Definition of a Fraction

Fractions are numbers that look like \frac{a}{b} where b\neq0 because division by 0 is meaningless.

a is the top of the fraction and is called the numerator.

b is the bottom of the fraction and is called the denominator.

Examples

All of these numbers are fractions

\frac{1}{3}  \frac{4}{5}  \frac{9}{8}  \frac{12}{1}

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Division: Fractions as Quotients

A fraction \frac{a}{b} is also known as a quotient of a to b. A quotient of a to b tells us to split the number a into b parts and it is equivalent to division.

We may also say that a is broken into b parts or that we divide the number a by b.

Example

Split a=4 into b=5 parts or perform the division of \frac{4}{5}

\frac{4}{5}=\frac{2}{2}(\frac{4}{5})=\frac{8}{10}=0.8

4 split into 5 parts is equivalent to saying x=\frac{4}{5} or 5x=4 where x=0.8

5\times0.8=0.8+0.8+0.8+0.8+0.8=\frac{4}{5}+\frac{4}{5}+\frac{4}{5}+\frac{4}{5}+\frac{4}{5}=\frac{20}{5}=4

Rational Numbers: Ratio of Integers

Many fractions are also Rational numbers. The Rational Numbers \mathbb{Q} are a special case of fractions where a is any integer and b is any integer besides 0. The Integers \mathbb{Z} are the positive and negative whole numbers along with 0, like on a number line.

\mathbb{Q}=[a,b\in\mathbb{Z}: b\neq0]

The Rational in Rational number means that there is a ratio between the two numbers a and b . The big \mathbb{Q} stands for quotient. The big \mathbb{Z} for the Integers is from the German word for numbers 'zahlen'.

There are many fractions that are not rational, not the ratio of 2 integers. These numbers are called irrational numbers and have a decimal value that is never-ending (infinite) and non-repeating.

We will be sticking to fractions that are Rational numbers throughout these lessons.

Negative Fractions

If a fraction is negative, we put the negative sign in the numerator or out in front of the fraction itself

-\frac{1}{4}=\frac{-1}{4}

If the negative sign happens to be in the denominator, don't write \frac{2}{-3}.

Move the negative sign up top or out front

\frac{2}{-3}\to\frac{-2}{3}

Fractional Units

A fractional unit is 'one part = numerator' of the 'whole'.

The 'whole' is the number that we 'split the number 1 into', which is the denominator.

Example

The green shaded box represents 'one part' of the entire thing, which is 'split' into 3 parts.

The 'whole' thing would be \frac{3}{3}=1.

Example

The green shaded box represents 1 unit out of 6 total units that make up the 'whole' thing.

There are 6 of these fractional units \frac{1}{6} that make up the object.

Fractional Values

Example

The green boxes represent \frac{2}{3} of the whole amount or 2 parts out of 3.

Example

The green boxes represent \frac{4}{6} of the whole amount or 4 parts out of 6.

Fractions of a Number

We can take fractional amounts of numbers other than 1. We can look at it as a multiplication exercise, a division exercise or a mixture of both.

Example

What is \frac{1}{2} of 44?

Multiplication

\frac{1}{2}\times44=22

Division

\frac{1}{2}\times44=\frac{44}{2}=22

Example

What is \frac{2}{3} of 120?

\frac{2}{3}\times120=\frac{2\times120}{3}=\frac{240}{3}=80

or

\frac{2}{3}\times120=2\times\frac{120}{3}=2\times40=80

Fractions: Ratios and Proportions

When comparing different amounts or types of the same thing, you can form fractions from the ratios or proportions.

Example

Suppose we have 10 crayons: 4 blue  3 red  2 green  1 yellow.

The 10 crayons can be placed in a 4:3:2:1 ratio of blue : red : green : yellow.

We can form 4 fractions: 1 for the amount that each color of crayon is of the whole amount of crayons 10

\frac{4}{10}=\frac{2}{5} are blue

\frac{3}{10} are red

\frac{2}{10}=\frac{1}{5} are green

\frac{1}{10} are yellow

Proper Fractions

Proper fractions have the values of their numerators less than the value of their denominators.

