# Arithmetic with fractions

## Introduction

This chapter is about arithmetic with fractions. Such a fraction may represent a rational number or a division, but for the arithmetic this makes no difference.

In mathematics fractions with a horizontal bar are widely used. The horizontal fraction bar in a way acts as a grouping symbol where normally brackets must be used.

$$\left(2+3\right)\xf7\left(1/3\right)=\frac{2+3}{\frac{1}{3}}$$

Note the different lengths of the fraction bars within the expression and the fact that the main bar is at the same horizontal level as the equal sign.

Sometimes a "bevelled" fraction with a slash fraction bar is written with the numerator slightly raised and the denominator slightly lowered. This reduces the height of the printout, but makes it less legible.

$$\raisebox{1ex}{$2+3$}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$}\right.$$

## Arithmetic operations

### Adding and subtracting fractions

Adding and subtracting fractions with the same denominator is straight forward: add or subtract both numerators to get the new numerator. The denominator stays the same.

$\frac{4}{7}-\frac{9}{7}=\frac{-5}{7}=-\frac{5}{7}$

When adding or subtracting fractions with different denominators, the fractions must be converted to the same denominator before being added or subtracted (also see Figure 1):

One unit (the pie) is divided in 3 × 4 = 12 parts. The 1/3 part is then divided in 4 one-twelfth parts and the 1/4 part is divided in 3 one-twelfth parts. Now we can add them up: 4/12 + 3/12 = 7/12.

### Multiplying fractions

First, a reminder that something divided by itself must always be equal to 1, and that something divided by 1 must always be equal to "something" itself.

$$\frac{7}{7}=1,\phantom{\rule{10px}{0ex}}\frac{3}{1}=3$$

We have seen already seen this in the chapter on division. Everyone gets 1 apple if 7 apples are equally divided among 7 persons. Jake gets 3 apples if 3 apples are shared between him and himself.

"Two times one third" (or "one third times two"), often expressed as "one third *of* two",
is the same as one third plus one third, since multiplication is repeated addition.
So we can just multiply the factor and the numerator. The denominator remains the same:

$$2\times \frac{1}{3}=\frac{2\times 1}{3}=\frac{2}{3}$$

In the same manner we can write:

$$3\times \frac{2}{8}=\frac{3\times 2}{8}=\frac{6}{8}$$

$$200\times \frac{1}{100}=\frac{200}{100}=2$$

$$3\times \frac{2}{5}=\frac{6}{5}=1\u2064\frac{1}{5}$$

How to calculate a fraction of a fraction, that is, a fraction times a fraction? The next pie diagram illustrates "one third of one fourth":

So, $\frac{1}{3}\times \frac{1}{4}=\frac{1}{3\times 4}$. This is illustrated above by dividing 1/4 pie in 3 equal parts, and continue this for the other 3 quarters of the pie. This divides the whole pie in 4 × 3 = 12 parts. So, one of the 3 equal parts a quarter pie is divided in must be 1/12 of a pie. Can you illustrate "One fourth of one third" using a pie diagram?

Now a little more difficult:

$$\begin{array}{l}\frac{2}{3}\times \frac{2}{4}=2\times 2\times \frac{1}{3}\times \frac{1}{4}\\ \phantom{\frac{2}{3}\times \frac{2}{4}}=4\times \frac{1}{12}\\ \phantom{\frac{2}{3}\times \frac{2}{4}}=\frac{4}{12}\end{array}$$

Can you see a way to reduce 4/12 by looking at the pie diagram above? Factorize the numerator and denominator and see if you can come to the same conclusion.

So when you multiply fractions, you need to multiply the numerators to get the new numerator and multiply the denominators to get the new denominator.

A fraction of a fraction (i.e., a fraction times a fraction) is called a *compound fraction*.

### Rules

There are a number of fraction rules that we can use when working with fractions. These rules always apply. The validity of the rules can be demonstrated by means of pie diagrams or other illustrative means, or they can be derived by using some algebra. Appendix A provides algebraic derivations of some of the arithmetic rules for fractions. It is important to learn these rules, by practicing a lot.

