# How do you order the following fractions from smallest to largest?: {18/9, 32/24, 25/20, 33/12}?

May 14, 2016

From smallest to largest, they are $\left\{\frac{25}{20} , \frac{32}{24} , \frac{18}{9} , \frac{33}{12}\right\}$

#### Explanation:

The best way to order fractions is to first convert them to decimal fractions. Now

$\frac{18}{9} = 2.000000$

$\frac{32}{24} = \frac{4}{3} = 1.333333$

$\frac{25}{20} = 1.250000$

$\frac{33}{12} = \frac{11}{4} = 2.750000$

Hence comparing them and ordering them from smallest to largest, they are

$\left\{\frac{25}{20} , \frac{32}{24} , \frac{18}{9} , \frac{33}{12}\right\}$

May 14, 2016

$\frac{25}{20} , \frac{32}{24} , \frac{18}{9} , \frac{33}{12}$

#### Explanation:

If we express these fractions using the same denominator, then we just need to put the numerators in ascending order.

If we first reduce them to simplest terms (but still leave as fractions), then we find that they can all be expressed as multiples of $\frac{1}{12}$:

$\frac{18}{9} = \frac{2 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{9}}}}{1 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{9}}}} = \frac{2}{1} = \frac{2 \times 12}{1 \times 12} = \frac{\textcolor{b l u e}{24}}{12}$

$\frac{32}{24} = \frac{4 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{8}}}}{3 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{8}}}} = \frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{\textcolor{b l u e}{16}}{12}$

$\frac{25}{20} = \frac{5 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{5}}}}{4 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{5}}}} = \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{\textcolor{b l u e}{15}}{12}$

$\frac{33}{12} = \frac{11 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{3}}}}{4 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{3}}}} = \frac{11}{4} = \frac{11 \times 3}{4 \times 3} = \frac{\textcolor{b l u e}{33}}{12}$

So from smallest to largest, the fractions are:

$\frac{\textcolor{b l u e}{15}}{12} , \frac{\textcolor{b l u e}{16}}{12} , \frac{\textcolor{b l u e}{24}}{12} , \frac{\textcolor{b l u e}{33}}{12}$

In original form:

$\frac{25}{20} , \frac{32}{24} , \frac{18}{9} , \frac{33}{12}$

$\textcolor{w h i t e}{}$
Footnote

When reducing a fraction like $\frac{32}{24}$ to simplest form, we need to find the greatest common factor (GCF) of the numerator $32$ and denominator $24$.

One way of finding the GCF of two numbers goes like this:

• Divide the larger number by the smaller number to give a quotient and remainder.

• If the remainder is $0$ then the smaller number is the GCF.

• Otherwise repeat with the smaller number and the remainder.

So in the case of $32$ and $24$ we find:

$\frac{32}{24} = 1$ with remainder $8$

$\frac{24}{8} = 3$ with remainder $0$

So the GCF of $32$ and $24$ is $8$ and we can simplify $\frac{32}{24}$ by dividing both the numerator and denominator by $8$:

$\frac{32}{24} = \frac{\left(\frac{32}{8}\right)}{\left(\frac{24}{8}\right)} = \frac{4}{3}$

or if you prefer:

$\frac{32}{24} = \frac{4 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{8}}}}{3 \times \textcolor{red}{\cancel{\textcolor{b l a c k}{8}}}} = \frac{4}{3}$

May 15, 2016

The ordered set is : {25/20;32/24;18/9;33/12}

#### Explanation:

We want to put this set of numbers in ascending order:

{18/9;32/24;25/20;33/12}

First you can notice that all those fractions are improper (their numerators are bigger than denominators), which means that they represent numbers greater than $1$ which can be written as mixed numbers.

If you do so you get:

$\frac{18}{9} = 2$

$\frac{32}{24} = 1 \frac{8}{24}$

$\frac{25}{20} = 1 \frac{5}{20}$

$\frac{33}{12} = 2 \frac{9}{12}$

From this you see that the last two numbers are {18/9;33/12}

To order the remaining number you have to either reduce them or expand so that either their numerators or denominators are the same.

In both cases their denominators are multiples of numerators so you can reduce both fractions to get numerators equal to $1$

$1 \frac{8}{24} = 1 \frac{1}{3}$

$1 \frac{5}{20} = 1 \frac{1}{4}$

If two fractions have equal numerators then the one is bigger which has lower denominator. So the ascending order of those numbers is:

{25/20;32/24}.

If we combine those result we get the final answer:

{25/20;32/24;18/9;33/12}