Fractions: adding, subtracting, multiplying and dividing

We can use a fraction to write any rational number. A fraction has two parts. The top part is called the numerator and the bottom part the denominator. There is also the complex (or compound) fraction, which is nothing more than a fraction that has another fraction in its numerator or denominator. And by the way, all the signs between fractions (plus, minus, equals and so on) are always written level with the fraction bar, not level with the numerator or the denominator. And anyway, did you know that five out of four people have trouble with fractions?

The basics

A fraction has the following form:

$$\frac{\text{numerator}}{\text{denominator}}$$

An example of a fraction is

$$\frac{2}{5},$$

that is, two fifths. The denominator gets its name because it names the fraction. Fifths, thirds, sixths… that is the main name of the fraction, and it comes from the number below the fraction bar. The numerator, on the other hand, tells you how many there are – in the previous example it was two fifths. So much for the names.

The numerator and the denominator can be more or less any number, or even another fraction, but most often we meet fractions where both are natural numbers.

A fraction is just another way of writing division, and you work out the value of a fraction by dividing the numerator by the denominator. So in general, if we have a fraction $\frac{a}{b}$, then the value of the fraction is the number a/b. The fraction above (two fifths) would have the value 2/5, which is 0.4.

Simplifying a fraction

We can work with fractions in various ways and change their form – writing equivalent fractions and simplifying them – without changing the value of the fraction at all. It is easy to picture in words: one half has the same value as two quarters or four eighths. This follows from the fact that a fraction is just division in disguise, and we can arrive at one half by dividing several different pairs of numbers. Four divided by eight is one half. Ten divided by twenty is also one half. That is why the fractions

$$\frac{1}{2}=\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\dots =\frac{10}{20}=\dots$$

all have the same value.

As you can see, we found more fractions with the same value by multiplying both the numerator and the denominator of the original fraction 1/2 by two. After multiplying, we got the fraction 2/4, two quarters. If we multiply the numerator and denominator of this fraction by two as well, we get the fraction 4/8, four eighths. At this point we were writing equivalent fractions (multiplying the numerator and denominator by the same number).

The opposite of this is simplifying (cancelling down) a fraction, where we divide both the numerator and the denominator by the same number. To simplify a fraction, we have to find a number that divides both the numerator and the denominator exactly. Simplifying fractions is used very often in practice, because it makes a fraction much simpler and easier to work with. Suppose we have the fraction 12/18. At a glance we see that both numbers are even, so they are divisible by two. We could certainly simplify the fraction by dividing by two, but a second look shows that the numerator and the denominator are both divisible by six as well. If we divide by six, we get a simpler fraction than if we had divided only by two. So we divide by six, and we get the fraction 2/3.

This fraction cannot be simplified any further: there is no number (greater than one) that divides both the numerator and the denominator exactly. A fraction that cannot be simplified any further is said to be in its simplest form. Always try to work with fractions in their simplest form. Whenever you work with fractions, first check whether any of them can be simplified. You will save yourself time. Here is one more example:

$$\frac{24}{42}$$

Which number divides both 24 and 42? They are both even, so certainly two. We divide the numerator and the denominator by two:

$$\frac{24}{42}=\frac{12}{21}$$

Is there a number that divides both of these numbers? Yes, this time it is three.

$$\frac{12}{21}=\frac{4}{7}$$

Is there still a number that divides both the numerator and the denominator? There is not, so the fraction is in its simplest form.

Multiplying fractions

You may be surprised, but multiplying and dividing is easier with fractions than adding and subtracting. To multiply two fractions, you simply multiply the numerator of the first fraction by the numerator of the second, and the denominator by the denominator. That is all. An example of multiplying fractions:

$$\frac{2}{3}\cdot \frac{5}{7}=\frac{2\cdot 5}{3\cdot 7}=\frac{10}{21}$$

When multiplying fractions there is another way to simplify them. You do not have to cancel only within a single fraction – you can cancel crosswise. If the numerator of the first fraction and the denominator of the second fraction can be cancelled, you can do so and make the multiplication easier (this is called cross-cancelling). Example (the numbers being cancelled are highlighted):

$$\frac{\boxed{\text{4}}}{5}\cdot \frac{3}{\boxed{\text{8}}}=\frac{1}{5}\cdot \frac{3}{2}=\frac{3}{10}$$

What did we do? We divided both the 4 and the 8 by four. The value of the product stayed the same. If we did not cancel now, we could still cancel after multiplying.

To multiply a fraction by an integer, you simply turn it into a multiplication of two fractions by writing the integer c as c/1:

$$\frac{3}{7}\cdot 5=\frac{3}{7}\cdot \frac{5}{1}=\frac{3\cdot 5}{7\cdot 1}=\frac{15}{7}$$

As you can see, when you multiply a fraction by an integer, it is enough to multiply the numerator of the fraction by that number.

$$c\cdot \frac{a}{b}=\frac{ac}{b}$$

Dividing fractions

Dividing fractions is practically the same as multiplying them. To divide one fraction by another, you turn the second fraction (the divisor) upside down and multiply the fractions as usual. Here is a simple example of dividing fractions (notice that after turning the fraction upside down we can cancel 12 and 6):

$$\frac{12}{7}÷\frac{6}{11}=\frac{12}{7}\cdot \frac{11}{6}=\frac{2}{7}\cdot \frac{11}{1}=\frac{22}{7}$$

