Page 93 - ArithBook5thEd ~ BCC

3.8.2 Adding and Subtracting Unlike Fractions

Here is the rule for adding any two fractions, like or unlike:

a c ad + bc
+ =
b d bd

Compared to the rule for multiplying fractions (Section 3.4), it looks a bit strange. But it is easily
proved, using (yet again) the fact that 1 = b/b and 1 = d/d, for any nonzero b or d.

a c a d c b
+ = · + ·
b d b d d b
ad bc
= +
bd bd
ad + bc
=
bd

2 3
Example 102. Add + .
5 7
Solution. Using the boxed rule,

2 3 2 · 7+ 5 · 3 14 + 15 19
+ = = = .
5 7 5 · 7 35 35

15 19
Example 103. Add + .
16 24
Solution. Using the boxed rule,

15 19 15 · 24 + 16 · 19 664
+ = = .
16 24 16 · 24 384

83
Cancelling the GCF, 8, we obtain the sum in lowest terms, .
48
The rule works for like fractions too:
4 3 7
Example 104. Use the boxed rule to show that + = .
5 5 5
Solution.
4 3 20 + 15 35 7
+ = = = .
5 5 25 25 5
Note that an extra factor of 5 was introduced, and then cancelled at the end. (A needless complication!)

Page 93