Introduction
We have learnt that a fraction is a number representing a part of a whole. The whole may be a single or a group of objects. Let us now learn about the addition and subtraction of like fractions. But, before that, we must recall what we mean by 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.
Fraction in its Simplest Form
It is also important to recall that a fraction is said to be in the standard form if the denominator is positive and the numerators have no common divisor other than 1.
In order to express a given fraction in the standard form, the following steps should be followed –
Step 1 – Check whether the given number is in the form of a fraction.
Step 2 – See whether the denominator of the fraction is positive or not. If it is negative, multiply or divide the numerator as well as the denominator by -1 so that the denominator becomes positive.
Step 3 – Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator.
Step 4 – Divide the numerator and the denominator of the given fraction by the GCD (HCF) obtained in step III. The fraction so obtained is the standard form of the given fraction.
This was important to understand because, after the addition or subtraction of fractions, we will need to reduce them in the simplest form.
Now, let us learn how to add two or more improper fractions. But before that let us recall what we mean by improper 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. Even the fraction $\frac{3}{3}$ is an improper fraction as in this case the numerator is equal to the denominator. So, we can say the condition for a fraction to be an improper fraction is that its numerator should be greater than or equal to its denominator. Let us now understand how to add or subtract improper fractions.
Addition of Improper Fractions
Addition of Improper Fractions is done based on the equivalence of their denominators, i.e. whether they have same different denominators. Let us discuss both the cases.
Improper Fractions with Same Denominators
Recall that, fractions with the same denominator are called like fractions. Therefore, if the denominators of the improper fractions are the same, then would be called like fractions. Therefore, in order to add two improper fractions, we follow the following steps:
- Obtain the numerators of the two given fractions and their common denominator
- Add the numerators obtained in the first step.
- Write a fraction whose numerator is the sum obtained in the second step and the denominator is the common denominator of the given fractions.
Let us understand it by an example.
Example
Suppose we want to add the fractions $\frac{8}{5} and \frac{13}{5}$
Solution
Here we can see that both the fractions have the same denominator, i.e. 5.
Therefore, we go by the above-defined steps.
We check the numerators of both the fractions. They are 8 and 13.
Then, we add these numerators and get 8 + 13 = 21.
Now, we write the sum of these fractions as $\frac{21}{5}$
Hence, $\frac{8}{5} + \frac{13}{5} = \frac{21}{5}$
In the above example we have seen how to add two improper fractions having the same denominator. What if their denominators are different?
Improper Fractions with Different Denominators
Improper fractions having different denominators are called unlike fractions. . For example, $\frac{9}{7} and \frac{11}{8}$ are unlike fractions as both the fractions have different denominators. But they are improper functions as well as their numerator is greater than the respective denominator.
Let us see how to add these improper fractions. We shall use the following steps to find the sum of improper fractions with different denominators.
- Obtain the fractions and their denominators. The denominators of the fractions should be such that they are not the same.
- Find the Least Common Multiple ( L.C.M) of the denominators. In other words, make the denominators the same by finding the Least Common Multiple (LCM) of their denominators. This step is exactly the same as finding the Least Common Denominator (LCD).
- Convert each function into an equivalent fraction having the same denominator equal to the L.C.M obtained in the previous step. This means that you need to rewrite each fraction into its equivalent fraction with a denominator that is equal to the Least Common Multiple that you found in the previous step.
- Since the fractions are now like fractions, add them as we do for like a fraction, i.e. add their numerators.
- Reduce the fraction to its simplest form, if required.
Let us understand the above steps through an example.
Example
Add $\frac{7}{3} and \frac{8}{7}$
Solution
We have been given the fractions, $\frac{7}{3} and \frac{8}{7}$. We can clearly see that the denominators of these fractions are different, hence they are unlike fractions. Therefore, we will proceed according to the steps defined above to obtain their sum.
Let us first find the L.C.M of 3 and 7.
The L.C.M of 3 and 7 = 21
So, we will convert the given fractions into equivalent fractions with denominator 21.
We will get,
$\frac{7}{3} = \frac{7 x 7}{3 x 7} = \frac{49}{21}$
Similarly,
$\frac{8}{7} = \frac{8 x 3}{7 x 3} = \frac{24}{21}$
Now, we two fractions, $\frac{49}{21} and \frac{24}{21}$ which have a common denominator 21 and are thus like fractions. So, we will add their numerators to get,
$\frac{49}{21} + \frac{24}{21} = \frac{73}{21}$
Hence,
$\frac{7}{3} + \frac{8}{7} = \frac{73}{21}$
Subtraction of Improper Fractions
Subtraction of improper fractions, is similar to the addition of Improper Fractions where the calculations are based on the equivalence of their denominators, i.e. whether they have same different denominators. Let us discuss both the cases.
