Question

Find the rational number exactly halfway between: \[\dfrac{1}{6}\] and \[\dfrac{1}{9}\].

Answer 1

Hint: In this problem, we have to find the rational number exactly half between \[\dfrac{1}{6}\] and \[\dfrac{1}{9}\]. We can find this by multiplying the number 30 for the fraction \[\dfrac{1}{6}\] in numerator and denominator and we can multiplying the number 30 for the fraction \[\dfrac{1}{9}\] in numerator and denominator as we can get fractions with same denominator, from which we can find the rational number between them. We can also use the mean formula by adding the given two fractions and dividing them by 2, we can get the result.

Complete step by step solution:
We know that the given fractions are,
\[\dfrac{1}{6}\] and \[\dfrac{1}{9}\].
We can now multiply number 30 on both the numerator and the denominator for the first fraction, we get
\[\Rightarrow \dfrac{1}{6}\times \dfrac{30}{30}=\dfrac{30}{180}\] …….. (1)
We can now multiply number 20 on both the numerator and the denominator for the second fraction, we get
\[\Rightarrow \dfrac{1}{9}\times \dfrac{20}{20}=\dfrac{20}{180}\] ……… (2)
We can now see that the fraction (1) and (2) have the same denominator, and the number between the numerators, 30 and 20 is 25.
\[\Rightarrow \dfrac{25}{180}=\dfrac{5}{36}\]

Therefore, the rational number between \[\dfrac{1}{6}\] and \[\dfrac{1}{9}\] is \[\dfrac{5}{36}\].

Note: e also have another method to solve this problem.
We should know that exactly halfway between the rational number is,
\[\Rightarrow \dfrac{a+b}{2}\] ……… (1)
We know that the given fractions are,
\[\dfrac{1}{6}\] and \[\dfrac{1}{9}\].
Where, a = \[\dfrac{1}{6}\] and b = \[\dfrac{1}{9}\].
We can now substitute the above value in (1), we get
\[\begin{align}
  & \Rightarrow \dfrac{\dfrac{1}{6}+\dfrac{1}{9}}{2}=\dfrac{\dfrac{3+2}{18}}{2} \\
 & \Rightarrow \dfrac{5}{18\times 2}=\dfrac{5}{36} \\
\end{align}\]
Therefore, the rational number between \[\dfrac{1}{6}\] and \[\dfrac{1}{9}\] is \[\dfrac{5}{36}\].