Adding, subtracting, multiplying and dividing fractions

We can use a fraction to write any rational number. A fraction consists of two parts. The top part is called the numerator and the bottom part is called the denominator. There is also a compound fraction, which is nothing but a fraction that has another fraction in the numerator or denominator. And by the way, all signs between fractions (plus, minus, equals, etc.) are written at the fraction line level as a matter of principle, not at the numerator or denominator level. And anyway, did you know that five out of four people have problems with fractions?

Background information

A fraction has the following form:

$$\frac{\mbox{ Reader }}{\mbox{ Denominator }}$$

An example of a fraction might be a fraction

$$\frac25,$$

that's two-fifths. The denominator is called the denominator because it names the fraction. A fifth, a third, a sixth... this is the main name of the fraction and is derived from the number that is below the fraction line. The numerator, on the other hand, specifies the number, in the previous example it was two fifths. So much for names.

In principle, the numerator and denominator can be any number or, again, a fraction, but most often we encounter a fraction where both the numerator and denominator are natural numbers.

A fraction is just a division written differently; we calculate 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 previous fraction (two fifths) would then have the value 2/5, which is 0,4.

Converting to the base form

We can work with fractions in various ways to change their shape - expanding and contracting them - without changing the value of the fraction. It is also easy to think of it in words, for example one half has the same value as two quarters or four eighths. It follows that a fraction is just a division in disguise. And we can get to the number one half by dividing several different numbers. So four divided by eight is one half. Ten divided by twenty is also one half. That's why fractions

$$\frac12=\frac24=\frac36=\frac48=\ldots=\frac{10}{20}=\ldots$$

have the same value.

As you can see, we arrived at other fractions with the same value by multiplying both the numerator and denominator in the original fraction 1/2 by two. After multiplying, the fraction came out to be 2/4, two fourths. If we multiply the numerator and denominator by two for this fraction as well, we get the fraction 4/8, four-eighths. At this point, we expanded the fraction.

The opposite operation to expanding is truncating fractions, where we divide the numerator and denominator by the same number. If we want to shorten a fraction, we must find a number by which both the numerator and denominator are divisible without remainder. Fractional reduction is very often used in practice because reduction makes the fraction much simpler and easier to work with. If we have a fraction 12/18, a cursory glance reveals that both numbers are even, hence divisible by two. Of course, we could truncate the fraction by two, but a second glance reveals that the numerator and denominator are also divisible by six. If we truncate the fraction by six, we get a simpler fraction than if we only truncate by two. Therefore, we truncate the fraction with six. The result is the fraction 2/3.

This fraction cannot be further shortened; there is no number that we can divide both the numerator and denominator by without remainder. A fraction that cannot be further reduced is said to be in base form. Always try to work with fractions in base form; if you are going to work with fractions somewhere, look first to see if some fractions can be truncated. It will save you time. One more example:

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

What number is divisible by 24 and 42 at the same time? They are both even numbers, so definitely two. Divide the numerator and denominator by two:

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

Is there any number that divides both numbers at the same time? Yes, three this time.

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

Is there still a number that divides both the numerator and denominator? No, the fraction is in base form.

Multiplying fractions

You may be surprised, but multiplication and division are easier for fractions than addition and subtraction. If you have to multiply two fractions, you simply multiply the numerator of the first fraction with the numerator of the second fraction and the denominator with the denominator. That's it. Example of multiplying fractions:

$$\frac23\cdot\frac57=\frac{2\cdot5}{3\cdot7}=\frac{10}{21}$$

When multiplying fractions, there is an additional way of truncating fractions. You don't have to multiply only within a fraction, but you can cross multiply. If you can truncate the numerator of the first fraction with the denominator of the second fraction, you can do that and simplify your multiplication. Example (truncated numbers are highlighted):

$$\frac{\fbox{4}}{5}\cdot\frac{3}{\fbox{8}}=\frac15\cdot\frac32=\frac{3}{10}$$

What did we do? We divided both four and eight by four. The value of the product remained unchanged. If we didn't truncate it now, we could truncate it after multiplying.

You can then convert multiplying a fraction by an integer to multiplying two fractions simply by writing the integer c as c/1:

$$\frac{3}{7}\cdot5=\frac37\cdot\frac51=\frac{3\cdot5}{7\cdot1}=\frac{15}{7}$$

As you can see, if you multiply a fraction by an integer, then you just multiply the numerator of the fraction by that number.

