Let A = 2, 3, 4, 5, ....., 30 and '≃' be an equivalence relation on A × A, defined by (a, b) ≃ (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 :

Solution

Maths sir · 4 steps

Read ad = bc as a/b = 4/3, then count the pairs (4k, 3k) that fit inside A.

1. Rewrite the condition. (a, b) ≃ (4, 3) means 3a = 4b, that is a/b = 4/3. Since 4 and 3 have no common factor, every such pair is (4k, 3k) for a natural number k.

   (a, b) = (4k, 3k), 2 ≤ 3k and 4k ≤ 30

2. Range of k. 3k ≥ 2 holds for every k ≥ 1. 4k ≤ 30 gives k ≤ 7.5, so k = 1, 2, …, 7.
3. List them. The pairs are (4,3), (8,6), (12,9), (16,12), (20,15), (24,18), (28,21). All entries lie between 2 and 30.
4. So. There are 7 ordered pairs, including (4, 3) itself. The answer is (d).

The idea to keep. A relation like ad = bc groups pairs by the ratio a/b. Reduce the ratio to lowest terms, write the pair as (pk, qk), and let the set's bounds fix how many values k takes.

Answer from the NTA final answer key. Solution by PrepWiser.