Question

If p and q are two non-zero positive rational numbers and p is not eqal to q then A) p+q/2<p B) p+q/2<qC) p+q/2=pD) p+q/2=q​

Answer 1

Given:

p and q are two non-zero positive rational numbers

p≠q

To Find:

The correct relation from:

A) $\frac{p +q}{2}$<p

B) $\frac{p + q}{2}$<q

C) $\frac{p + q}{2}$=p

D) $\frac{p + q}{2}$=q​

Solution:

• What are rational numbers?

"A rational number is a number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, a numerator p and a non-zero denominator q."

• Now, we will analyze every option:

A) $\frac{p +q}{2}$<p

• Simplifying the expression further:

                     ⇒p + q <2p

                     ⇒ q < p

• From the given conditions, we know that

           p≠q.

So, A may be true.

B) $\frac{p + q}{2}$<q

• Simplifying this expression further:

                         ⇒ p + q < 2q

                         ⇒ p < q

Since from part A), we have q <p, we can conclude that A) cannot be true if B) is true and vice versa.

C) $\frac{p + q}{2}$=p

•        Further simplifying the expression:

                     ⇒ p + q = 2p

                     ⇒ q = p

•      But, according to the question,

                        p≠q

So, C cannot be correct.

D) $\frac{p + q}{2}$=q​

• On further simplification, we get:

                   ⇒ p + q = 2q

                   ⇒ p = q

•       But according to the question, we know

                     p≠q

So, D) cannot be true.

Hence, we can conclude that for p and q, either A) or B) is correct, but both cannot be correct for the same rational numbers.