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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

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.

Three oranges.

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

One and one half oranges.

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

=

N

P

M

M

1

N

P

=

MP

1    N

=

MP

N

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

=

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