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Composing Fractions

Introduction

In mathematics we come across different kinds of numbers such as natural numbers, whole numbers, decimals, fractions and more. It is important to understand the composition of these numbers, i.e. how the numbers in these sets are formed. Let us learn about composition of fractions. But, before that let us recall what we mean by fractions.

What are Fractions?

A fraction is a number representing a part of a whole. The whole may be a single or a group of objects.  This means that when one whole is divided into equal parts, each part is a fraction. The whole may be a single object or a group of objects. A whole one can be divided into 2 halves or 3 thirds or 4 quarters and so on.

Types of Fractions

Before we start the discussion on composing fractions, it is important to understand some key terms and definitions that are integral to fractions. Following are the types of fractions we come across when performing various operations on fractions. 

Unit Fractions

A fraction in which the numerator is always 1 is called a unit fraction. A unit fraction is the base unit of any fraction.

Like Fractions – Fractions that have the same denominator are called like fractions. For example, the fractions, $\frac{4}{9}, \frac{13}{9}, \frac{1}{9}$ and $\frac{5}{9}$ are all like fractions having the common denominator 9. 

Unlike Fractions – Fractions with different denominators are called unlike fractions. For example, $\frac{3}{7}$ and $\frac{5}{8}$ are unlike fractions as both the fractions have different denominators. 

Proper Fractions – Fractions with the numerator less than the denominator are called proper fractions. For example, consider the fractions $\frac{1}{2}$ and $\frac{2}{3}$. In both of these fractions, the numerators are less than their respective denominators. Hence they are proper fractions.

Improper Fractions – Fractions with the numerator either equal or greater than the denominator are called improper fractions. For example, consider the fractions $\frac{5}{2}$ and $\frac{7}{3}$. In both these fractions, the numerators are greater than their respective denominators. Hence they are improper fractions.

Mixed Fractions – A combination of a whole number and a proper fraction is called a mixed fraction. Example 4$\frac{2}{3}$. 

Equivalent Fractions – A given fraction and various fractions obtained by multiplying or dividing its numerator and denominator by the same non-zero number are called equivalent fractions.

How to Compose Fractions

We can compose fractions by taking two more fractions and combining them together to form a larger fraction.

Composing Fractions using Two Halves

Let us learn how to compose fractions using two halves. We know that a whole can be represented as – 

Now let us divide it into two equal halves. We will have,

We can see that each part of the two halves of the whole part is a fraction and is denoted by 12. Hence, we can say that 2 halves when composed together make a whole. Hence, we can say that 2 halves when compose together make a whole. This means that – 

$\frac{1}{2} + \frac{1}{2} = 1$

In the above case, we can see that the numerator of both the fractions was 1 while the denominator was also the same. So, it was easy to add the two fractions and compose them into a larger fraction. 

Composing Fractions using 3 Thirds

Let us learn how to compose fractions using 3 thirds. Again, we know that a whole can be represented as – 

Now let us divide it into three equal halves. We will have,

We can see that each part of the three halves of the whole part is a fraction and is denoted by 13. Hence, we can say that 3 thirds when composed together make a whole. This means that – 

$\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = 1$

Composing Fractions using 4 Quarters

Let us learn how to compose fractions using 4 quarters. Again, we know that a whole can be represented as – 

Now let us divide it into three equal halves. We will have,

We can see that each part of the three halves of the whole part is a fraction and is denoted by $\frac{1}{4}$. Hence, we can say that 4 quarters when composed together make a whole. This means that – 

$\frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1$

Composing Fractions using Mathematical Operators

Now that we have understood how to compose simple fractions, let us understand composing fractions using different mathematical operators such as addition, subtraction, multiplication and division. 

Composing Fractions Through Addition

Suppose we have the fractions, $\frac{2}{8}, \frac{3}{8}$ and $\frac{1}{8}$ and we want to add these fractions. We can see that all three fractions have the same denominator, i.e. 8. This means that they are like fractions. So, in order to add these fractions, we will simply add their numerators, i.e. 2, 3 and 1. We will get 2 + 3 + 1 = 6. Therefore,

$\frac{2}{8} + \frac{3}{8} + \frac{1}{8} = \frac{6}{8}$

Now, let us represent each of these fractions, graphically.

The fraction $\frac{2}{8}$ will graphically be represented as 

We can see that the above fraction has 2 shaded parts out of a total of 8 equal parts.

The fraction $\frac{3}{8}$ will graphically be represented as 

We can see that the above fraction has 3 shaded parts out of a total of 8 equal parts.