Let
`a`,
`b`,
`c` and
`d` be any number
and a denominator never be zero,
then the general arithmetic rules for fractions are:

$$\frac{a}{a}=1,\phantom{\rule{10px}{0ex}}\frac{a}{1}=a,\phantom{\rule{10px}{0ex}}\frac{0}{a}=0$$

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

$$\frac{a}{b}=\frac{a\times c}{b\times c}$$

$$\frac{a}{b}\pm \frac{c}{b}=\frac{a\pm c}{b}$$

$$\frac{a}{b}\pm \frac{c}{d}=\frac{\left(a\times d\right)\pm \left(c\times b\right)}{b\times d}$$

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

$$-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}$$

The "±" sign in a rule means that this rule applies to both addition and subtraction. You can read this rule with everywhere a plus sign or with everywhere a minus sign.

Rule 2 also applies to an integer multiplied by a fraction:

$$\begin{array}{l}3\times \frac{2}{11}=\frac{3}{1}\times \frac{2}{11}\\ \phantom{3\times \frac{2}{11}}=\frac{3\times 2}{1\times 11}=\frac{6}{11}\end{array}$$

Rule 4 is used for adding or subtracting fractions. In this case both denominators are
equal. When fractions with *unequal* denominators are to be added or subtracted,
both fractions need to be converted to the same denominator first. Both fractions need to have a *common denominator*.
Rule 5 can be used to convert to the same denominator.
This always works but is not always the most effective way. Later more about this.

$$\begin{array}{l}\frac{3}{4}-\frac{2}{3}=\frac{3\times 3}{4\times 3}-\frac{2\times 4}{3\times 4}\\ \phantom{\frac{3}{4}-\frac{2}{3}}=\frac{9}{12}-\frac{8}{12}\\ \phantom{\frac{3}{4}-\frac{2}{3}}=\frac{1}{12}\end{array}$$

An often made mistake is that rule 2 and rule 4 (or multiplying and adding) are confused:

$$\frac{2}{5}\times \frac{3}{5}\ne \frac{2\times 3}{5}$$

$$\frac{2}{5}+\frac{3}{5}\ne \frac{2+3}{5+5}$$

In the chapter about negative numbers we learned that a negative times a negative equals a positive. Can you, given the rules above, derive a general rule for a negative divided by a negative: $\frac{-a}{-b}=\mathrm{?}$

The *reciprocal* of a fraction is another fraction with the numerator and denominator exchanged.

The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ (and vice versa).

The reciprocal of 3 is $\frac{1}{3}$ (and vice versa).

A number multiplied by its reciprocal always yields 1 (why?). Dividing by a fraction means multiplying by the reciprocal of that fraction. This is often called "flip and multiply" or "invert and multiply". Rule 6 describes this.

$$\frac{2}{\frac{1}{3}}=2\times 3=6$$

Note the main fraction bar being slightly longer than the fraction bar in the denominator. When only using the operator "÷", you must use brackets to write $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\frac{1}{3}$}\right.$.

The division above could describe a real-live problem like: if a cake is decorated with two cherries per each 1/3 of the cake, how many cherries do you need to decorate a whole cake? Each one third of the cake is decorated with two cherries, so, the whole cake is decorated with 3 × 2 = 6 cherries: 6 cherries per cake.

Some examples:

$$\begin{array}{l}\frac{\frac{2}{3}+\frac{1}{7}}{\frac{4}{5}-\frac{2}{11}}=\frac{\frac{2\times 7+1\times 3}{3\times 7}}{\frac{4\times 11-2\times 5}{5\times 11}}=\frac{\frac{17}{21}}{\frac{34}{55}}\\ \phantom{\frac{\frac{2}{3}+\frac{1}{7}}{\frac{4}{5}-\frac{2}{11}}}=\frac{17}{21}\times \frac{55}{34}=\frac{17}{21}\times \frac{55}{17\times 2}\\ \phantom{\frac{\frac{2}{3}+\frac{1}{7}}{\frac{4}{5}-\frac{2}{11}}}=\frac{17}{17}\times \frac{55}{2\times 21}=\frac{55}{42}=1\u2064\frac{13}{42}\end{array}$$

$$\begin{array}{l}\frac{2+\frac{1}{3}}{\frac{1}{3}\times \frac{1}{5}}=\frac{\frac{2\times 3}{3}+\frac{1}{3}}{\frac{1}{3\times 5}}=\frac{\frac{7}{3}}{\frac{1}{15}}\\ \phantom{\frac{2+\frac{1}{3}}{\frac{1}{3}\times \frac{1}{5}}}=\frac{7}{3}\times 15=\frac{7\times 3\times 5}{3}\\ \phantom{\frac{2+\frac{1}{3}}{\frac{1}{3}\times \frac{1}{5}}}=7\times 5=35\end{array}$$

