May 20th
May 20th
Adding and Subtracting Like Fractions
Let’s first discuss how to add and subtract like
fractions. Suppose that we want to add  and  . A diagram can help us understand what is
involved. First we shade one-fifth of the diagram, then another
three-fifths.
We see in the diagram that the total shaded area is
four-fifths, so  . Note that we added the original
numerators to get the numerator of the answer but that the
denominator stayed the same.
The diagram shows how to subtract like fractions by computing  . If we shade four-fifths of the diagram
and then remove the shading in one-fifth, three-fifths remain
shaded. Therefore  . Note that we could have gotten this
answer simply by subtracting numerators without changing the
denominator.
The following rule summarizes how to add or subtract
fractions, provided that they have the same denominator.
To Add (or Subtract) Like Fractions
- first add (or subtract) the numerators,
- then use the given denominator, and
- finally write the answer in simplest form.
EXAMPLE 1
Add: 
Solution
Applying the rule, we get
TIP
Be careful not to add the denominators when adding like
fractions.
EXAMPLE 2
Add:  .
Solution
So the sum of  .
EXAMPLE 3
Find the difference between  .
Solution
EXAMPLE 4
In the following diagram, how far is it from the college to
the library via city hall?
Solution
Examining the diagram, we see that
- the distance from the college to city hall is
 mile and that
- the distance from city hall to the library is
 mile
To find the distance from the college to the library via city
hall, we add.
The distance is 1 mile.
EXAMPLE 5
According to one study, of the people who exercise regularly live
at least to age 70, in contrast to only of the people who do not exercise
regularly. What is the difference between these two fractions?
Solution
Subtracting, we get  . Therefore the difference between these
fractions is  . We can check our answer by adding to to get .
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