In this chapter we will learn methods to add improper fractions.
But let us start with the basics.
What is improper fraction?
The fraction in which numerator is greater than denominator is called improper fraction.
Hence in improper fraction, numerator > denominator.
How to add improper fractions?
There are three scenarios of adding improper fractions;
(a) adding improper fractions with same denominator.
(b) adding improper fraction with different denominator.
(c) cross multiplication method
Let us learn each of the cases in detail.
Adding Improper Fraction with same denominator
When two fractions with same denominators are given then simply add the numerators and keep the same denominator.
Example 01
Add the fractions; \mathtt{\frac{7}{6} \ +\ \ \frac{9}{6}}
Solution
Both 7/6 and 9/6 are improper fraction with same denominator.
Simply add the numerator and retain the same denominator.
Hence, 16/6 is the solution of above addition.
Example 02
Add the fractions; \mathtt{\frac{13}{11} \ +\ \ \frac{15}{11}}
Solution
Both the improper fractions have same denominator.
Here we simply add the numerators.
Hence, 28/11 is the solution of above addition.
Adding Improper fractions with different denominator
In this case we have to make denominator equal by using help of LCM concept.
Follow the below steps for addition.
(i) Find the LCM of denominators.
(ii) Multiply the fractions to make denominator equal to LCM
(iii) Now simply add the numerator and retain the denominator
Given below are some example for your understanding.
Example 01
Add the fractions; \mathtt{\frac{5}{3} \ +\ \ \frac{7}{4}}
Solution
Both the improper fractions have different denominator.
Follow the below steps;
(a) Find LCM of denominators.
LCM (3, 4) = 12
(b) Multiply the fractions to make denominator 4
Fraction 5/3
Multiply numerator and denominator by 4
\mathtt{\frac{5\times 4}{3\times 4} \ =\ \frac{20}{12}}
Fraction 7/4
Multiply numerator and denominator by 3
\mathtt{\frac{7\times 3}{4\times 3} \ =\ \frac{21}{12}}
Now both the fractions 20/12 and 21/12 have same denominator.
(c) Add the numerators and retain the denominator
Hence, 41/12 is the solution of addition.
Example 02
Add the improper fractions; \mathtt{\frac{9}{7} \ +\ \ \frac{15}{14}}
Solution
Both 9/7 and 15/14 are improper fractions.
Follow the below steps;
(a) Find LCM of denominators
LCM ( 7, 14 ) = 14
(b) Multiply the fractions to make denominator 14
Fraction 9/7
Multiply numerator and denominator by 2
\mathtt{\frac{9\times 2}{7\times 2} \ =\ \frac{18}{14}}
Fraction 15/14
The denominator is already 14 so don’t need to do anything.
The fractions 18/14 and 15/14 now have the same denominator.
(c) Add the numerator and retain the denominator.
Hence, 33/14 is the solution.
Example 03
Add the improper fractions; \mathtt{\frac{8}{6} \ +\ \ \frac{13}{8}}
Solution
Both 8/6 and 13/8 are improper fractions with different denominator.
Follow the below steps;
(a) Find LCM of denominators
LCM ( 6, 8 ) = 24
(b) Multiply the fractions to make denominator 24.
Fraction 8/6
Multiply numerator and denominator by 4
\mathtt{\ \frac{8\times 4}{6\times 4} \ =\ \frac{32}{24}}
Fraction 13/8
Multiply numerator and denominator by 3
\mathtt{\ \frac{13\times 3}{8\times 3} \ =\ \frac{39}{24}}
Now both the fractions 32/24 and 39/24 have the same denominator.
(c) Add the numerator and retain the denominator.
Hence, 71/24 is the solution.
Cross Multiplication Method of Adding Fractions
This is a shortcut method of adding fractions.
To add the fractions using this method you should have great multiplication skills.
If a/b and c/d are the two given fractions for addition.
The addition can be done by following below steps;
(a) Final Numerator
Cross Multiply the numerator and denominator of the fractions.
(b) Final Denominator
Multiply the given denominators.
Hence, the final addition looks like this;
If you are good at multiplication this method would be cake walk for you.
Example 01
Add the improper fractions; \mathtt{\frac{3}{2} \ +\ \ \frac{4}{3}}
Solution
Adding the fractions using cross multiplication method.
(a) Getting Numerator
Cross multiply the fractions to get numerator.
(b) Getting denominator
Multiply the two denominators
Hence, 17/6 is the solution of above addition.
Example 02
Add the improper fractions \mathtt{\frac{5}{7} \ +\ \ \frac{6}{9} \ }
Solution
Follow the below steps;
(a) Getting numerator
Cross Multiply the fraction to get numerator.
(b) Getting denominator
Multiply the existing denominator.
Hence, 87/63 is the solution.
