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# Dividing Fractions

A fraction is a mathematical result of division that represents a portion of something. The denominator of a fraction can never equal zero. A zero denominator is undefined math.

Suppose we divided 96 by 14 using long division:

6 R 12

14/  96

The result has a remainder of 12, a fractional amount of the denominator.

Using fraction division instead of long division:

96/14 = 6 12/14 = 6 6/7

## Dividing Complex Fractions

(3/1) / (2/1) = 3/1 • 1/2 = 3/2 = 1 1/2

When we are dividing fractions there is a fraction numerator divided by fraction denominator. With a complex fraction we are dividing fractions by fractions, each has its own numerator and denominator.

Any number can be divided by 1 without changing its value. Dividing a numerator or denominator by 1 makes that fraction a complex fraction:

5/8 = (5/1) / (8/1) = (5/1) • (1/8) = 5/8

Therefore, when M and N are real numbers:

M/N = (M/1) / (N/1) = (M/1) • (1/N) = M/N

This is because multiplying and dividing fractions are opposite math operations.

Fraction division becomes multiplication by taking the reciprocal fraction of the denominator to multiply by the numerator.

When dividing fractions we can always perform the division using complex fractions.

=

N

P

M

M

1

N

P

=

MP

1    N

=

MP

N

=

2

4

3

3

1

2

4

=

34

1    2

=

12

2

=

6

## Dividing Mixed Fractions

Dividing mixed fractions consists of either or both the numerator or denominator being a mixed number.

Integer divided by a mixed number:

1. Convert mixed numbers to improper fractions:
5 / (1 3/4) = (5) / (7/4) =
2. Convert integer numerator to a fraction:
(5) / (7/4) = (5/1) / (7/4) =
3. Convert fraction division to fraction multiplication by the reciprocal of the denominator:
(5/1) / (7/4) = (5/1) • (4/7) =
4. Solve by fraction multiplication:
(5/1) • (4/7) = 20/7 = 2 6/7

(5/1) / (1 3/4) = (5) / (7/4) = (5/1) • (4/7) = 20/7 = 2 6/7

Mixed number divided by a mixed number:

2 3/5 / 1 1/3 = (13/5) / (4/3) = (13/5) • (3/4) = 39/20 = 1 19/20

3 3/7 / 1 2/3 = (24/7) / (5/3) = (24/7) • (3/5) = 72/35 = 2 2/35