Operations With Fractions
To add or subtract fractions, use a common denominator. The common denominator is the least common multiple of the denominators.
Example 1
Simplify
2
3
+
3
5
.
2 thirds , plus , 3 fifths , .
2
3
+
3
5
=
2
3
⋅
5
5
+
3
5
⋅
3
3
For
3
and
5
,
the least common multiple is
15
.
=
10
15
+
9
15
write
2
3
and
3
5
as equivalent fractions with denominators of
15
.
=
19
15
or
1
4
15
Add the numerators.
table with 3 rows and 3 columns , row1 column 1 , 2 thirds , plus , 3 fifths , column 2 equals , 2 thirds , dot , 5 fifths , plus , 3 fifths , dot , 3 thirds , column 3 cap for , 3 , and , 5 comma . theleastcommonmultipleis . 15 . , row2 column 1 , , column 2 equals , 10 over 15 , plus , 9 fifteenths , column 3 write . 2 thirds , and , 3 fifths . asequivalentfractionswithdenominatorsof . 15 . , row3 column 1 , , column 2 equals , 19 over 15 . or , 1 , and 4 fifteenths , column 3 cap addthenumerators. , end table
Example 2
Simplify
5
1
4
−
3
2
3
.
5 , and 1 fourth , minus 3 , and 2 thirds , .
5
1
4
−
3
2
3
=
5
3
12
−
3
8
12
Write equivalent fractions.
=
4
15
12
−
3
8
12
Write
5
3
12
as
4
15
12
so you can subtract the fractions.
=
1
7
12
Subtract the fractions
. Then subtract the whole numbers
.
table with 3 rows and 3 columns , row1 column 1 , 5 , and 1 fourth , minus 3 , and 2 thirds , column 2 equals 5 , and 3 twelfths , minus 3 , and 8 twelfths , column 3 cap writeequivalentfractions. , row2 column 1 , , column 2 equals 4 , and 15 over 12 , minus 3 , and 8 twelfths , column 3 cap write , 5 , and 3 twelfths , as , 4 , and 15 over 12 . soyoucansubtractthefractions. , row3 column 1 , , column 2 equals 1 , and 7 twelfths , column 3 cap subtractthefractions . .cap thensubtractthewholenumbers . . , end table
To multiply fractions, multiply the numerators and multiply the denominators. You can simplify by using a greatest common factor.
Example 3
Simplify
3
4
⋅
8
11
3 fourths , dot , 8 elevenths
-
Method 1
3
4
⋅
8
11
=
24
44
=
24
÷
4
44
÷
4
=
6
11
3 fourths , dot , 8 elevenths , equals , 24 over 44 , equals . fraction 24 divides 4 , over 44 divides 4 end fraction . equals , 6 elevenths
Divide 24 and 44 by 4, their greatest common factor.
-
Method 2
3
4
⋅
8
2
11
=
6
11
fraction 3 , over cross out enclosing , 4 , end crossout end fraction , dot . fraction cross out enclosing , 8 , end crossout 2 , over 11 end fraction . equals , 6 elevenths
Divide 4 and 8 by 4, their greatest common factor.
To divide fractions, use a reciprocal to change the problem to multiplication.
Example 4
Simplify
3
1
5
÷
1
1
2
3 , and 1 fifth , divides 1 , and 1 half
3
1
5
÷
1
1
2
=
16
5
÷
3
2
Write mixed numbers as improper fractions
.
=
16
5
⋅
2
3
Multiply by the reciprocal of the divisor
.
=
32
15
or
2
2
15
Simplify
.
table with 3 rows and 3 columns , row1 column 1 , 3 , and 1 fifth , divides 1 , and 1 half , column 2 equals , 16 over 5 , divides , 3 halves , column 3 cap writemixednumbersasimproperfractions . . , row2 column 1 , , column 2 equals , 16 over 5 , dot , 2 thirds , column 3 cap multiplybythereciprocalofthedivisor . . , row3 column 1 , , column 2 equals , 32 over 15 . or , 2 , and 2 fifteenths , column 3 cap simplify , . , end table
Exercises
Perform the indicated operation.
-
3
5
+
4
5
3 fifths , plus , 4 fifths
-
1
2
+
2
3
1 half , plus , 2 thirds
-
4
1
2
+
2
1
3
4 , and 1 half , plus 2 , and 1 third
-
5
3
4
+
4
2
5
5 , and 3 fourths , plus 4 , and 2 fifths
-
2
3
−
3
7
2 thirds , minus , 3 sevenths
-
5
1
2
−
3
2
5
5 , and 1 half , minus 3 , and 2 fifths
-
7
3
4
−
4
4
5
7 , and 3 fourths , minus 4 , and 4 fifths
-
3
4
5
⋅
10
3 , and 4 fifths , dot 10
-
2
1
2
⋅
3
1
5
2 , and 1 half , dot 3 , and 1 fifth
-
6
3
4
⋅
5
2
3
6 , and 3 fourths , dot 5 , and 2 thirds
-
1
2
÷
1
3
1 half , divides , 1 third
-
6
5
÷
3
5
6 fifths , divides , 3 fifths
-
8
1
2
÷
4
1
4
8 , and 1 half , divides 4 , and 1 fourth
-
8
9
−
2
3
8 ninths , minus , 2 thirds
-
5
1
4
⋅
8
5 , and 1 fourth , dot 8
