Lesson 25
ADDING AND SUBTRACTING FRACTIONS AND MIXED NUMBERS
A FRACTION IS A NUMBER we need for measuring (Lesson 20); therefore we sometimes have to add or subtract them. Now, to add or subtract anything, the names of what we are counting  the units  must be the same.
2 apples + 3 apples = 5 apples.
We cannot add 2 apples plus 3 oranges  at least not until we call them "pieces of fruit."
In the name of a fraction  "2 ninths," for example  ninths are what we are adding. (Lesson 21, Example 3.)
2 ninths + 3 ninths = 5 ninths.
That unit appears as the denominator.

Example 1. 
5 8 
+ 
2 8 
= 
7 8 
. 
"5 eighths + 2 eighths = 7 eighths."
The denominator of a fraction has but one function, which is to name what we are counting. In this example, we are counting eighths.
Example 2. 
5 8 
− 
2 8 
= 
3 8 
. 
Fractions with different denominators
To add or subtract fractions, the denominators must be the same. Before continuing, then, the student should know how to convert one fraction to an equivalent one, by multiplying the numerator and the denominator. See Lesson 22, Examples 1, 2, 3, and especially Example 4.

We choose a common multiple of the denominators because we change a denominator by multiplying it. Lesson 22.
Solution. The lowest common multiple of 3 and 4 is their product, 12. (Lesson 22, Question
4.)
We will convert each fraction to an equivalent fraction with denominator 12.
2 3 
+ 
1 4 
= 
8 12 
+ 
3 12 




= 
11 12 
. 
We converted 
2 3 
to 
8 12 
by saying, "3 goes into 12 four times. Four 
times 2 is 8."
(In that way, we multiplied both 2 and 3 by the same number, namely 4. See Lesson 22, Question 3.)
We converted 
1 4 
to 
3 12 
by saying, "4 goes into 12 three times. Three

times 1 is 3." (We multiplied both 1 and 4 by 3.)
The fact that we say what we do shows again that arithmetic is a spoken skill.
In practice, it is necessary to write the common denominator only once:
2 3 
+ 
1 4 
= 
8 + 3 12 
= 
11 12 
. 
Solution. The LCM of 5 and 15 is 15. Therefore,
4 5 
+ 
2 15 
= 
12 + 2 15 
= 
14 15 
. 
We changed 
4 5 
to 
12 15 
by saying, "5 goes into 15 three times. Three 
times 4 is 12."
We did not change 
2 15 
, because we are not changing the denominator 
15.
Example 5. 
2 3 
+ 
1 6 
+ 
7 12 
Solution. The LCM of 3, 6, and 12 is 12.
2 3 
+ 
1 6 
+ 
7 12 
= 
8 + 2 + 7 12 
2 3 
+ 
1 6 
+ 
7 12 
= 1 
5 12 
. 
We converted 
2 3 
to 
8 12 
by saying, "3 goes into 12 four times. Four

times 2 is 8."
We converted 
1 6 
to 
2 12 
by saying, "6 goes into 12 two times. Two

times 1 is 2."
We did not change 
7 12 
, because we are not changing the 
denominator 12.
Finally, we changed the improper fraction 
17 12 
to 1 
5 12  by dividing 17 
by 12. (Lesson 20.)
"12 goes into 17 one (1) time with remainder 5."
Solution. The LCM of 6 and 9 is 18.
5 6 
+ 
7 9 
= 
15 + 14 18 
= 
29 18 
= 1 
11 18 
. 
We changed 
5 6 
to 
15 18 
by multiplying both terms by 3. 
We changed 
7 9 
to 
14 18 
by multiplying both terms by 2. 
Example 7. Add mentally 
1 2 
+ 
1 4 
. 
Answer. 
1 2 
is how many 
1 4 
's? 
Just as 1 is half of 2, so 2 is half of 4. Therefore,
The student should not have to write any problem in which one of
the fractions is 
1 2 
, and the denominator of the other is even. 
For example,
Example 8. In a recent exam, one eighth of the students got A, two fifths got B, and the rest got C. What fraction got C?
Solution. Let 1 represent the whole number of students. Then the question is:
Now,
1 8 
+ 
2 5 
= 
5 + 16 40 
= 
21 40 
. 
The rest, the fraction that got C, is the complement of 
21 40 
. 

Example 9. 4 
3 8 
+ 2 
2 8 
= 6 
5 8 
. 
Example 10. 3 
2 5 
+ 1 
4 5 
= 4 
6 5 
. 
Therefore,
Solution. When the denominators are different, we may arrange the work vertically; although that is not necessary.
To add the fractions, the denominators must be the same. The LCM
of 4 and 8 is 8. We will change 
3 4 
to 
6 8 
 by multiplying both terms by 2: 
We added 6 + 3 = 9. 
6 8 
+ 
5 8 
= 
11 8 
= 1 
3 8 
. 
At this point, please "turn" the page and do some Problems.
or
Continue on to Section 2: Subtracting mixed numbers
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