They have a value greater than 0 and less than 1 if positive

0<\frac{1}{5}<\frac{2}{5}<\frac{3}{5}<\frac{4}{5}<1

They have a value less than 0 and greater than -1 if negative

-1<-\frac{4}{5}<-\frac{3}{5}<-\frac{2}{5}<-\frac{1}{5}<0

Improper Fractions

Improper fractions are fractions that have their numerators larger than their denominators, which means they have a value greater than 1.

Examples

\frac{3}{2}   \frac{8}{3}   \frac{101}{100}

Mixed Numbers

A mixed number is the union of an integer and a fractional amount less than 1.

They are equivalent to their improper fractional form.

Example

Express \frac{43}{5} as a mixed number.

Divide the numerator 43 by the denominator 5 to obtain the highest whole number amount that will divide into it evenly.

Here it is 8 because \frac{40}{5}=8.

The remainder 3 is then the numerator of the fraction less than 1 that we attach next to the whole number, \frac{3}{5}.

\frac{43}{5}=\frac{40}{5}+\frac{3}{5}=8 \frac{3}{5}

Decimal Fractions

Decimal fractions have a power of 10 in the denominator.

Example

0.37284=\frac{37284}{100000}=\frac{3}{10}+\frac{7}{100}+\frac{2}{1000}+\frac{8}{10000}+\frac{4}{100000}=\frac{3}{10^{1}}+\frac{7}{10^{2}}+\frac{2}{10^{3}}+\frac{8}{10^{4}}+\frac{4}{10^{5}}

Equivalent Fractions

2 fractions \frac{a}{b} and \frac{c}{d} are equivalent if and only if the product of the extremes equals the product of the means

\frac{a}{b}=\frac{c}{d}\iff ad=bc

a and d are known as the extremes, while b and c are known as the means

Example

Determine whether \frac{a}{b}=\frac{4}{5} is equivalent to \frac{c}{d}=\frac{12}{15}

a=4, b=5, c=12 and d=15

ad=4\times 15=60 and bc=5\times 12=60\implies\frac{4}{5}=\frac{12}{15}

Simplifying Fractions

Most often we want fractions to be in simplest form. This means that the fraction is in lowest terms.

Lowest terms means that the numerator and denominator have no common factors or are relatively prime.

Example

\frac{4}{8}, \frac{3}{6}  and \frac{2}{4} are not in lowest terms but are all equal to \frac{1}{2}.

Both the numerators and denominators of each fraction have a common factor.

In \frac{4}{8}, both the 4 and the 8 have 4 as a common factor.

In \frac{3}{6}, both the 3 and the 6 have 3 as a common factor.

In \frac{2}{4}, both the 2 and the 4 have 2 as a common factor.

The simplified form of each of these fractions is \frac{1}{2}.

Example

Is \frac{15}{120} in lowest terms?

In \frac{15}{12o}, both 15 and 120 have the common factor of at least 5.

Actually, they both have the factor 15 in common

\frac{15}{120}=\frac{3}{24}=\frac{1}{8}

We first factored out a 5 to make \frac{3}{24} and then factored out another 3 to make \frac{1}{8} and 3\times5=15.

Example

Is \frac{12}{84} in lowest terms?

\frac{12}{84}=\frac{2\times2\times3}{2\times2\times3\times7}=\frac{1}{7}

We took the Prime Factorization of both 12=2\times2\times3 and 84=2\times2\times3\times7 and cancelled out the common factors of 2, 2 and 3.

Irreducible Fractions

Irreducible fractions are those that cannot be simplified any further (the numerator and denominator are in lowest terms). The numerator and denominator do not have any common factors and are called co-prime.

Examples

All of these fractions are irreducible or in lowest terms

\frac{2}{5}  \frac{6}{7}  \frac{11}{10}

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Patrick