Question

Find four rational numbers between $\dfrac{3}{7}$ and $\dfrac{5}{7}$.

Answer 1

Hint: In order to find the four rational numbers between two numbers a and b we will do the sum of a and b and then divide it by 2 then we will get the rational number bet a and b the same procedure we do with a or b and the answer obtained in first operation four times we will get the four rational numbers between the numbers a and b.

Complete step-by-step answer:
Rational numbers -
The number 3.14 is a rational number. A rational number is a number that can be written as a fraction, $\dfrac{a}{b}$, where a and b are integers or any number which can be expressed in the form of $\dfrac{p}{q}$ where $q \ne 0$ are rational numbers.
We need to find the rational numbers between $\dfrac{3}{7}$ and $\dfrac{5}{7}$ we will sum $\dfrac{3}{7}$ and $\dfrac{5}{7}$ then on dividing it by 2 we will get the right answer.
So, the rational number between $\dfrac{3}{7}$ and $\dfrac{5}{7}$ is $\dfrac{{\dfrac{5}{7} + \dfrac{3}{7}}}{2} = \dfrac{8}{{14}} = \dfrac{4}{7}$…..(1)
The rational number between $\dfrac{3}{7}$ and $\dfrac{4}{7}$ is $\dfrac{{\dfrac{4}{7} + \dfrac{3}{7}}}{2} = \dfrac{7}{{14}} = \dfrac{1}{2}$…..(2)
The rational number between $\dfrac{3}{7}$ and $\dfrac{1}{2}$ is $\dfrac{{\dfrac{4}{7} + \dfrac{1}{2}}}{2} = \dfrac{{15}}{{28}}$…..(3)
The rational number between $\dfrac{3}{7}$ and $\dfrac{{15}}{{28}}$ is $\dfrac{{\dfrac{4}{7} + \dfrac{{15}}{{28}}}}{2} = \dfrac{{31}}{{56}}$…..(4)
Therefore the four rational numbers between $\dfrac{3}{7}$ and $\dfrac{5}{7}$ are $\dfrac{4}{7}$, $\dfrac{1}{2}$, $\dfrac{{15}}{{28}}$, $\dfrac{{31}}{{56}}$.
Hence, the answer to this problem is $\dfrac{4}{7}$, $\dfrac{1}{2}$, $\dfrac{{15}}{{28}}$, $\dfrac{{31}}{{56}}$.

Note: In such problems first we need to know that rational numbers are those numbers which can be expressed as p by q form where q is not equal to zero. And we need to find 4 rational numbers between two numbers so the average of those two numbers will give the numbers between the two numbers and finding 4 times the average between any one of the given numbers and the answer obtained from the first number will give us the right answer. Whenever you need to find the rational numbers between two numbers you can find by this method.