In the two examples above, the fractions are reduced to their simplest form. More about this in the next section.

## Reducing a fraction

Division can be thought of as reducing a fraction to its *simplest form*. What this means will be explained in this section.

A division can be explained as a quantity *per* other quantity. The result of the division is supposed to represent the *quantity per unit*.
For example, 15 apples per 3 persons (15 apples divided equally between 3 people) results in 5 apples *per person*.
One person is a unit in this example. The fraction 15/3 is reduced to 5/1 = 5.
The value of the fraction remained the same; 15/3 and 5 are *equivalent numbers*.

$$\begin{array}{l}\frac{15}{3}=\frac{3\times 5}{3}\\ \phantom{\frac{15}{3}}=\frac{3}{3}\times 5\\ \phantom{\frac{15}{3}}=1\times 5=5\end{array}$$

In the example above we used rule 3. Factors that the numerator and denominator share can be "canceled out". Above, 3 is canceled as a common factor. Another example:

$$\frac{45}{15}=\frac{\overline{)5}\times 9}{\overline{)5}\times 3}=\frac{9}{3}$$

In the above fraction common factor 5 is canceled out. So, the fraction is reduced,
but not to its simplest form, or *in lowest terms*. The fraction 9/3 can again be reduced to 3.

$$\frac{45}{15}=\frac{3\times \overline{)3}\times \overline{)5}}{1\times \overline{)3}\times \overline{)5}}=3$$

Both the numerator and denominator are factorized and then all common factors are canceled out.

A fraction is *irreducible* or *in lowest terms* or *in its simplest form*
if and only if the numerator and denominator do not share common prime factors.

This also holds for proper fractions:

$$\frac{30}{45}=\frac{2\times \overline{)3}\times \overline{)5}}{3\times \overline{)3}\times \overline{)5}}=\frac{2}{3}$$

Remember that reducing a fraction is the same thing as a division:

$$\frac{182}{13}=\frac{2\times 7\times \overline{)13}}{\overline{)13}}=14$$

1 | 4 | |||

13 | ) | 1 | 8 | 2 |

1 | 3 | ↓ | ||

5 | 2 | |||

5 | 2 | |||

0 |

But what about an improper fraction like 145/13? Numerator (145 = 5 × 29) and denominator do not share prime factors, so 145/13 is in lowest terms: it cannot be reduced. However, if we use long division, we get a quotient of 11 and a remainder of 2. This is the same as converting an improper fraction to a mixed number, as we have seen before. The remainder divided by the original divisor forms a proper fraction that may be reducible.

1 | 1 | |||

13 | ) | 1 | 4 | 5 |

1 | 3 | ↓ | ||

1 | 5 | |||

1 | 3 | |||

2 |

Converting an improper fraction to a mixed number is nothing more than determining the quotient and remainder.

$$\begin{array}{l}\frac{145}{13}=\frac{11\times 13+2}{13}\\ \phantom{\frac{145}{13}}=\frac{11\times 13}{13}+\frac{2}{13}\\ \phantom{\frac{145}{13}}=11\u2064\frac{2}{13}\end{array}$$

