Question

Three rational numbers between 5/31 and 6/31

Answer 1

`(26)/(155)`,
`(27)/(155)`, and
`(28)/(155)`.

Explanation:

What is a rational number? By definition, a rational number can be represented as the fraction of two integers.

The goal is to find three fractions in the form
`(p)/(q)` between
`(5)/(31)` and
`(6)/(31)`.

`(5)/(31) < (p)/(q) < (6)/(31)`.

At this moment, there doesn't seems to be a number that could fit. The question is asking for three of these numbers. Multiple the numerator and the denominator by a number greater than three (e.g., five) to obtain

`(25)/(155) < (p)/(q) < (30)/(155)`.

Since
`p` and
`q` can be any integers, let
`q = 155`.

`(25)/(155) < (p)/(155) < (30)/(155)`.

`\implies 25 < p < 30`.

Possible values of
`p` are 26, 27, and 28. That corresponds to the fractions

`(26)/(155)`,
`(27)/(155)`, and
`(28)/(155)`.

These are all rational numbers for they are fractions of integers.