Properties of fractions

Important fraction rules

There are a few fraction rules you’ll want to keep straight before you do anything more complicated.

Numerator equal to zero

If the numerator of a fraction is $0$, then the entire fraction equals $0$ (as long as the denominator is not $0$). In algebra, this means that any value of $x$ that makes the numerator equal to $0$ will make the whole fraction equal to $0$.

Take a look at this fraction:

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

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

Since a fraction equals $0$ when its numerator equals $0$, set the numerator equal to $0$ and solve:

$$3x-6=0$$

Denominator equal to zero

If the denominator of a fraction is $0$, then the fraction is undefined. In other words, having $0$ in the denominator is “against the rules” in math.

A helpful way to think about this is to remember that a fraction is division. For example:

• Cutting a pizza into $8$ equal pieces gives pieces that are each $\frac{1}{8}$ of the pizza.
• Cutting a sheet of paper into $2$ equal pieces gives pieces that are each $\frac{1}{2}$ of the paper.

Now try to imagine dividing something into $0$ equal pieces. That doesn’t make sense as a division process, so any fraction with $0$ in the denominator is undefined or infinite.

Remembering the difference

If you ever mix up which rule applies to which part of the fraction, try these simple calculator checks:

$0/1$ 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 equals $1$. This is always true except for zero.

Zero divided by zero is still undefined because you are dividing by zero.

Canceling out common numbers

A common way to simplify fractions is to remove a factor that appears in both the numerator and the denominator.

How can we simplify this fraction?

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

You can’t combine $x$ and $y$, but you can notice that both the numerator and denominator have a factor of $2$. That common factor cancels.

If you want to see why this works, here’s the idea written out:

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

So the $2$ cancels from the numerator and denominator. You don’t need to memorize the full breakdown - just remember that common factors cancel.

Canceling out fractions within a fraction

Sometimes you’ll see a fraction inside another fraction. A quick way to simplify is to notice that numerators cancel with numerators, and denominators cancel with denominators.

For example:

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

This looks complicated, but notice that $(x+4)$ appears in the denominator of the top fraction and also in the denominator of the bottom fraction. Those cancel.

That leaves:

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

Canceling with addition and subtraction

When you cancel factors, the factor you cancel must be a factor of the entire numerator and a factor of the entire denominator.

That’s why canceling works nicely when the common factor is outside parentheses:

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

Here, the $2$ is multiplied by every term in the numerator and every term in the denominator, so it cancels.

Compare that to this fraction:

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

In this case, you can’t cancel the $2$ because it is part of an addition/subtraction expression. It is not a factor of the entire numerator $(2x-1)$.

So, you can only cancel common numbers if they are outside of addition and subtraction (that is, outside parentheses).

Multiplication and division with fractions

To multiply fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator.

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

• Multiply the numerators: $1\ast 2$.
• Multiply the denominators: $3\ast 4$.
• Then simplify $\frac{2}{12}$ to $\frac{1}{6}$.

Division is similar, with one extra step. Instead of dividing by the second fraction, you multiply by its reciprocal (the fraction flipped upside down).

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

Now multiply:

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

Addition and subtraction with fractions

When you add or subtract fractions, you need a common denominator. A common denominator lets you combine the fractions by adding or subtracting only the numerators.

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 rewrite the expression as:

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

This equals $1$.

Notice what happened when we changed $\frac{2}{4}$ to $\frac{1}{2}$:

• The denominator was divided by $2$ (from $4$ to $2$).
• The numerator was also divided by $2$ (from $2$ to $1$).

Not every problem is this simple, so you need a reliable method for creating a common denominator. One common method is cross multiplying.

Once the denominators match, you add or subtract only the numerators. You should never add or subtract the denominators. Afterward, you may be able to simplify the result.

Cross multiplying

To rewrite two fractions with a common denominator using cross multiplication:

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.