OPERATIONS WITH FRACTIONS
ADDITION AND
SUBTRACTION
If the
denominators are the same, add or subtract and simplify your answer.
Example:
Example:
Note that we
couldn't subtract 
from 
so we
"borrowed" 
from 14.
Trick: You can just subtract 1 from 14 and then add
the numerator and denominator together for the new numerator.
Then 
Another
Trick: Break the problem into two
parts, one with whole numbers and one with fractions. Then just subtract without borrowing, keeping track of the
negative part of your answer.
Whole
number: 14 - 7 = 7
Fraction: 
-
=
Answer: 
One last
trick: You can use "equal
additions"
= 
=
In addition and
subtraction, if the denominators are not the same you need to find a
common denominator
Example:
To find the
common denominator of 4 and 6, look for common multiples:
Multiples of
4: 4, 8, 12, 16, 20, 24...
Multiples of
6: 6, 12, 18, 24, 30...
Common
multiples: 12, 24...
Least common
multiple: 12
Star
problem: Subtract 
from 
.
Hint: First find the common denominator. Answer: 
MULTIPLICATION
To multiply
fractions, multiply numerators and multiply denominators, and then reduce your
answer.
Example:
Shortcut: Simplify before you multiply by dividing
numerators and denominators by common factors.
Before
multiplying we usually change mixed numbers into improper fractions.
Example:
Shortcut: Use the distributive rule.
Best
shortcut: Change 16
to 16.5 and plug into your calculator.
DIVISION
To divide,
invert the second term (the divisor) and follow the rules for
multiplication. If you have mixed
numbers change them to fractions first.
(Note that the
"reciprocal" of a fraction 
is 
. When we divide we
multiply by the reciprocal of the divisor.)
Example:
Danger: Do not "cancel" in division!
Example: 
Shortcut: Did you know that you can divide numerators
and divide denominators?
Example: AB is a line segment with midpoint C, and D
is the midpoint of AC. If
DB = 8, what is the length of AC?
Answer: Draw a picture. It may help to also picture the midpoint of CB, say x, in order
to see that there are four equally spaced sections on AB. Then DB is three sections long and AC is two
sections long, so AC is (2/3) of DB or (2/3) x 8 = 16/3.
Example: Angela and Seth split a pie in half. Angela eats half of her part and saves the
rest in the refrigerator. Later she
eats half of the piece that she put in the refrigerator. The rest she gives to Seth, who was still
humgry. What fractional part of the pie
did Angela eat?
A. 3/16
B. 1/4
C. 3/8
D. 1/2
E. 3/4
Answer: C
Angela gets half
the pie. She eats half of her part and
saves the other half, which means she eats 
and saves 
. Later, she eats
half of what she had saved, or 
. Altogether, she ate
and 
which is 
of the pie.
Another
solution: Draw a picture. In the first picture the shaded area represents
Angela's original share. (Note that in
this problem, pie are square.)
Next we
illustrate the part that Angela ate first.
Finally she also
eats half of her remaining piece.
By comparing
equal pieces, we see that the shaded area is 3/8 of the whole amount.