Some examples:

$$\frac{105}{3}=\frac{7\times 3\times 5}{3}=7\times 5=35$$

$$\frac{3}{6}=\frac{3}{3\times 2}=\frac{3}{3}\times \frac{1}{2}=\frac{1}{2}$$

$$\frac{4}{12}=\frac{4}{4\times 3}=\frac{1}{3}$$

$$\frac{63}{21}=\frac{9\times 7}{3\times 7}=\frac{3}{1}=3$$

$$\frac{54}{81}=\frac{6\times 9}{9\times 9}=\frac{2}{3}$$

$$\frac{72}{96}=\frac{9\times 8}{12\times 8}=\frac{3\times 3}{4\times 3}=\frac{3}{4}$$

$$\frac{72}{96}=\frac{{2}^{3}\times 3\times 3}{{2}^{3}\times 3\times 4}=\frac{3}{4}$$

$$\frac{100}{101}=\frac{{2}^{2}\times {5}^{2}}{101}=\frac{100}{101}$$

$$\frac{100}{102}=\frac{{2}^{2}\times {5}^{2}}{2\times 51}=\frac{50}{51}$$

$$\begin{array}{l}\frac{21}{6}=\frac{3\times 6+3}{6}=\frac{3\times 6}{6}+\frac{3}{6}\\ \phantom{\frac{21}{6}}=3+\frac{3}{6}=3\u2064\frac{1}{2}\end{array}$$

or:

$$\begin{array}{l}\frac{21}{6}=\frac{7}{2}=\frac{2\times 3+1}{2}\\ \phantom{\frac{21}{6}}=\frac{2\times 3}{2}+\frac{1}{2}=3\u2064\frac{1}{2}\end{array}$$

$$\frac{55}{42}=\frac{1\times 42+13}{42}=1\u2064\frac{13}{42}$$

In some occasions, before applying rule 5 it is easier to reduce the fractions (or one of the fractions) first:

$$\begin{array}{l}\frac{21}{28}-\frac{6}{9}=\frac{3\times 7}{4\times 7}-\frac{2\times 3}{3\times 3}\\ \phantom{\frac{21}{28}-\frac{6}{9}}=\frac{3}{4}-\frac{2}{3}\\ \phantom{\frac{21}{28}-\frac{6}{9}}=\frac{9}{12}-\frac{8}{12}\\ \phantom{\frac{21}{28}-\frac{6}{9}}=\frac{1}{12}\end{array}$$

## Comparing fractions

We can use the rules to rewrite a fraction in different ways while maintaining the fractional value. In this way, we can try to write two fractions in the same notation and find out whether the two fractions have the same fractional value. But what if we want to know whether a fraction is greater or less than an other fraction?

When we order fractions from least to greatest or from greatest to least we distinguish 3 possibilities:

- the denominators of the fractions are the same
- the numerators of the fractions are the same
- numerators and denominators are different.

If the denominators of the fractions are the same, it is easy to compare the fractions: 2/3 is greater than 1/3 because 2 is greater than 1. The equal portions the whole is divided in are the same, thus 2 of those portions is more than 1 portion, in other words, 2/3 is more of the whole than 1/3.

$$$$ because $$$$ because $$$$

If the numerators of the fractions are the same, we look at the denominators. If the denominator is greater than the denominator of an other fraction, the whole fraction is less than the other fraction, and vice versa.

$$$$

If the whole is divided into 3 equal parts, one such part must be smaller than a part if the whole is divided into only two equal parts. In other words, one third is less than a half.

$$\frac{3}{4}>\frac{3}{8}$$ because $$\left(3\times \frac{1}{4}\right)>\left(3\times \frac{1}{8}\right)$$ because $$\frac{1}{4}>\frac{1}{8}$$

3/4 is greater than 3/8 because 3 equal portions is greater than 3 smaller equal portions.

If the numerators and denominators are different, fractions usually need to be converted to a common denominator so that they can be compared.

$$\begin{array}{rcl}\frac{2}{18}& ?& \frac{1}{15}\\ \frac{2\times 15}{18\times 15}& ?& \frac{1\times 18}{15\times 18}\\ \frac{30}{18\times 15}& >& \frac{18}{15\times 18}\\ \frac{2}{18}& >& \frac{1}{15}\end{array}$$

In some cases, it is easier to reduce the fractions (or one of the fractions) first:

$$$$

Sometimes we can also reason to an answer:

$$$$ 1/18 < 1/15, thus 1/18 is certainly less than 2/15.

$$\frac{6}{10}>\frac{1}{3}$$ 6/10 is more than a half, 1/3 is less than a half.