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Using Rational Numbers

How to add, subtract, multiply and divide rational numbers

A rational number is a number that can be written as a simple fraction (i.e. as a ratio).

Examples:

So, a rational number looks like this:

p / q

But q cannot be zero, as that would be dividing by zero.

How to Add, Subtract, Multiply and Divide

If the rational number is something simple like 3, or 0.001, then just use mental arithmetic, or your calculator!

But if it is still in the form p / q, then read on to find how to handle it.

I will start with multiplication, as that is the easiest.

Multiplication

To multiply two rational numbers, just multiply the tops and bottoms separately, like this:

Here is an example:

 

Division

To divide two rational numbers, first flip the second number over (make it a reciprocal) and then do a multiply like above:

Here is an example:

 

Addition and Subtraction

I will cover Addition and Subtraction in one go, as they are the same method.

Before you can add or subtract, the rational numbers should have the same bottom number (called a Common Denominator).

The easiest way to do this is to

Multiply both parts of each number by the bottom part of the other

Like this (note that I use the dot · to mean multiply):

Here is an example of addition:

And an example of subtraction (I skipped the middle step to make it quicker):

Simplest Form

Sometimes you will have a rational number like:

That is not as simple as it can get!

We can actually divide both top and bottom by 5 to get:

÷ 5

÷ 5

Now it is in the "simplest form".

Be Careful With "Mixed Fractions"

You may be tempted to write an Improper Fraction (a fraction that is "top-heavy", i.e. where the top number is bigger then the bottom number) as a Mixed Fraction:

For example ⁷/₄ = 1 ³/₄, shown here:

Improper Fraction		Mixed Fraction
7/4		1 3/4
	=

But for mathematics the "Improper" form (such as ⁷/₄) is actually better.

Because Mixed fractions (such as 1 ³/₄) can be confusing when you write them down in a formula, as it can look like a multiplication:

 

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