Question

Which of the following statements is always true?

• A.
  $\frac{(x-y)}{2}$ is a rational number between x and y.
• B.
  $\frac{(x\times y)}{2}$ is a rational number between x and y.
• C. $\frac{(x÷y)}{2}$ is a rational number between x and y.
• D.
  $\frac{(x+y)}{2}$ is a rational number between x and y.

Answer 1

The correct option is D

$\frac{(x+y)}{2}$ is a rational number between x and y.
After reading all the statements, we recall that for any two given rational numbers, we can find a rational number between them by taking the average of the two given rational numbers. Let us illustrate this by taking two rational numbers a and b where a > b.

Given a > b. Adding a to both sides, we get
a + a > b + a
$\Rightarrow$ 2a > b + a
$\Rightarrow$ a > $\frac{(b+a)}{2}$ ... (i)

Let us again take a > b. Adding b to both sides, we get
a + b > b + b
$\Rightarrow$ a + b > 2b
$\Rightarrow$ $\frac{(a+b)}{2}$ > b ... (ii)
From (i) and (ii), we see that
a > $\frac{(b+a)}{2}$ > b
So, statement (B) is CORRECT.

Let us now see other statements as well. We will check why these statements are INCORRECT.
To prove the same, let us take x = $\frac{2}{5}$ and y = $\frac{1}{5}$.
Statement A says $\frac{(x-y)}{2}$ is a rational number between x and y.
$\frac{(x-y)}{2}$ = $\frac{(2-1)}{2\times 5}$ = $\frac{1}{10}$
Expressing x and y with denominator 10:
x = $\frac{2\times 2}{5\times 2}$ = $\frac{4}{10}$ and y = $\frac{1\times 2}{5\times 2}$ = $\frac{2}{10}$
It is clear that $\frac{1}{10}$ does not lie between $\frac{4}{10}$ and $\frac{2}{10}$.
So, $\frac{(x-y)}{2}$ does not lie between x and y. Hence, statement A is INCORRECT.

Statement C says $\frac{x\times y}{2}$ is a rational number between x and y.
$\frac{x\times y}{2}$ = $\frac{2\times 1}{2\times 5\times 5}$ = $\frac{2}{50}$, x = $\frac{2\times 10}{5\times 10}$ = $\frac{20}{50}$ and y = $\frac{1\times 10}{5\times 10}$ = $\frac{10}{50}$
It is clear that $\frac{2}{50}$ does not lie between $\frac{20}{50}$ and $\frac{10}{50}$.
So, $\frac{x\times y}{2}$ does not lie between x and y. Hence, statement C is INCORRECT.

Statement D says $\frac{x÷y}{2}$ is a rational number between x and y.
$\frac{x÷y}{2}$ = $\frac{2\times 5}{2\times 5}$ = 1
1 is greater than $\frac{2}{5}$ and $\frac{1}{5}$, hence it does not lie between them.
So, $\frac{x÷y}{2}$ does not lie between x and y. Therefore, statement D is INCORRECT as well.