Properties of fractions

Important fraction rules

There are a few fraction rules that we need to clarify before we do anything too complicated:

Numerator equal to zero

When the numerator of a fraction is equal to $0$, then the whole fraction is equal to $0$. For algebra, this means that if the value of $x$ causes the numerator to equal $0$, then that value of $x$ will cause the whole fraction to equal $0$.

Take a look at this fraction:

$$\frac{(3x-6)}{(14x^2)}$$

For what value of $x$ does the fraction equal $0$?

Well, we know that if the numerator equals $0$ then the whole thing equals $0$. So, what value of $x$ would make the numerator $(3x-6)$ equal $0$?

Denominator equal to zero

When the denominator of a fraction is equal to $0$, then the fraction is undefined. In other words, it is good to remember that having $0$ as a denominator is “against the rules” in math. This can be a little confusing to some students. “Why is a zero on top allowed,” they ask, “but not on the bottom?” If you think about fractions as division, it can be a little easier to imagine.

If you have a pizza, that pizza can be easily divided into $8$ pieces, so that each slice becomes $\frac{1}{8}$ of a pizza, right? And if you cut a piece of paper into $2$, you now have two $\frac{1}{2}$ pieces of paper. Now try to imagine dividing something into $0$ pieces. Obviously, it won’t work. You can’t cause something not to exist by breaking it up into smaller pieces! That’s why any fraction with a $0$ on the bottom is undefined or infinite.

Remembering the difference

If you ever get confused as to which rule applies to what part of the fraction, try putting an easy example into your calculator:

$0/1$ evaluates to $0$ when input into a calculator.

$1/0$ evaluates to “Error” when input into a calculator.

A number divided by itself

A number divided by itself will always equal $1$. This is always the case except for zero. Zero divided by zero is still undefined because you are dividing by zero.

Canceling out common numbers

Something that you will encounter with fractions is that the most effective way to simplify them is by removing a number that exists on both the numerator and denominator. This topic is difficult to illustrate with words alone, so we will turn to some examples as we discuss:

How can we simplify this fraction?

$$\frac{2x}{2y}$$

Although we can’t do anything to combine $x$ and $y$, we can notice that there is a $2$ in both the numerator and denominator, which means we can actually remove that number from both parts. If this is a strange concept to you, check out this further analysis:

$$\begin{array}{rl}\frac{2x}{2y}& =\frac{2}{2}\ast \frac{x}{y}\\ & \\ \frac{2}{2}\ast \frac{x}{y}& =1\ast \frac{x}{y}\\ & \\ 1\ast \frac{x}{y}& =\frac{x}{y}\end{array}$$

By this process, we removed the $2$ from the numerator and denominator! You do not, however, need to remember this whole process. It is fine to just know that in this case they both cancel out.

Canceling out fractions within a fraction

Something else that you can run into is the occasional fraction within a fraction. One thing that can help you quickly simplify a difficult fraction like this is that the numerators cancel with numerators, and the denominators cancel with denominators. That’s a mouthful, but let’s take a look at this example:

$$\frac{(\frac{3}{(x+4)})}{(\frac{9}{(x+4)})}$$

This can look scary at first, but notice how there’s an $x+4$ in both the denominator on top and the denominator on bottom? Those cancel out just like the example above! So what we’re left with instead is:

$$\frac{3}{9}=\frac{1}{3}$$

See? That wasn’t so bad after all!

Canceling with addition and subtraction

An important detail to note about this process of simplifying fractions is that the number being taken from the numerator and denominator must be multiplied into every term. In other words, if there is addition and subtraction in a portion of the fraction, the term being canceled out must be factored to the outside of the parentheses. Here is an example:

$$\frac{2(x-1)}{2(y+4)}=\frac{(x-1)}{(y+4)}$$

Here, we cancel out the $2$s because they exist outside of the addition and subtraction. Compare this to the following:

$$\frac{(2x-1)}{(2y+4)}$$

In this case, we would no longer be able to cancel out that $2$ because it is taking part in the addition and in the subtraction. So remember, we can only cancel out common numbers if they exist outside of addition and subtraction, or in other words outside of the parentheses.

