Multiplication of Fractions
- Multiplying Two Fractions
Suppose that you are making lemonade and have found a recipe that requires
three-quarters of a cup of sugar to make one pitcher of lemonade. You
want to make a half pitcher of lemonade. How much sugar do you need?
Solution
The operation of arithmetic needed here is multiplication, that is, what
is
3 1
x
4
2
The pictures below show how to proceed.
We draw three shaded regions of four possible and than cut the shaded
regions in half.
The result is that there are still three shaded regions, but their are 4 x 2
total pieces.
We conclude that
3
1
3 x
1
3
x
=
=
4
2
4 x
2
8
In general, to multiply two fractions,
multiply the numerators and multiply the denominators.
Example
-
2
4
2 x
4
8
x
=
=
3
5
3 x
5
15
-
4
15
4 x 15
x
=
9
28
9 x 28
2 x 2 x 3 x
5
5
=
=
We have canceled common factors
3 x 3 x 2 x 2 x
7 21
-
8
3
8 x 3
x
=
Improper fractions multiply similarly
3
16
3 x 16
2 x 2 x 2 x 3
1
=
=
If all factors factors, we are left with a
one
3 x 2 x 2 x 2
x 2 2
-
2
5 2
5 x
=
A whole number can always be put over 1.
3
1 3
10
=
3
Notice that we always cancel out common factors
before multiplying
Exercises
Multiply the following fractions
-
5 3
x
7
11
Hold mouse over the yellow rectangle for the
solution
-
55 18
x
27
22
Hold mouse over the yellow rectangle for the
solution
-
25 24
x
6
10
Hold mouse over the yellow rectangle for the
solution
- Multiplying Mixed Numbers
If we need to multiply mixed numbers, we must first write them as
improper fractions.
Example
Multiply
3
1
2
x 1
4
3
Solution
First we write the two fractions as improper fractions
3
1
2 x 4 +
3
1 x 3 + 1
2
x 1 =
x
4
3
4
3
11
4
11 x 2 x 2
x =
4
3
2 x 2 x
3
11
3
Now we can convert back to a mixed fraction by doing long division
3
3 |
11
- 9
2
We can conclude that
3
1
2
2
x 1 =
3
4 3 3
Exercises
Multiply
-
1
3
3
x 2
8
5
Hold mouse over the yellow rectangle for the
solution
-
1
3
5
x 3
3
4
Hold mouse over the yellow rectangle for the
solution
- Applications
You just bought a 2 1/3 acre lot overlooking Lake Tahoe. The Tahoe
Regional Planning Agency has determined that because the land contains
environmentally sensitive soil, you are only allowed to build on 1/5 of your
land. How many acres of land can your new house
cover?
Solution
This is a multiplication problem, so
1 1
2 x 3 + 1 1
2
x =
x
3 5
3
5
7
1
7
=
x
=
3
5
15
You can build on exactly 7/15
of an acre
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page
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Department page
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