The fraction $\frac{1}{8}$ will graphically be represented as 

We can see that the above fraction has 1 shaded part out of a total of 8 equal parts.

If we sum up all the shaded parts in the three fractions, we can see that we will have 2 + 3 + 1 = 6 shaded parts. This will be represented graphically as – 

The above fraction will be written as $\frac{6}{8}$ which is the result that we had obtained after we added the three fractions above.

Composing Fractions Through Subtraction

Suppose we have the fractions, $\frac{5}{8}$ and $\frac{1}{8}$ and we want to compose the fractions by finding the difference between these fractions. If we go by the steps that we have defined above for subtraction of like fractions, we can see that the two fractions have the same denominator, i.e. 8. So, in order to subtract these fractions, we will simply subtract the numerators, i.e. 5 and 1. We will get 5 – 1 = 4 Therefore,

$\frac{5}{8} – \frac{1}{8} = \frac{4}{8}$

Now, let us represent each of these fractions, graphically.

The fraction $\frac{5}{8}$ will graphically be represented as 

We can see that the above fraction has 5 shaded parts out of a total of 8 equal parts.

The fraction $\frac{1}{8}$ will graphically be represented as 

We can see that the above fraction has 1 shaded part out of a total of 8 equal parts.

If we subtract the two shaded parts in these fractions, we can see that we will have 5 – 1 = 4 shaded parts. This will be represented graphically as – 

The above fraction will be written as $\frac{4}{8}$ which is the result that we had obtained after we subtracted the two fractions above.

Composing Fractions Through Multiplication

Suppose we want to evaluate $\frac{2}{3} x \frac{4}{5}$ with the help of pictorial representation. The following steps will be followed for this purpose – 

Step 1 – We first make a rectangle which we shall use as a whole number. Let the rectangle be represented in the below manner –

Step 2 – To mark $\frac{4}{5}$ we first divide the rectangle into 5 equal horizontal parts. We will now have

We can see that each horizontal part is now one of the five parts which can also be written as $\frac{1}{5}$

Step 3 – Next, we shade four of these 5 parts as we have a numerator as 4 in our fraction. We will get  –

So, we have now plotted our first fraction. Next, we will plot our second fraction in the same rectangle to obtain our result for the multiplication of the two fractions.

Step 4 – To mark $\frac{2}{3}$, we divide the same rectangle into 3 equal parts, but this time it will be a vertical division. We will get,

We can see that every vertical part represents one out of the total three parts when divided vertically which can also be written as $\frac{1}{3}$

Step 5 – Now we shade 2 out the three vertical parts, we will get – 

Now, identify the part that has been shaded twice. We can see that the parts that have been shaded twice have two colours while the parts that have been shaded once have one colour only.

Count the number of parts that have been shaded twice. We get 8 parts out of a total of 15 parts. Therefore, we can say that

 $\frac{2}{3} x \frac{4}{5} = \frac{8}{15}$

Composing Fractions Through Division

Before understanding composing fractions through division we must recall that the division of a fraction $\frac{a}{b}$ by a non-zero fraction $\frac{c}{d}$ is defined as the product of $\frac{a}{b}$ with the multiplicative inverse of $\frac{c}{d}$.  Now, where what do we mean by the multiplicative inverse. The multiplicative inverse is the other name for the reciprocal of a fraction.

Therefore, if we two fractions say $\frac{a}{b}$ and $\frac{c}{d}$ and we want to divide the fraction $\frac{a}{b}$ by $\frac{c}{d}$, we will have 

$\frac{a}{b} ÷ \frac{c}{d} = \frac{a}{b} x \frac{d}{c} = \frac{a d}{b c}$  

For example, suppose we two fractions, say $\frac{3}{5}$ and $\frac{5}{9}$ and we want to divide the $\frac{3}{5}$ by $\frac{5}{9}$, we will have 

$\frac{3}{5} ÷ \frac{5}{9} = \frac{3}{5} x \frac{9}{5} = \frac{27}{25}$  

So, the steps involved in the division of fractions can be defined as under-  

  1. Identify the divisor and the dividend of the two fractions. This is important as we know that division is not commutative which means that a ÷ b is not equal to b ÷ a. 
  2. Take the reciprocal of the fraction from which the dividend is being dividend.
  3. Replace the divisor with its reciprocal and change the sign of division between the two fractions to the sign of multiplication.
  4. Proceed with the multiplication of fractions.
  5. Multiply your numerators to get your new numerator
  6. Multiply your denominators to get your new denominator
  7. Simplify the final fraction, if possible
  8. The fraction thus obtained is your result.

Let us understand it with an example.

