1. Understand the equivalence relation

We are given: $A = \{2,3,4,\dots,30\}$ and on $A \times A$ the relation $(a,b) \simeq (c,d) \iff ad = bc.$

We need the number of ordered pairs $(a,b) \in A \times A$ such that $(a,b) \simeq (4,3).$

So put $(c,d)=(4,3)$. Then $(a,b) \simeq (4,3) \iff a\cdot 3 = b\cdot 4,$ that is, $3a = 4b.$

2. Solve the Diophantine equation

We need integer solutions $(a,b)$ with $a,b \in \{2,3,\dots,30\}$ satisfying $3a=4b.$

Since $\gcd(3,4)=1$:

• $4 \mid a$, so let $a=4k$
• $3 \mid b$, so let $b=3k$

Thus all solutions are of the form $(a,b)=(4k,3k)$ for some positive integer $k$.

3. Apply the range restriction

Because $a,b \in A$, $2 \le 4k \le 30, \qquad 2 \le 3k \le 30.$

From these:

• $4k \le 30 \Rightarrow k \le 7.5$
• $3k \le 30 \Rightarrow k \le 10$

So the effective restriction is $k \le 7.$ Also $k\ge 1$ works, because then $(4,3) \in A\times A$.

Hence possible values are $k=1,2,3,4,5,6,7.$

4. List the ordered pairs

These are:

Total number of ordered pairs $=7$.

5. Check options

The correct option is: $\boxed{7}$ which is Option D.

6. Compare with stored answer

Stored correct answer: D

Our derived answer: D

So they agree.