Dividing Fractions

How to divide fractions?

No big deal, if you’re already familiar with multiplying fractions – then dividing fractions won’t be a problem.

Process of multiplying and dividing fractions is actually very similar. We could say division of fractions is literary one step away from multiplying.

Let me show you how:

$Dividing Fractions Regularly$

If we switch the numerator/denominator places of the second (divisor) fraction – we get good old multiplication.

Let’s examine the following cases:

Dividing whole numbers by fractions:

numerator > 1, denominator = 1
numerator = 1, denominator > 1
numerator and denominator > 1

Now, the other way around.

Dividing fractions by whole numbers:

numerator > 1, denominator = 1
numerator = 1, denominator > 1
numerator and denominator > 1

Dividing fractions (by fractions):

numerators = 1, denominator > 1
numerators > 1, denominators > 1
dividing mixed fractions

Complex fractions:

Simplifying Complex fractions
Solving Complex fractions

Before we start with dividing fractions, it’s important for you to:

Remember this:
All whole numbers have denominator = 1

Dividing whole numbers by fractions

numerator > 1, denominator = 1

4	÷	2	=	4	×	1	=	2×2×1	=	2	= 2
1		1		1		2		1×2		1

We’re actually dividing whole numbers written in fraction form. Of course, we reduce the yellow highlighted common factors.

We can see that the numerator of the divisor (second) fraction ends up below, in the “division area”.

numerator = 1, denominator > 1

4	÷	1	=	4	×	2	=	4×2	=	8	= 8
1		2		1		1		1×1		1

The question for this case would be: if a 4 is one half, how much would be the whole thing?

We see that the denominator of the divisor (second) fraction ends up above, in the “multiplication area”.

numerator > 1, denominator > 1

4	÷	2	=	4	×	3	=	2×2×3	=	6	= 6
1		3		1		2		1×2		1

Again we factored 4 to 2×2 to reduce it with the 2 below. No point in multiplying and dividing by the same number.

Dividing fractions by whole numbers

Let’s use the same fractions from the previous 3 examples but we them, but change the place of divisor and dividend.

Because of that, we’ll get the reciprocal values to those of previous 3 examples.
Again, if we want to treat whole numbers as fractions we must put 1 as their denominator.

numerator > 1, denominator = 1

2	÷	4	=	2	×	1	=	2×1	=	1
1	÷	1	=	1	×	4	=	1×2×2	=	2

numerator = 1, denominator > 1

1	÷	4	=	1	×	1	=	1×1	=	1
2	÷	1	=	2	×	4	=	2×4	=	8

numerator > 1, denominator > 1

2	÷	4	=	2	×	1	=	2×1	=	1
3	÷	1	=	3	×	4	=	3×2×2	=	6

Dividing fractions (by fractions)

numerators = 1, denominators > 1

1	÷	1	=	1	×	3	=	1×3	=	3
2	÷	3	=	2	×	1	=	2×1	=	2

This is what should you write down if someone would ask you something like this: “if a 1/2 is a 1/3 of some number, what would that number be?” (or something like that).

numerators > 1, denominators > 1

proper fractions division

3	÷	2	=	3	×	3	=	3×3	=	9
4	÷	3	=	4	×	2	=	4×2	=	8

improper fractions division

40	÷	56	=	40	×	12	=	8×5	×	4×3	=	5×4	=	20
9		12		9		56		3×3		8×7		3×7		21

The remaining numbers are primes, i.e. can’t be factored, so we just multiply.

dividing mixed fractions

First we have to turn our mixed fractions into improper ones.

2	1	÷	1	3	=	2×3+1	÷	1×4+3	=	7	÷	7	=
	3			4		3		4		3		4

After that we go down the same road as before:

7	÷	7	=	7	×	4	=	4	=	1	1
3		4		3		7		3			3

We eliminated the common 7…

And finally:

Complex fractions

In a case of a complex fraction we have the same story:

$Dividing Complex Fractions$

The product of outer (red) numbers (“a” and “d”) is going above as the quotient numerator, and the product of inner (blue) numbers (“b” and “c”) is going below as the quotient denominator.

This is how you simplify complex fractions:

$Dividing Complex Fractions$

We can simplify:

I with the III, and
II with the IV row

Of course, as before, we can individually simplify every fraction by itself:

I and II, and
III and IV row.

If the above was simplifying, now follows further solving, i.e. multiplying:

$Dividing Complex Fractions$

Therefore, we only aren’t allowed to reduce outer and inner “multiplication rows”.

Do NOT mix multiplying with simplifying!

At the end here are the 2 examples on how to turn “incomplete” complex fractions into complete ones.

There are usually 3 lines in complex fractions, and if there are 2, we must input the third in there somewhere. We must always be aware of the main line (in the middle), that’s the one in line with an equal sign (wow, this rhymes).