Adding and Subtracting Fractions
Addition (when the denominators are the same)
Add the numerators. Keep the same denominator.
Subtraction (when the denominators are the same)
Subtract the numerators. Keep the same denominator.
Since we can change the form of a fraction by multiplying the
top and bottom by the same thing, we can force any two fractions
to have the same denominator. The trick is to give both
denominators all the same factors.
Note that if the denominators we want to match are 3 and 5, we
can multiply 3 by 5 and 5 by 3 to convert both fractions to
15ths. (We actually didnt change the fractions in the
process, because we multiplied both top and bottom of each
fraction by the same number in each case. Thats what keeps
it all legal.)
Least Common Denominators
If two denominators have some factors in common, simply
multiplying the denominators together will give you needlessly
big numbers that have to be reduced later. Instead, spread out
and compare the factors of both denominators and multiply each
denominator by the factors it lacks.
Notice that both denominators contain 2 Ã— 2 (marked in blue),
but one has an extra 3 and the other has two more extra 2s.
If the first fraction is multiplied (top and bottom) by 4 ( = 2
Ã— 2 ) and the second fraction is multiplied (top and bottom) by
3, all the factors in both denominators will match.
The little bit of extra work breaking the denominators into
factors at this stage is easier than multiplying 12 by 16 and 16
by 12 (= 192), because you would then have to turn right around
and reduce the resulting fractions which would by then involve
much larger numbers.
Addition or Subtraction when the denominators are NOT the
Convert the fractions to equivalent fractions having the same
denominator, then add or subtract the numerators as before.