Improper Fractions with Same Denominators
Recall that, fractions with the same denominator are called like fractions. Therefore, if the denominators of the improper fractions are the same, then would be called like fractions. Therefore, in order to add two improper fractions, we follow the following steps:
- Obtain the numerators of the two given fractions and their common denominator
- Find the difference in numerators obtained in the first step. It is important to check which numerator is the subtrahend and which one is minuend as we know that in case of subtraction a – b might not always be equal to b – a.
- Write a fraction whose numerator is the difference of the sum obtained in the second step and the denominator is the common denominator of the given fractions.
Let us understand it by an example.
Example
Suppose we want to find the difference between fractions $\frac{9}{4} and \frac{5}{4}$
Solution
Here we can see that both the fractions have the same denominator, i.e. 4.
Therefore, we go by the above-defined steps.
We check the numerators of both the fractions. They are 9 and 5.
Then, we subtract 5 from 9, we will get 9 – 5 = 4
Now, we write the difference of these fractions as $\frac{4}{3}$
Hence, $\frac{9}{4} – \frac{5}{4} = \frac{4}{3}$
Let us now understand how to find the difference between improper fractions having different denominators.
Improper Fractions with Different Denominators
Again, recall that, fractions with the same denominator are called like fractions. Therefore, if the denominators of the improper fractions are the same, then would be called like fractions. Therefore, in order to add two improper fractions, we follow the following steps:
- Obtain the fractions and their denominators. The denominators of the fractions should be such that they are not the same.
- Find the Least Common Multiple ( L.C.M) of the denominators. In other words, make the denominators the same by finding the Least Common Multiple (LCM) of their denominators. This step is exactly the same as finding the Least Common Denominator (LCD).
- Convert each function into an equivalent fraction having the same denominator equal to the L.C.M obtained in the previous step. This means that you need to rewrite each fraction into its equivalent fraction with a denominator that is equal to the Least Common Multiple that you found in the previous step.
- Since the fractions are now like fractions, subtract them as we do for a like fraction, i.e. subtract their numerators.
- Reduce the fraction to its simplest form, if required.
Let us understand the above steps through an example.
Example
Solve $\frac{33}{4} – \frac{17}{6}$
Solution
We have been given the fractions, $\frac{33}{4} and \frac{17}{6}$
We can clearly see that the denominators of these fractions are different, hence they are unlike fractions. Therefore, we will proceed according to the steps defined above to obtain their difference.
We will first find the L.C.M of 4 and 6
L.C.M of 4 and 6 = 12
So, we will convert the given fractions into equivalent fractions with denominator 12.
We will get,
$\frac{33}{4} = \frac{33 x 3}{4 x 3} = \frac{99}{12}$
Similarly,
$\frac{17}{6} = \frac{17 x 2}{6 x 2} = \frac{34}{12}$
Now, we two fractions, $\frac{99}{12} and \frac{34}{12}$ which have a common denominator 12 and are thus like fractions. So, we will subtract their numerators to get,
$\frac{99}{12} – \frac{34}{12} = \frac{99- 34}{12} = \frac{65}{12}$
Hence, $\frac{33}{4} – \frac{17}{6} = \frac{65}{12}$
Graphical representation of Addition and Subtraction of Improper Fractions
Let us how to represent the addition and subtraction of in a graphical manner. For this purpose, let us consider two fractions, $\frac{5}{4} and \frac{6}{4}$
Let us draw a picture to understand how the fractions would like in pictorial form. Since we have 4 in the denominator of these fractions, this means that each whole has been divided into four equal parts.
Therefore, $\frac{5}{4}$ can be represented as –
Similarly, $\frac{6}{4}$ can be represented as –
If we sum of these shaded parts above we will get, 5 +6 = 11 shaded parts.
