Ordering and comparing rational numbers

To order / compare rational numbers;

a )You can order rational numbers either logically, based on its division.

b) You can think of a rational number as a fraction and then sort it. The way
we sort fractions is the same way to sort rational numbers.

We recommend that you also look at the topic of ordering in fractions.

Order   \( \displaystyle \frac{8}{5} \) and \( \displaystyle \frac{12}{5} \)

\( \displaystyle \frac{8}{5} \)   “8 divided by 5” , divide 8 by 5 

\( \displaystyle \frac{12}{5} \)  “12 divided by 5” , divide 12 by 5

You can find the solution to the equation either using a calculator or manual and compare the results or you can use logic to find out which one is larger or smaller.

Let’s do this by dividing ;

\( \displaystyle \frac{8}{5}=1,6 \)

\( \displaystyle \frac{12}{5}=2,4 \)

it is clear that \( \displaystyle \frac{12}{5}>\frac{8}{5}\)

Now let’s use logic;

Let’s say we have 8 units long stick. We need to divide this to 5 equal parts. We also have 12 units long stick in our other hand, and we must also divide this by 5. Note
that we are dividing both sticks into 5 equal parts, but the lengths of the sticks are different...
As a result, the longer the stick, the longer the divided pieces will be.

\( \displaystyle \frac{12}{5}>\frac{8}{5}\)

Compare \( \displaystyle \frac{15}{6}\) and \( \displaystyle \frac{15}{10}\)

We have 15 units long stick and we are dividing it into 6 equal parts;
We also have 15 units long stick in the other hand and we are dividing this into 10 equal parts - which one is larger?

The more you divide the sticks to equal lengths, the more pieces you will have, and they will get
smaller after each division. Therefore;

\( \displaystyle \frac{15}{6}>\frac{15}{10}\) 

Let’s find the result using division;

\( \displaystyle \frac{15}{6}=2,5\)  We have divided 15 with 6

\( \displaystyle \frac{15}{10}=1,5\) We have divided 15 with 10.

\( \displaystyle \frac{15}{6}>\frac{15}{10}\) 

Compare \( \displaystyle \frac{3}{4}\) and \( \displaystyle \frac{5}{6}\)

Firstly; neither the number that is divided nor the number that is used for division are
the same, also both rational numbers are between 0 and 1. If we can make the numbers that
are being divided equal, we can find a solution using the same logic from the previous section that you have just learnt

Let us equalize the denominators. Just like expanding in fractions…

 \( \displaystyle \frac{3}{4}.\frac{3}{3}=\frac{9}{12}\) There is no difference between dividing 3 by 4 and 9 by 12; the results are the same.

 \( \displaystyle \frac{5}{6}.\frac{2}{2}=\frac{10}{12}\)  In the same way there is no difference between dividing, 5 by 6 or 10 by 12. The results are the same.

The problem is now as follows;

\( \displaystyle \frac{9}{12}\) and \( \displaystyle \frac{10}{12}\)

Imagine that we have a stick of 9 units and that we are going to divide this into 12
equal parts; we also have a stick of 10 units and will divide it into12 equal parts.
Which one will be larger? I am dividing both sticks by 12, so however long our stick,
the longer the individual parts.

\( \displaystyle \frac{10}{12}>\frac{9}{12}\)

\( \displaystyle \frac{5}{6}>\frac{3}{4}\)

Compare \( \displaystyle \frac{4}{6}\) and \( \displaystyle \frac{5}{8}\) 

Both numbers are between, 0 and 1 and the divided numbers are not the same as the numbers used for dividing; so let us equalize one or the other.

This time let us equalise the numbers being divided. I equalise through expanding;

\( \displaystyle \frac{4}{6}.\frac{5}{5}=\frac{20}{30}\)

\( \displaystyle \frac{5}{8}.\frac{4}{4}=\frac{20}{32}\)

We have a stick of 20 units and we divide this into 30equal parts, we also have another
stick of 20 units and this time we divide by 32. The more we divide the stick; the pieces
will get smaller; for bigger pieces we need to divide less. So;

\( \displaystyle \frac{20}{30}>\frac{20}{32}\)

So the result is ;

\( \displaystyle \frac{4}{6}>\frac{5}{8}\)

For numbers where it is not clear whether  it is larger or smaller; for rational numbers where the dividend and divisor numbers are different –  you can use logic to find the result by equalizing.

\( \displaystyle 2\frac{1}{4}\) and \( \displaystyle 1\frac{3}{5}\)

You do not have to pair in anyway; simply, 2 as a whole number with a fraction component, is larger than 1 with a fraction component.

\( \displaystyle 2\frac{1}{4}\) It is between 2 and 3.

\( \displaystyle 1\frac{3}{5}\) It is between 1 and 2.

\( \displaystyle 2\frac{1}{4}>1\frac{3}{5}\) 

\( \displaystyle 1\frac{3}{5}\) and \( \displaystyle 1\frac{1}{4}\)

Both numbers are made up of single whole numbers and fractional components. To figure out which one is larger, we need to compare the fractions.

\( \displaystyle \frac{3}{5}\)  Larger than half

\( \displaystyle \frac{1}{4}\)   Quarter
By equalizing the dividend or divisor number, you can
find out which rational number is larger than the other.

\( \displaystyle 1\frac{3}{5}>1\frac{1}{4}\)

Comparison in negative rational numbers

The rules for sorting integers are the same for rational numbers.
For example ; When \( \displaystyle 4>3\)
 , with negative integers it is \( \displaystyle -4<-3\) 

In the same way, half negative is smaller than a quarter negative .

\( \displaystyle \frac{1}{2}>\frac{1}{4}\) becomes \( \displaystyle -\frac{1}{2}<-\frac{1}{4}\)

\( \displaystyle -\frac{3}{4}\) and \( \displaystyle -\frac{5}{6}\)

\( \displaystyle \frac{5}{6}>\frac{3}{4}\) Let us sort according to positive rational numbers, we have explained this above.

 \( \displaystyle -\frac{5}{6}<-\frac{3}{4}\)  Now, let us add the - sign, let s turn the ‘smaller than’ and ‘greater than’ symbols on their head.

Both rational numbers are between, 0 and -1 but \( \displaystyle -\frac{5}{6}\) is closer to -1. So it is smaller.

\( \displaystyle -\frac{5}{6}<-\frac{3}{4}\)

\( \displaystyle -2\frac{1}{4}\) and \( \displaystyle -1\frac{3}{5}\)

If you have learnt between which two numbers the rational numbers should be in,
there is then no need to sort according to positive integers.

\( \displaystyle -2\frac{1}{4}\)  Between -2 and -3

 \( \displaystyle -1\frac{3}{5}\)  Between -1 and -2

\( \displaystyle -2\frac{1}{4}< -1\frac{3}{5}\)

Ordering Positive Rational Numbers and
Negative Rational Numbers

\( \displaystyle \frac{1}{2}\) and \( \displaystyle -1\frac{3}{5}\)

Positive numbers, no matter what the  number are always bigger than negative numbers.

So it is  \( \displaystyle \frac{1}{2}> -1\frac{3}{5}\)