Let $A = {2, 3, 4, 5, ....., 30}$ and '$\simeq$' be an equivalence relation on $A \times A$, defined by $(a, b) \simeq (c, d)$, if and only if $ad = bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4, 3)$ is equal to :
Correct Answer:
D
Explanation
Given that $(a,b) \simeq (4,3)$, we have $3a=4b$, which means $a = \frac{4}{3}b$. Since $a$ must be an integer, $b$ must be a multiple of $3$. The possible values of $b$ are ${3, 6, 9, 12, 15, 18, 21, 24, 27, 30}$. For each value of $b$, we can find the corresponding value of $a$:
If $b=3$, $a=\frac{4}{3}(3) = 4$.
If $b=6$, $a=\frac{4}{3}(6) = 8$.
If $b=9$, $a=\frac{4}{3}(9) = 12$.
If $b=12$, $a=\frac{4}{3}(12) = 16$.
If $b=15$, $a=\frac{4}{3}(15) = 20$.
If $b=18$, $a=\frac{4}{3}(18) = 24$.
If $b=21$, $a=\frac{4}{3}(21) = 28$.
If $b=24$, $a=\frac{4}{3}(24) = 32$. But $a$ must be less than or equal to $30$, so we stop here.
If $b=27$, $a=\frac{4}{3}(27) = 36$, which is also greater than $30$.
If $b=30$, $a=\frac{4}{3}(30) = 40$, which is also greater than $30$.
The pairs $(a, b)$ that satisfy the condition are $(4, 3), (8, 6), (12, 9), (16, 12), (20, 15), (24, 18), (28, 21)$. There are $7$ such ordered pairs.