Question

How do you order the rational numbers from least to greatest: $0.11,-{\displaystyle \frac{1}{9}},-0.5,{\displaystyle \frac{1}{10}}$?

Answer 1

Hint: We will first separate out the positive and negative numbers, then we will make the denominator of the numbers the same, so that we can easily compare the numerators easily.

Complete step by step solution:
We are given that we are required to order the rational numbers from least to greatest: $0.11,-{\displaystyle \frac{1}{9}},-0.5,{\displaystyle \frac{1}{10}}$.
Now, we can clearly observe that we have two positive numbers and two negative numbers.
Negatives are surely less than the positive ones.
Let us look at those for once now: $-{\displaystyle \frac{1}{9}},-0.5$.
Now, we can write – 0.5 as $-{\displaystyle \frac{5}{10}}$.
Now, let us multiply and divide $-{\displaystyle \frac{1}{9}}$ by 10, so that we have the following equation with us:-
$\Rightarrow -{\displaystyle \frac{1}{9}}=-{\displaystyle \frac{1}{9}}\times {\displaystyle \frac{10}{10}}=-{\displaystyle \frac{10}{90}}$
Multiplying and dividing $-{\displaystyle \frac{5}{10}}$ by 9, we will obtain the following equation with us:-
$\Rightarrow -{\displaystyle \frac{5}{10}}=-{\displaystyle \frac{5}{10}}\times {\displaystyle \frac{9}{9}}=-{\displaystyle \frac{45}{90}}$
Comparing both the numbers now, we can clearly see that:-
$\Rightarrow -{\displaystyle \frac{45}{90}}<-{\displaystyle \frac{10}{90}}$
Writing these numbers in the original form, we will then obtain the following equation with us:-
$\Rightarrow -0.5<-{\displaystyle \frac{1}{9}}$
Now, let us compare the positive numbers which are 0.11 and ${\displaystyle \frac{1}{10}}$ which is equal to 0.1
Now, we can clearly see that 0.1 < 0.11
Therefore, we have the following equation with us:-
$\Rightarrow {\displaystyle \frac{1}{10}}<0.11$
Thus, we have the final order given by the following equation:-
$\Rightarrow -0.5<-{\displaystyle \frac{1}{9}}<{\displaystyle \frac{1}{10}}<0.11$

Note:
The students must notice that we have reversed the order in negatives as compared to the numerators.
If we have a number a < b for any real numbers a and b, then we have – a > - b.
Thus, we got the following expression as the result when we solved above.
$\Rightarrow -0.5<-{\displaystyle \frac{1}{9}}$
The students must also notice the underlying fact we used here which is given by the following equation:-
$\Rightarrow$Negative Number < 0 < Positive Number
Therefore, we got the result as stated in the above solution given by the following expression:-
$\Rightarrow -0.5<-{\displaystyle \frac{1}{9}}<{\displaystyle \frac{1}{10}}<0.11$