We changed the division into a multiplication by turning the fraction $\frac{6}{11}$ upside down to get $\frac{11}{6}$ and multiplying it by the other, unchanged fraction. Why does this work? Just think of a simpler example, such as dividing by one half. If we are working out

$$\frac{10}{1/2}=?,$$

what are we actually doing? We are finding how many times one half fits into ten. One half fits into each unit exactly twice (two times one half is one), so it fits into ten twenty times. We get the same result if we multiply two by ten – again twenty. And what value do we get when we turn one half upside down? Two over one, which is two.

$$\frac{1}{2}\to \frac{2}{1}=2$$

So in the end we have:

$$\frac{10}{1/2}=10\cdot \frac{2}{1}=10\cdot 2=20.$$

Adding fractions

Adding fractions tends to be a little more complicated. We can only add fractions if they have the same denominator. If they do not, we have to rewrite them with a common denominator. After that we proceed just as simply as in multiplication: we add the numerator of the first fraction to the numerator of the second fraction. Unlike in multiplication, though, we keep the same denominator. First, an example of adding fractions with the same denominator:

$$\frac{1}{2}+\frac{5}{2}=\frac{6}{2}=3$$

Does that make sense? One half plus five halves equals six halves – that makes quite a lot of sense.

If the fractions do not have the same denominator, which is the more common case, we have to give them a common denominator, which means writing one or both of them as equivalent fractions so that we get the same denominator. Let’s add these two fractions:

$$\frac{2}{3}+\frac{5}{2}=?$$

The first fraction has a 3 in the denominator and the second has a 2. It is hard to add them as they are, but if we multiply the numerator and denominator of the first fraction by two, it will have a 6 in the denominator, and if we multiply the numerator and denominator of the second fraction by three, it will also have a 6 in the denominator. Now both fractions have a common denominator and we can simply add them. So in the first step we rewrite the fractions so that they have the same denominator:

$$\frac{2}{3}+\frac{5}{2}=\frac{2\cdot 2}{3\cdot 2}+\frac{5\cdot 3}{2\cdot 3}=\frac{4}{6}+\frac{15}{6}$$

And now we just add the numerators and keep the denominator the same:

$$\frac{4}{6}+\frac{15}{6}=\frac{19}{6}$$

Watch out: when adding, we cannot cancel across fractions as we do in multiplication. For example, after rewriting we had a 6 in the denominator of the first fraction and a 15 in the numerator of the second. Even so, we cannot cancel by three:

$$\frac{4}{\boxed{\text{6}}}+\frac{\boxed{\text{15}}}{6}\ne \frac{4}{2}+\frac{5}{6}$$

We could only allow ourselves this cancelling if we were multiplying the fractions:

$$\frac{4}{\boxed{\text{6}}}\cdot \frac{\boxed{\text{15}}}{6}=\frac{4}{2}\cdot \frac{5}{6}$$

How do we give two fractions a common denominator in general? We multiply the numerator and denominator of the first fraction by the denominator of the second, and the numerator and denominator of the second fraction by the denominator of the first. In general we can write it like this:

$$\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}$$

We multiplied the numerator and denominator of the first fraction by d, which is the denominator of the second fraction. We multiplied the numerator and denominator of the second fraction by b, which is the denominator of the first fraction. A specific example:

$$\frac{7}{4}+\frac{9}{8}=\frac{7\cdot 8}{4\cdot 8}+\frac{9\cdot 4}{8\cdot 4}$$

We multiplied the numerator and denominator of the first fraction by eight, and those of the second by four. After that, we can add the fractions.

The general formula for adding fractions would then look like this:

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$

Subtracting fractions

Subtracting fractions works exactly like adding them, except that we subtract the resulting numerators instead of adding them. So we change the general formula for addition above like this:

$$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$$

A specific example; first we give the fractions a common denominator:

$$\frac{7}{9}-\frac{2}{5}=\frac{7\cdot 5}{9\cdot 5}-\frac{2\cdot 9}{5\cdot 9}=\frac{35}{45}-\frac{18}{45}$$

And now we just subtract the numerators and keep the denominator the same:

$$\frac{35}{45}-\frac{18}{45}=\frac{35-18}{45}=\frac{17}{45}$$

Cross-multiplication

When solving equations we can use cross-multiplication, which simplifies an equation between two fractions. It is really nothing more than two steps that keep the equation equivalent (doing the same to both sides). So let’s take an equation with two fractions

$$\frac{a}{b}=\frac{c}{d}.$$

We can rearrange this equation by multiplying it by the denominator of the first fraction and then by the denominator of the second. First we multiply the equation by the denominator of the first fraction, that is, by b. We get

$$\frac{ab}{b}=\frac{bc}{d}.$$

Then we multiply the equation by d and get

$$\frac{abd}{b}=\frac{bcd}{d}.$$

In the first fraction we cancel b and in the second we cancel d. The result is the equation

$$ad=bc.$$

So we can treat this as a formula: if we have an equation in the form of two fractions, we can rewrite it as follows:

$$\begin{array}{rcl}\frac{a}{b}& =& \frac{c}{d}\to \\ ad& =& bc\end{array}$$

An example follows. In the first step we rewrite the fractions using the formula above, and in the second step we just tidy up the expressions on both sides.

$$\begin{array}{rcl}\frac{x+2}{x}& =& \frac{3x}{x+2}\\ (x+2)(x+2)& =& x\cdot 3x\\ (x+2{)}^2& =& 3x^2\end{array}$$

The equation would then be solved like any other quadratic equation, but that is beyond the scope of this article.

Related topics

Fractions often appear when simplifying expressions, which is covered in the article on algebraic fractions. There is also an algorithm for dividing a polynomial by a polynomial.