Example

Divide $\frac{4}{9}$ by $\frac{2}{3}$

Solution

We have been given two fractions $\frac{4}{9}$ and $\frac{2}{3}$ and we need to find $\frac{4}{9}$ by $\frac{2}{3}$. We will proceed in accordance with the steps we have defined above. We will have,

$\frac{4}{9} ÷ \frac{2}{3}$

Here, $\frac{2}{3}$ is the divisor. Therefore we will take its reciprocal. We will get $\frac{3}{2}$

Now, we will replace the divisor with its reciprocal and change the sign of division between the two fractions to the sign of multiplication. We will get,

$\frac{4}{9}  x \frac{3}{2}$

Now, this is a case of multiplication of fractions, we will see if the numerators and the denominator have anything in common. We will get,

$\frac{4 2}{9 3}  x \frac{3}{2} = \frac{2}{3}$

Hence, $\frac{4}{9} ÷ \frac{2}{3} = \frac{2}{3}$

Solved Examples

Example 1 Maria painted 1/5 of the wall space in her room. Her brother Romil helped and painted 3/5 of the wall space. How much did they paint together? How much of the room is left unpainted?

Solution We have been given that Maria painted 1/5 of the wall space in her room. Her brother Romil helped and painted 3/5 of the wall space. We are required to find out – 

  1. How much did they paint together?
  2. How much of the room is left unpainted?

Let us find the above information one by one.

First of all, we need to understand the fractions given to us. We have,

Fraction of wall space painted by Maria = 1/5

Fraction of wall space painted by Romil = 3/5

It can be clearly seen that both the fractional values have the same denominator. Therefore the first thing to be noticed is that they are like fractions. Now in order to find out how much they painted together, we will have to find the sum of these fractions. Since they are like fractions, we just need to add the numerators of the fractions as we have learnt above. Therefore, we have,

Wall space painted by both = 1/5 + 3/5

=  (1+3)/5 = 4/5

Therefore, the wall space painted by them together = 4/5 which is the answer to the first problem. 

Now, let us consider the second problem, how much of the room is left unpainted? To find the answer to this question we need to subtract the fraction of the wall painted from the total wall pace. We have,

Here, the total wall space will be 1 as it is a whole.

Also, we have already found that the wall space painted by Maria and Romil together = 4/5

Therefore

Wall Space left to be painted =  Total wall space  – Wall space painted 

We get the unpainted space = (5 – 4)/ 5 = 1/5

Therefore, Maria and Romil painted 4/5 of the wall space together and the room space left unpainted is 1/5.

Example 2 Daniel bought 2 ½ kg sugar whereas Sam bought 3 ½ kg of sugar. Find the total amount of sugar bought by both of them.

Solution We have been given that Daniel bought 2 ½ kg of sugar whereas Sam bought 3 ½ kg of sugar. We are required to find the total amount of sugar bought by both of them.

To find the answer to the above problem, let us first understand the fraction values given to us.

Sugar bought by Daniel = 2 ½ Kg

Sugar bought by Sam = 3 ½ kg

It can be clearly seen that both the fractional values have the same denominator, i,e 2. Therefore the first thing to be noticed is that they are like fractions.

Now, to find the value of the total sugar bought by them, we will need to add the two fractions. We shall proceed with adding these fractions by adding the numerators as we have learnt above.

But, before, that, it is important to understand that the given fractions are mixed fractions. We will first need to convert them to improper fractions. Therefore,

2 ½ kg = 5/2 kg and 3 ½ kg = 7/2 kg

So the total sugar bought by both of them = Sugar bought by Daniel + Sugar bought by Sam

Now, adding their numerators we have, 5 + 7 = 12

Therefore, 5 / 2 + 7 / 2 = 12 / 2 = 6 kg

Hence, Daniel and Sam together bought 6 kg of sugar.

Key Facts and Summary 

  1. A fraction is a number representing a part of a whole. The whole may be a single or a group of objects.  
  2. A fraction in which the numerator is always 1 is called a unit fraction. A unit fraction is the base unit of any fraction.
  3. Fractions that have the same denominator are called like fractions.
  4. Fractions with different denominators are called unlike fractions.
  5. Fractions with the numerator less than the denominator are called proper fractions.
  6. Fractions with the numerator either equal or greater than the denominator are called improper fractions.
  7. A combination of a whole number and a proper fraction is called a mixed fraction.
  8. A given fraction and various fractions obtained by multiplying or dividing its numerator and denominator by the same non-zero number are called equivalent fractions.
  9. 2 halves when composed together make a whole. Similarly, 3 thirds when composed together make a whole and 4 quarters when composed together make a whole.

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