Multiplication and division with fractions

Performing multiplication with fractions should feel rather simple as we discuss it. The principle is that you multiply the numerators together from the two fractions to form one final numerator, and then perform the same process to get a final denominator. Putting these final portions together gets you the final product:

$$\begin{array}{rl}x& =\frac{1}{3}\ast \frac{2}{4}\\ & \\ x& =\frac{(1\ast 2)}{(3\ast 4)}\\ & \\ x& =\frac{2}{12}\\ & \\ x& =\frac{1}{6}\end{array}$$

We first rewrote the fraction so that we were multiplying the numerators together.

Then, we did the same thing for the denominators.

Finally, we put the new numerator and denominator together for our final answer $(\frac{2}{12})$.

We did an extra step, simplifying $\frac{2}{12}$ to $\frac{1}{6}$.

Division involves a similar process of multiplication, but with one new step: Instead of simply multiplying the two fractions together, we take the dividing fraction and flip it upside down. This is called the reciprocal. So, if we have our first fraction divided by our second fraction, we will instead multiply our first fraction by the upside down version (reciprocal) of our second fraction. Here is a visualization:

$$\begin{array}{rl}x& =\frac{1}{2}÷\frac{3}{4}\\ & \\ x& =\frac{1}{2}\ast \frac{4}{3}\end{array}$$

Now we perform our multiplication as we learned above:

$$\begin{array}{rl}x& =\frac{1}{2}\ast \frac{4}{3}\\ & \\ x& =\frac{(1\ast 4)}{(2\ast 3)}\\ & \\ x& =\frac{4}{6}\\ & \\ x& =\frac{2}{3}\end{array}$$

Addition and subtraction with fractions

When adding or subtracting with fractions, we have to remember a very important step: find the least common denominator. What this means is that we modify each fraction in our problem so that it has the same denominator. However, if we change the denominator then we have to change the numerator, too. Let’s explore this:

How do we add $\frac{1}{2}+\frac{2}{4}$?

You might recognize that $\frac{2}{4}$ is equal to $\frac{1}{2}$, so you can modify the equation to be

$$\frac{1}{2}+\frac{1}{2}$$

This is equal to $1$. We modify the second fraction in order to make it easier to interact with. In this case, it was so we could add it to a fraction with a different denominator. You’ll note that as we divided the denominator of the second fraction by $2$ to get $2$, we also divided the numerator by $2$ to get $1$.

Not every case is as easy as this example. So, we have a reliable way to find out how to modify fractions to get a common denominator: cross multiplying.

When you have a common denominator, you can add or subtract only the numerators. You should never add or subtract the denominators of fractions. However, after having added or subtracted the numerators, you may want to simplify the fraction into a smaller form.

Cross multiplying

In order to modify our fractions through cross multiplication, we need to multiply the bottom of the first fraction by the top of the second fraction, then multiply the bottom of the second fraction by the top of the first fraction. What this does is create the new numerators for our fractions. Then, we can find the common denominator by simply multiplying the denominators together. Follow these steps as you look at the example below:

1. Multiply the bottom of the first fraction by the top of the second fraction (this makes the new numerator for the second fraction)
2. Multiply the bottom of the second fraction by the top of the first fraction (this makes the new numerator for the first fraction)
3. Multiply the two denominators together (this creates the new denominator for both fractions)
4. Put all the new parts together in the equation and add or subtract the numerators

What is the sum of $\frac{3}{4}+\frac{5}{7}$?

Key points

Numerator equals zero. When the numerator equals zero, the whole fraction is equal to zero.

Denominator equals zero. When the denominator equals zero, the whole fraction is undefined.

Simplifying fractions by canceling numbers. We cancel out numbers that are on both the numerator and denominator of a fraction. Please note: You can only do this if they are outside of any parentheses.

Multiplication and division. Multiply by finding the product of the numerators and putting it over the product of the denominators. Do the same for division, except using the reciprocal (upside-down version) of the second fraction.

Addition and subtraction. Find a common denominator for the fractions and adjust the numerators accordingly. The easiest way to do this is through cross multiplication to find the new numerators, and multiplication of the denominators to find the new denominators.