Hence, the sum of, $\frac{5}{4} and \frac{6}{4}$ will be $\frac{11}{4}$
If we were to find the difference between $\frac{5}{4} and \frac{6}{4}$, then to find the value of $\frac{6}{4} – \frac{5}{4}$ we would have subtracted the shaded parts of the two fractions to get, 6 – 5 = 1
Hence, $\frac{6}{4} – \frac{5}{4} = \frac{1}{4}$
Solved Examples
Example 1 Amy bought $\frac{5}{2}$ kg sugar whereas Patricia bought $\frac{7}{2}$ kg of sugar. Find the total amount of sugar bought by both of them.
Solution We have been given that Amy bought $\frac{5}{2}$ kg sugar whereas Patricia bought $\frac{7}{2}$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 Amy = $\frac{5}{2}$ kg
Sugar bought by Patricia = $\frac{7}{2}$ 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.
So the total sugar bought by both of them = Sugar bought by Amy + Sugar bought by Patricia
Now, adding their numerators we have, 5 + 7 = 12
Therefore, 5 / 2 + 7 / 2 = 12 / 2 = 6 kg
Hence, Amy and Patricia together bought 6 kg of sugar.
Example 2 Rosa ate 15/7 pizzas and she gave 17/14 pizzas to her mother. How many pizzas did Rosa have initially ?
Solution We have been given that Rosa ate 15/7 pizzas and she gave 17/14 pizzas to her mother. We are required to find how many pizzas Rosa had initially.
Let us first summarise the fractions given to us.
Fraction of pizza eaten by Rosa = $\frac{15}{7}$
Fraction of pizza eaten by Rosa’s mother = $\frac{17}{14}$
Pizzas that were available with Rosa initially will be given by adding these two fractions. So, we need to find, $\frac{15}{7} + \frac{17}{14}$
Note, both the fractions have different denominators; hence we will first have to take the L.C.M of the denominators so as to make them equivalent fractions.
L.C.M of 7 and 14 = 14
Now, we will convert the given fractions into equivalent fractions with denominator 14.
Note, that the fraction $\frac{17}{14}$ already has the denominator as 8, so we need not convert it any further.
$\frac{15}{7} = \frac{15 x 2}{7 x 2} = \frac{30}{14}$
Now,
$\frac{15}{7} + \frac{17}{14} = \frac{30}{14} + \frac{17}{14} = \frac{47}{14}$
Hence, total pizzas that were available with Rosa initially = $\frac{47}{14}$
Example 3 Emily has 11/3 acres of land. She gives 5/4 acres of land to her friend. How many acres of land does Emily have now ?
Solution We have been given that Emily has 11/3 acres of land. She gives 5/4 acres of land to her friend. We need to find out the total land left with Emily now. To calculate this, first, we will summarise the fractions given to us.
Land available with Emily = $\frac{11}{3}$ acres
Land Emily gives to her friend = $\frac{5}{4}$ acres
Land left with Emily can be found out by finding the difference between these two fractions, i.e. by finding the value of $\frac{11}{3} – \frac{5}{4}$
Note, both the fractions have different denominators; hence we will first have to take the L.C.M of the denominators so as to make them equivalent fractions.
L.C.M of 3 and 4 is 12
Therefore,
$\frac{11}{3} = \frac{11 x 4}{3 x 4} = \frac{44}{12}$ and
$\frac{5}{4} = \frac{5 x 3}{4 x 3} = \frac{15}{12}$
Now,
$\frac{11}{3} – \frac{5}{4} = \frac{44}{12} – \frac{15}{12} = \frac{44 – 15}{12} = \frac{29}{12}$
Hence, Emily has $\frac{29}{12}$ acres of land left with her.
Key Facts and Summary
- A fraction is a number representing a part of a whole. The whole may be a single or a group of objects.
- A fraction is said to be in the standard form if the denominator is positive and the numerators have no common divisor other than 1.
- To reduce a fraction to its simplest form Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator. Divide the numerator and the denominator of the given fraction by the GCD (HCF) obtained in step III. The fraction so obtained is the standard form of the given fraction.
- The number that is subtracted is called the subtrahend while the number from which the subtrahend is subtracted is called minuend. The result of this subtraction is called the difference.
- Fractions with the numerator either equal or greater than the denominator are called improper fractions.
- Fractions with the same denominator are called like fractions while fractions with different denominators are called unlike fractions.
- In order to add two or more improper fractions with the same denominators, we add the fractions in the same manner as we do for like fractions.
- In order to add two or more improper fractions with different denominators, we first convert them into the corresponding equivalent like fractions and then they are added in the same manner as we do for like fractions.
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