Question

Find two rational numbers between $ \dfrac{-2}{9} $ and $ \dfrac{5}{9} $ .

Answer 1

Hint: A rational number is a real number that can be expressed in the form $ \dfrac{p}{q} $ , where p and q are both integers but $ q\ne 0 $ . When we want to find the rational numbers between two numbers of the same denominator, we have to check the numerators of both. If the magnitude of the difference is more than the number of rational numbers we want to find, then we can simply go on incrementing the numerator of the smaller number by 1.

Complete step-by-step answer:
Let us first see what is a rational number.
A rational number is a real number that can be expressed in the form $ \dfrac{p}{q} $ , where p and q are both integers but $ q\ne 0 $ .
For example, the number 2.5 is a rational number because it can be written as $ \dfrac{5}{2} $ and this satisfies the condition of a rational number.
The given two numbers, $ \dfrac{-2}{9} $ and $ \dfrac{5}{9} $ are also rational numbers.
If we take any two rational numbers, then there exist infinite rational numbers between them.
When we want to find the rational numbers between two numbers of the same denominator, we have to check the numerators of both. If the magnitude of the difference is more than the number of rational numbers we want to find, then we can simply go on incrementing the numerator of the smaller number by 1.
In this case, the difference between the two numerators is $ 5-(-2)=7 $ and we have to find 2 rational number between $ \dfrac{-2}{9} $ and $ \dfrac{5}{9} $ .
We know that 7 > 2. Therefore, the above condition is satisfied.
Now, let us increment the numerator of the smaller number $ \left( i.e.\dfrac{-2}{9} \right) $ by one two times.
i.e. $ \dfrac{-2+1}{9}=\dfrac{-1}{9} $ and $ \dfrac{-1+1}{9}=\dfrac{0}{9}=0 $
This means that two rational between $ \dfrac{-2}{9} $ and $ \dfrac{5}{9} $ are $ \dfrac{-1}{9} $ and 0.
It is not like that there are only two rational numbers between the two given numbers. There are infinitely many such numbers.
So, the correct answer is “ $ \dfrac{5}{9} $ are $ \dfrac{-1}{9} $ and 0.”.

Note: If the denominators of the two numbers are not the same, then we have to make their denominators the same. This can be done making both the denominators equal to their L.C.M (least common multiple).
For example, consider the numbers $ \dfrac{1}{2} $ $ \dfrac{1}{2} $ and $ \dfrac{2}{3} $ .
The LCM of 2 and 3 is 6.
To make the denominator of $ \dfrac{1}{2} $ as 6, multiply and divide by 3.
 $ \Rightarrow \dfrac{1\times 3}{2\times 3}=\dfrac{3}{6} $ .
To make the denominator of $ \dfrac{2}{3} $ as 6, multiply and divide by 2.
 $ \Rightarrow \dfrac{2\times 2}{2\times 2}=\dfrac{4}{6} $
On doing this the value of the original numbers do not change and it becomes easy to find the rational numbers between these two.