Question

Find three rational numbers between \[\dfrac{-3}{14}\] and \[\dfrac{6}{14}\]

Answer 1

Hint: We solve this problem by using the comparisons of fractions.
If the denominators of some fractions are equal then based on the numerator we can say that the fraction is smaller or greater. The fraction with less numerator value will be less and the fractions with more numerator value will be more.
We are asked to find the three rational numbers between \[\dfrac{-3}{14}\] and \[\dfrac{6}{14}\]
Let us assume that the required rational numbers also have the same denominator 14 as follows
\[\dfrac{x}{14},\dfrac{y}{14},\dfrac{z}{14}\]

Complete step by step answer:
We know that if the denominators of some fractions are equal then based on the numerator we can say that the fraction is smaller or greater
Here, we can see that we need to find the rational numbers between \[\dfrac{-3}{14}\] and \[\dfrac{6}{14}\]
By using the above two conditions we can say that the value of \[x,y,z\] lie between -3 and 6
Let us take any three values between -3 and 6 as \[x,y,z\] then we get the required numbers as
\[\begin{align}
  & \Rightarrow \dfrac{-1}{14},\dfrac{2}{14},\dfrac{5}{14} \\
 & \Rightarrow \dfrac{-1}{14},\dfrac{1}{7},\dfrac{5}{14} \\
\end{align}\]
Therefore we can conclude that the required rational numbers between \[\dfrac{-3}{14}\] and \[\dfrac{6}{14}\] as
\[\dfrac{-1}{14},\dfrac{1}{7},\dfrac{5}{14}\]


Note:
 We can find the required rational numbers in other methods.
We are asked to find the three rational numbers between \[\dfrac{-3}{14}\] and \[\dfrac{6}{14}\]
We have the condition that if we increase the denominator of a positive fraction leaving the numerator as it is then that number will be less than the fraction.
Here, we can see that there is a positive fraction that is \[\dfrac{6}{14}\]
Now, let us increase denominator in the above fraction leaving the numerator as it is then we get the three rational numbers as
\[\Rightarrow \dfrac{6}{17},\dfrac{6}{19},\dfrac{6}{29}\]
Here, we can see that the above fractions are all positive that are greater than \[\dfrac{-3}{14}\]
Therefore we can conclude that the required rational numbers between \[\dfrac{-3}{14}\] and \[\dfrac{6}{14}\] as
\[\dfrac{6}{17},\dfrac{6}{19},\dfrac{6}{29}\]