Fractions Review Sheet
1. To add fractions we need a common denominator:
a
bc
+ dc = ad
+ bd
= ad+bc
b
bd
bd
3
7
+
5
4
=
3·4
7·4
+
7·5
7·4
=
3·4+7·5
7·4
Add the following fractions:
(a)
(b)
(c)
(d)
1
+ 15 =
3
5
+ 32 =
4
8
7
+ 15
=
2
15
7
+ 35
=
36
2. We should reduce the fractions before we do any operations to simplify the problem:
ae
bc
+ bd
= ad+bc
+ dcff = ab + dc = ad
bd
bd
be
16
72
+
33
110
=
2·8
9·8
+
3·11
10·11
=
2
9
+
3
10
=
2·10
9·10
+
9·3
9·10
=
2·10+9·3
9·10
Add the following fractions:
(a)
(b)
(c)
(d)
100
+ 85
=
50
34
12
3
+ 18
=
144
8
72
+ 15
=
144
16
7
+ 42 =
36
3. We don’t have to use the denominators to get a common denominator, only their factors not
in common:
a
c
ad
bc
+ de
= bde
+ bde
= ad+bc
be
bde
2
15
+
3
35
=
2
3·5
+
3
7·5
=
2·7
3·5·7
+
3·3
3·5·7
=
2·7+3·3
3·5·7
Add the following fractions:
(a)
(b)
(c)
(d)
1
1
+ 15
=
6
3
2
+ 24
=
16
72
8
+ 15
=
144
16
7
+ 42 =
36
4. Subtraction follows the same rules as addition.
5. When multiplying fractions we multiply straight across: numerator to numerator and denominator to denominator.
a c
· = ac
b d
bd
5
7
·
3
2
5·3
7·2
=
Multiply the following fractions (don’t forget you can reduce the fractions first):
3 7
· =
4 5
12 5
· =
17 2
72
· 8 =
144 15
7
· 16 =
36 42
(a)
(b)
(c)
(d)
6. We can also cancel factors that are in common between the numerator of one fraction and the
denominator of the other:
a
· cde = ab · dc = ac
bd
be
2
9
·
3
14
2
3·3
=
·
3
2·7
=
2
3
·
1
7
=
2
3·7
Multiply the following fractions:
(a)
(b)
(c)
(d)
3 2
· =
4 5
24 34
· =
17 18
80
· 12
=
144
8
14
· 6 =
42 56
7. When we multiply a fraction by a whole number, we just multiply the numerator:
a · cb = a1 · cb = a·b
c
3·
5
7
=
3
1
·
5
7
=
3·5
7
Multiply the following fractions (don’t forget rule 5):
(a) 5 ·
(b) 3 ·
2
3
5
6
(c) 12 ·
(d) 25 ·
=
=
13
18
6
35
=
=
8. When we divide two fraction
we multiply the dividend by the reciprocal of the divisor:
a
a
c
a d
a·d
a d
b
÷ d = b · c = b·c or c = b · c = a·d
b
b·c
d
3
5
2
7
or ( 35 ÷ 27 ) =
3
5
·
7
2
=
3·7
5·2
Divide the following fractions:
(a)
(b)
4
÷ 67 =
5
13
÷ 11
=
3
2
(c)
16
5
12
3
(d)
5
9
8
3
=
=
9. When dividing we can cancel common factors in the numerators or common factors in the
denominators:
ae
b
ce
d
= abe ·
a
bf
c
df
=
d
ce
a
b
c
d
=
· dcf =
a
b
f
ae
bf
ce
df
= bafe · dcfe =
22
7
11
14
7
=
= 11
2·11
2·7
2
1
1
2
or without the intermediate step
ae
b
ce
d
a
bf
c
df
=
a
b
c
d
, similarly
a
b
c
d
or without the intermediate step =
a
b
c
d
a
b
c
d
ae
or without the intermediate step bcfe =
df
a
b
c
d
, and both together
=4
Divide the following fractions:
(a)
(b)
14
5
3
13
(c)
18
7
12
35
(d)
132
9
144
3
÷
÷
7
=
6
4
=
26
=
=
10. When we divide a fraction by a whole number we can just multiply the denominator by that
number:
a
a
a 1
b
b
=
· = bca
c =
c
b c
1
3
5
7
=
3
5·7
Divide the following fractions:
(a)
(b)
(c)
(d)
14
5
3
13
18
7
4
132
9
55
÷3=
÷6=
=
=
11. When raising fractions to a power we can distribute to the numerator and the denominator:
n
( ab )n = abn
( 25 )2 =
22
52
Compute the following (don’t forget to reduce if necessary):
(a) ( 11
)2 =
5
(b) ( 32 )3 =
1
(c) ( 49 ) 2 =
40 2
(d) ( 16
) =
)2 =
(e) ( 55
25
1
(f) ( 54
)2 =
24
12. Now if we combine some of these operations:
n
n
an
b · ( ab )n = b · abn = a b bn−1 = bn−1
n
>
b
52
16 · ( 45 )2 = 42 · = 52
42
Compute the following (don’t forget to reduce if necessary):
)2 =
(a) 6 · ( 13
6
(b) 9 · ( 23 )4 =
1
(c) 9 · ( 25
)2 =
9
45 2
(d) 2 · ( 18
) =
)2 =
(e) 8 · ( 75
30
1
(f) 3 · ( 512
)2 =
72