Question

If x & y are two rational numbers then ( x + y)/2 is a rational number between x& y

Answer 1

Answer:

I hope, this picture will be solve your problem.

Answer 2

Concept:

A rational number is a sort of real number that has the form p/q and q does not equal zero.

Given:

x and y are two rational numbers.

Prove:

$\frac{(x+y)}{2}$ is a rational number between x and y.

Solution:

If x and y are rational numbers, then x and y may be written in the form of $\frac{p}{q}$ where p and q ∈ Z.

Let x=ab and let y=cd where a,b,c,d ∈ Z.

Add x and y

$x+y=\frac{a}{b} +\frac{c}{d}$

$x+y=\frac{ad+bc}{bd}$

ad+bc ∈ Z and bd ∈ Z.

∵ The multiplication and addition of two integers is again an integer.

Therefore $x+y=\frac{a}{b} +\frac{c}{d}$  is a rational number.

Check $\frac{(x+y)}{2}$ lies between x and y

Assume $x=2$ and  $y=4$

$\frac{(x+y)}{2}=\frac{4+2}{2}$

$\frac{(x+y)}{2}=\frac{6}{2} =3$

which lies between 2 and 4.

Hence if x and y are two rational numbers then $\frac{(x+y)}{2}$ is a rational number between x and y.