Adding fractions


Before learning about adding fractions, please read the primer on fractions.


The method used to add fractions depends on whether their denominators are the same (like denominators) or different (unlike denominators).


Adding fractions with like denominators


Let's start with some easy examples.


For instance,

1/4  +  1/4


Don’t mess with the denominators when they’re the same - just add the numerators

1/4  +  1/4   =   2/4


Notice how the denominator remains 4. You can also simplify by dividing numerator and denominator by 2



In summary

1/4  +  1/4   =   2/4   =   1/2


A quarter plus a quarter equals a half.



How about 1/2 + 1/2 ?

 

Again, leave the denominators alone - simply add the numerators 

1/2  +  1/2   =   2/2

 

2 divided by 2 equals 1 so you can also write

1/2  +  1/2   =   2/2   =   1

 

A half plus a half equals one.



Let's step it up a little - consider 1/3 + 1/3 + 2/3


Follow the same rules regarding the denominators (do nothing!) and the numerators (add!)

1/3  +  1/3  +  2/3  =   4/3


A third plus a third plus two thirds equals four thirds.


Adding fractions with unlike denominators


This is easy once you know how!


Take a look at

1/3  +  1/6


You need to find a way of expressing these fractions so that they have like denominators.

 

Ask yourself what’s the lowest number into which both 3 and 6 will divide exactly? The answer of course is 6 – this is the least common denominator.

 

The fraction 1/6 can be left alone – its denominator is already 6.

 

But how can 1/3 be expressed as a fraction with denominator 6? To get from denominator 3 to denominator 6 you need to multiply by 2. But you’ll also need to times the numerator 1 by the same factor 2 to get a fraction of equivalent value.

 


So

1/3  +  1/6   =   2/6  +  1/6


Now just follow the rules for addition of fractions with like denominators (don't change the denominators, add the numerators)

2/6  +  1/6   =   3/6


To simplify divide numerator and denominator by 3

3/6   =   1/2


Bringing everything together

1/3  +  1/6   =   2/6  +  1/6   =   3/6  =   1/2



Let’s work out 1/6 + 1/4 + 2/3

 

First express as fractions with like denominators. Ask yourself what’s the least common denominator, i.e. the lowest number into which 6, 4 and 3 divide exactly? The answer’s 12.


Consider 1/6 - to get from denominator 6 to denominator 12, you need to times by 2. You also need to multiply the numerator 1 by the same factor 2.



1/6 = 2/12.


What about 1/4? Multiply by 3 to get from denominator 4 to denominator 12. Times the numerator 1 by the same factor 3.

 


1/4 = 3/12.


Next consider 2/3 - to get from denominator 3 to denominator 12, times by 4. Multiply the numerator 2 by the same factor 4.



2/3 = 8/12.


Therefore

1/6  +  1/4  +  2/3   =   2/12  +  3/12  +  8/12

 

Now you’re adding fractions with like denominators

2/12  +  3/12  +  8/12   =   13/12

 

Putting it all together

1/6  +  1/4  +  2/3   =   2/12  +  3/12  +  8/12   =   13/12


A quick summary on how to add fractions ...


... with like denominators - leave the denominators alone, add the numerators, then simplify if possible.

 

... with unlike denominators - first express as fractions with like denominators by finding the least common denominator, then follow the rules for adding fractions with like denominators.


For more on fractions click:

Math fractions

Equivalent fractions

Simplifying fractions

How to subtract fractions

Multiplying fractions

How to divide fractions

Improper fractions

Mixed fractions (mixed numbers)

Comparing fractions

Ordering fractions

Fraction to decimal

Fraction to decimal chart

Decimal to fraction

Fraction to percent

Percent to fraction


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