Question

Find \[5\] rational numbers between \[\dfrac{1}{8}\] and \[\dfrac{1}{5}\].

Answer 1

Hint: We have to first find the least common multiple of the denominators by taking LCM of \[8\] and \[5\]. Then we will multiply both numerator and denominator with a number to make the denominator equal to LCM of \[8\] and \[5\]. And then we will multiply \[\dfrac{6}{6}\] to both the numbers to find five rational numbers between them.

Complete step-by-step answer:
Rational numbers are the numbers that can be written in the form of a fraction where numerator and denominator are integers. Numerator and denominator should be coprime and denominator should not be equal to zero.
In this question, the given numbers are \[\dfrac{1}{8}\] and \[\dfrac{1}{5}\]. We have to find five rational numbers between these numbers.
For this we will find the least common multiple (LCM) of the denominators \[8\] and \[5\].
We can write, \[{\text{LCM of 8 and }}5 = 8 \times 5\]
\[ = 40\]
Now, we will multiply both the numerator and denominator with a number to make the denominator equal to \[40\]i.e., \[\dfrac{5}{5}\] with \[\dfrac{1}{8}\] and \[\dfrac{8}{8}\] with \[\dfrac{1}{5}\].
Hence, we can write the given numbers as
\[ \Rightarrow \dfrac{1}{8} = \dfrac{1}{8} \times \dfrac{5}{5}\] and \[\dfrac{1}{5} = \dfrac{1}{5} \times \dfrac{8}{8}\]
Therefore, we can write \[\dfrac{1}{8}\] as \[\dfrac{5}{{40}}\] and \[\dfrac{1}{5}\] as \[\dfrac{8}{{40}}\].
To find five rational numbers between these numbers, we will multiply both the numbers by \[\dfrac{6}{6}\], therefore we get
\[ \Rightarrow \dfrac{5}{{40}} = \dfrac{5}{{40}} \times \dfrac{6}{6}\] and \[\dfrac{8}{{40}} = \dfrac{8}{{40}} \times \dfrac{6}{6}\] i.e., \[\dfrac{5}{{40}} = \dfrac{{30}}{{240}}\] and \[\dfrac{8}{{40}} = \dfrac{{48}}{{240}}\]
As numerator and denominator should be coprime i.e., the only positive integer that is a divisor of both the numerator and denominator should be \[1\].
Therefore, we can write five rational numbers between \[\dfrac{1}{8}\] and \[\dfrac{1}{5}\] as \[\dfrac{{31}}{{240}}\], \[\dfrac{{37}}{{240}}\], \[\dfrac{{41}}{{240}}\], \[\dfrac{{43}}{{240}}\] and \[\dfrac{{47}}{{240}}\] .
So, the correct answer is “\[\dfrac{{31}}{{240}}\], \[\dfrac{{37}}{{240}}\], \[\dfrac{{41}}{{240}}\], \[\dfrac{{43}}{{240}}\] and \[\dfrac{{47}}{{240}}\]”.

Note: In this question, we need to find five rational numbers between the given rational numbers. That’s why we have multiplied the number by \[\dfrac{6}{6}\]. But, in other cases according to the number of rationals required, we will multiply the respective fraction.