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The Mathematics 11
Competency Test
Dividing With Fractions
First, we need some terminology. When we wish to divide one quantity by another, the quantity
that we are dividing by is called the divisor, and the quantity being divided is call the dividend.
The result of the operation is sometimes called the quotient (although the word “quotient” is used
in other ways in mathematics as well).
432 ÷ 6 = 72
dividend
divisor
quotient
symbol for
“divide by”
When we invert a fraction, interchanging the numerator and denominator, we get the reciprocal
of the original fraction:
b
a
is the reciprocal of
a
b
Thus, for example
2
3
is the reciprocal of
3
2
7
5
is the reciprocal of
5
7
and
From the rule for multiplying two fractions together, you see that when we multiply a fraction by its
reciprocal, we get the result 1:
a b a×b
× =
=1
b a b×a
(Some people use the term “inverse” to refer to the reciprocal of a fraction, but this should not be
done in general, because the word “inverse” actually usually means something else in
mathematics.)
The procedure for dividing by a fraction is now easy to state in words: to divide by a fraction, you
just invert this divisor and multiply. So dividing by a fraction is done as multiplying by its
reciprocal. The multiplication is done exactly as described in the previous note in this series. So,
for example,
5 2 5 3 5 × 3 15
÷ = × =
=
7 3 7 2 7 × 2 14
In symbols, we can write
David W. Sabo (2003)
Dividing With Fractions
Page 1 of 3
a c a d a×d
÷ = × =
b d b c b ×c
Examples:
6 3 6 4 6×4
2 × 3 ×2×2 4
=
÷ =
× =
=
10 4 10 3 10 × 3
5
2 ×5× 3
4 5 4 8 4 × 8 32
÷ = × =
=
5 8 5 5 5 × 5 25
2 4 2 15 2 × 15 2 × 3 × 5 5
=
÷
= ×
=
=
3 15 3 4
3×4
3 × 2 ×2 2
In the first and third examples here, we factored the factors after the multiplication step to check
for possible simplification of the result. In the second example, we could see that there was no
possibility of simplification, because 5 is obviously not a factor of either 4 or 8.
Common Error
Quite often people have a vague recollection that division by a fraction involves flipping the
fraction and maybe something like multiplying numerators and denominators together, or
something like that. Then they end up doing something along the following lines:
5 2 5 3 5 × 2 10
÷ = × =
=
7 3 7 2 7 × 3 21
⇐
WRONG
mixing numerators and denominators in the multiplication step. This gives entirely the wrong
answer. You must do the multiplication step just like you would multiply any other pair of fractons
together.
Another Way of Writing Division by a Fraction
Since the form of a fraction represents the result of division of the numerator by the denominator,
we can also represent division by a fraction as a fractional expression in which the numerator is
the dividend value and the denominator is the divisor value. For example
5 2
÷ =
7 3
5
7
2
3
Later in these notes, we will indicate that expressions such as the one on the right above are
called complex fractions because they are a fraction whose parts contain other fractions. This
form does give a way to demonstrate that the rule for dividing by a fraction is consistent with
properties we’ve previously discovered about fractions.
David W. Sabo (2003)
Dividing With Fractions
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To try to simplify the complex fraction above, start by multiplying the numerator and denominator
by the reciprocal of the denominator:
5
7
2
3
=
5 3
×
7 2
2 3
×
3 2
This gives an equivalent fraction, since we are multiplying the numerator and denominator by the
same value. The thing is, the new denominator, which is the product of the original denominator
and its reciprocal, simplifies to 1:
2 3 2×3
=1
× =
3 2 3×2
and so
5
7
2
3
=
5 3
×
7 2
2 3
×
3 2
=
5 3
×
7 2
1
=
5 3 5 × 3 15
× =
=
7 2 7 × 2 14
when we drop the denominator of 1 (which can always be done because of the property that
b
= b for any number b). We get exactly the same result as we would have obtained using the
1
original rule for dividing by a fraction. This example illustrates why the invert and multiply rule
works for dividing by a fraction.
Division With Mixed Numbers
As has been true of all other arithmetic operations, division with mixed numbers requires you to
first convert the mixed numbers to pure fractions and then apply the methods appropriate for
fractions.
Example:
2
3
5 11 13
÷1 =
÷
4
8 4
8
David W. Sabo (2003)
=
11 8
×
4 13
=
11 × 8 11× 2 × 4 22
=
=
4 × 13
13
4 × 13
Dividing With Fractions
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