arithmetically it is:

\(\begin{align}\frac{2}{{\left( {\frac{1}{3}} \right)}} = 2 \times 3 = 6\end{align}\)

Let’s paint this picture:

Intuitively you know that there are 6 parts of \({1 \over 3}\) that make up 2. So;

\(\begin{align}\frac{1}{3} \times 6 = 2\end{align}\);

Hence, we take the reciprocal of the denominator and multiply!

Division of fraction by another fraction

The trick in dividing one fraction by another is to first find out the reciprocal of the second fraction, and then multiply the first fraction by the reciprocal of the second fraction. Yes, that’s all there is to it. 

However, simply telling you how to divide fractions is not the Cuemath way. The “Why” behind the “What” is where the magic lies. Let’s take a trip down memory lane back to when we read about Multiplication and Division. We described division as either repeated subtraction or distribution in equal parts, we shall focus on the second description of division to make it clear as to WHY the reciprocal of the second fraction (let’s call it the divisor fraction) is taken in the first place. 

Suppose I have 10 chocolates, and I have to make groups of 2 chocolates to give away, how many people get chocolates? The answer is 5 and that can be done via repeated subtraction. The other way to frame the same problem is to say, each person is to receive an equal number of chocolates, in which case we consider the part size, which is 2.

Similarly, when we talk about the division of fractions, say \({1 \over 4}\) divided by \({1 \over 2}\), we will say, how many parts of \({1 \over 4}\) make up \({1 \over 2}\). The answer is 2. This is why we take the reciprocal and multiply.

It always helps to remember fractions as a PART OF A WHOLE. Thus, the approach to division is also partative as opposed to repeated subtractions. Here’s a little graphic to help you understand:
 

This is a pictorial representation of \({1 \over 4}\) divided by \({1 \over 2}\). What about this picture jumps straight out at you when you look at it? It’s that 2 parts of \({1 \over 4}\) makes up one half. Once again, and we can’t stress this point enough, fractions are a part of a whole and hence;

Essentially, the take away from this exercise is to consider the following statement: When it comes to the division of two fractions it’s best to look at it like; How many of the denominator make up the numerator.

Let us say you wish to divide \({3\over 4}\) by \({5 \over 7}\). This is how you would do the steps:

\(\begin{align}& \frac{{\left( {\frac{3}{4}} \right)}}{{\left( {\frac{5}{7}} \right)}}\\   &= \left( {\frac{3}{4}} \right) \times \left( {{\rm{reciprocal}}\;{\rm{of}}\;\frac{5}{7}} \right)\\   &= \left( {\frac{3}{4}} \right) \times \left( {\frac{7}{5}} \right)\\   &= \frac{{21}}{{20}}  \end{align}\)

Let us see if anything changes if one of the fractions is negative. Ok, we already told you that, nothing does, except the sign. This is how you calculate the division of  \({3\over 4}\) by \(-{5 \over 7}\).

\(\begin{align} &  \frac{{\left( {\frac{3}{4}} \right)}}{{\left( { - \frac{5}{7}} \right)}}\\   & = \left( {\frac{3}{4}} \right) \times \left( { - \frac{7}{5}} \right)\\   & =  - \frac{{21}}{{20}}   \end{align}\)

Division of a fraction by a whole number

We just saw how to divide one negative or positive fraction by another negative or positive fraction. Let us now see how to tackle the division of a positive or negative fraction by a positive or negative integer.

As earlier, you need to find the reciprocal of the whole number. The reciprocal of any whole number is 1 divided by that number. So, the reciprocal of 4 is \({1 \over 4}\), and the reciprocal of -5 is \(-{1 \over 5}\) .

Once you know this, the division of a fraction becomes very simple. Let us see three examples of this.

\(\begin{align}  &1.\;\;\;\frac{{\left( { - \frac{3}{7}} \right)}}{5} = \left( { - \frac{3}{7}} \right) \times \left( {\frac{1}{5}} \right) =  - \frac{3}{{35}}\\\\  &2.\;\;\;\frac{{\left( { - \frac{4}{3}} \right)}}{{ - 3}} = \left( {\frac{4}{3}} \right) \times \left( { - \frac{1}{3}} \right) =  - \frac{4}{9}\\\\  &3.\;\;\;\frac{{\left( { - \frac{5}{8}} \right)}}{{ - 6}} = \left( { - \frac{5}{8}} \right) \times \left( { - \frac{1}{6}} \right) = \frac{5}{{48}}  \end{align}\)

Division of a whole number by a fraction

The other way round also presents itself sometimes. When you have a whole number to be divided by a fraction, the result is simply that whole number multiplied by the reciprocal of the fraction. Let us take the 3 examples again.

\(\begin{align} 1.\;\;\;\;\,\frac{5}{{\left( { - \frac{3}{7}} \right)}} &= 5 \times \left( { - \frac{7}{3}} \right) = - \frac{{35}}{3} \\&= - \left( {11 + \frac{2}{3}} \right)\\\\  2.\;\;\; - \frac{3}{{\left( {\frac{4}{3}} \right)}} &= - 3 \times \left( {\frac{3}{4}} \right)   - \frac{9}{4}\\\\  3.\;\;\;\;\,\frac{6}{{\left( { - \frac{5}{8}} \right)}} &= - 6 \times \left( { - \frac{8}{5}} \right) = \frac{{48}}{5} \\&= 9 + \frac{3}{5}  \end{align}\)

Common mistakes or misconceptions

  • Children may simply replace the division symbol with a multiplication symbol and solve the problem.
    This happens when they try and “remember” the rule instead of understanding the concept. They know that the operation changes but do not understand why and end up blindly replacing the symbol and getting an incorrect answer.
  • Children may “cancel” common factors across the two fractions before taking the reciprocal.
    Students are expected to take the reciprocal of the second fraction and switch the division sign with multiplication. While simplifying each fraction before this step won't change the answer, simplification across the two fractions (the numerator of one and the denominator of the other) will lead to an error.

Tips and Tricks

  • Tip: Always take the reciprocal first and switch the division sign with multiplication before simplifying. That way you dont have to consider whether you are simplifying one of the given fractions of across both of them.
  • Remember, after taking the reciprocal you have a multiplication statement. Now simplification can also be done across the two fractions. E.g. if there is a common factor between the numerator of one of the fractions and the denominator of the other fraction, you can simplify them and proceed.
    E.g. \(\begin{align}\frac{5}{28}\times\frac{7}{9}\end{align}\) can be simplified to \(\begin{align}\frac{5}{4}\times\frac{1}{9}\end{align}\) before multiplying.

 

Fractions and Division
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