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

Dividing fractions

Division of fractions is practically the same as multiplication. If you want to divide one fraction by another, you reverse one of the fractions and multiply the fractions normally. A simple example of dividing fractions (note that after reversing the fraction, we can multiply 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 converted division to multiplication by flipping the fraction $\frac{6}{11}$ into the fraction $\frac{11}{6}$ and multiplying it with the other unchanged fraction. Why does this work? Just imagine it with a simpler example, such as dividing by one half. If we divide

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

what are we actually doing? We're finding out how many times one half fits into ten. In each unit, the half fits just twice (two times one half is one), so it fits twenty times in ten. We get the same result when we multiply two times ten - also twenty. And what value do we get when we flip one half? Two ones, which is two.

$$\frac12\rightarrow\frac21=2$$

In the result, we have:

$$\frac{10}{1/2}=10\cdot\frac21=10\cdot2=20.$$

Adding fractions

Adding fractions is a little more complicated. We can only add fractions if the fractions have the same base, i.e. the same denominator. If the fractions do not have the same denominator, we have to convert them to the same denominator. We then proceed simply as in the case of multiplication, simply adding the numerator of the first fraction to the numerator of the second fraction. Unlike multiplication, however, we keep the denominator the same. First, an example of adding fractions with the same base:

$$\frac12+\frac52=\frac62=3$$

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

If the fractions don't have the same base, which is more often the case, we have to convert the fractions to the same base, which means expanding one or both fractions to get the same denominator. Let's want to add these two fractions:

$$\frac23+\frac52=?$$

The first fraction has a three in the denominator, the second fraction has a two. At this point, it's hard to add them, but if we expand the first fraction by two, it will have a six in the denominator, and if we expand the second fraction by three, it will also have a six in the denominator. Now both fractions have the same base and we can simply add them. So in the first step, we expand the fractions so that they have the same denominator:

$$\frac23+\frac52=\frac{2\cdot2}{3\cdot2}+\frac{5\cdot3}{2\cdot3}=\frac{4}{6}+\frac{15}{6}$$

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

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

Note that when adding, we can't multiply across fractions like we can with multiplication. For example, after the adjustment we had a six in the denominator of the first fraction and a fifteen in the numerator of the second fraction. Yet we can't multiply by three:

$$\frac{4}{\fbox{6}}+\frac{\fbox{15}}{6}\ne\frac42+\frac56$$

We could only afford this truncation if we were multiplying the fractions:

$$\frac{4}{\fbox{6}}\cdot\frac{\fbox{15}}{6}=\frac42\cdot\frac56$$

How do we generally convert two fractions to a common denominator? We expand the first fraction by the denominator of the second fraction, and we expand the second fraction by the denominator of the first fraction. In general, we can write it like this:

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

We expanded the first fraction with the expression d, which is the denominator of the second fraction. We expanded the second fraction with the expression b, which is the denominator of the first fraction. A concrete example:

$$\frac74+\frac98=\frac{7\cdot8}{4\cdot8}+\frac{9\cdot4}{8\cdot4}$$

We expanded the first fraction by an eight, and the second fraction by a four. After this we can add the fractions.

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

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

Subtraction of fractions

Subtraction of fractions is exactly the same as addition of fractions, except that we do not add the resulting numerators, but subtract them. So we modify the previous general addition formula as follows:

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

A concrete example; we convert to the common denominator first:

$$\frac79-\frac25=\frac{7\cdot5}{9\cdot5}-\frac{2\cdot9}{5\cdot9}=\frac{35}{45}-\frac{18}{45}$$

Now we just subtract the numerators, leaving the denominators the same:

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

The cross rule

In the case of equations, we can use the cross rule, which simplifies the equality of two fractions. In practice, it is nothing more than two equivalent modifications of a given equation. So let's have an equation of two fractions

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

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

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

Then we multiply the equation by the expression d to get

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

We truncate b in the first fraction and d in the second. The result is the equation

$$ad=bc.$$

So we can consider this as a formula, if we have an equation in the form of two fractions, we can modify them as follows:

$$\begin{eqnarray} \frac{a}{b}&=&\frac{c}{d}\rightarrow\\ ad&=&bc \end{eqnarray}$$

An example follows. In the first step we break down the fractions according to the previous formula and in the second step we just slightly modify the expressions on both sides.

$$\begin{eqnarray} \frac{x+2}{x}&=&\frac{3x}{x+2}\\ (x+2)(x+2)&=&x\cdot3x\\ (x+2)^2&=&3x^2 \end{eqnarray}$$

Next, the equation would be calculated as a classical quadratic equation, but that is beyond the scope of this article.

Related topics

Fractions tend to be a common problem when simplifying expressions, this is dealt with in the article fractional expressions where appropriate, and specifically there is an algorithm for dividing polynomials by polynomials.