Adding and Subtracting Fractions
with Like Denominators
Objective Learn how to add and subtract
fractions with like denominators.
This concept is extremely important, both from a conceptual
point of view and as a basic skill. Fractions represent portions
of a whole, and you will need to be able to manipulate fractions
arithmetically. Here we will discuss arithmetic operations with
fractions having the same denominators. It is very important that
you keep in mind that the methods in this lesson apply only to
fractions with like denominators.
Adding Fractions with Like Denominators
There are algorithms (procedures) for adding and subtracting
fractions. In this lesson we will only be using the algorithm to
add or subtract two fractions that have the same denominator.
First let’s visually interpret the sum of two fractions.
Example 1
What is the sum of  and  ?
Solution
The fraction can be thought of as 1
of 5 blocks that are the same size, which together make up a
whole.
Then the fraction can be thought of as 2
of 5 blocks just like those above.
Combined together, we have 3 blocks, which represents  of the whole.
This shows that  .
In order to add fractions with like denominators one may
simply add the numerators of the two fractions and write the sum
over the shared (common) denominator.
Key Idea
When adding fractions with the same denominator, add the
numerators and retain the denominator.
Example 2
Find  .
Solution
Why Does the Algorithm Work?
Suppose we want to add  . Think of it as if we
had a block that is cut into c equal pieces. So each piece is  of the whole. We are
adding a of these pieces to b of the pieces, for a total of a + b
pieces, each of which is of the whole. So we
have  parts of the whole. In
symbols,
Subtract Fractions with Like Denominators
The algorithm for subtracting fractions with like denominators
is similar to the algorithm for the addition of two fractions
with like denominators.
Key Idea
When subtracting fractions with the same denominator, subtract
the numerators and retain the denominator.
Example 3
Find  .
Solution
The algorithms given in this lesson only work when adding and
subtracting fractions whose denominators are the same.
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