If S={a ∈ R:|2 a-1|=3[a]+2{a }}, where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t, then 72 Σa ∈ S a is equal to _________.

Solution

Maths sir · 5 steps

Write a = n + f, open the modulus in two cases, and see which case gives a valid n and f.

1. Split a. Let a = n + f, where n = [a] is an integer and f = {a} satisfies 0 ≤ f < 1. The right side becomes 3n + 2f. It equals a modulus, so it cannot be negative. That forces n ≥ 0, because n = -1 gives -3 + 2f < 0.
2. Case one. If 2a - 1 ≥ 0, the equation is 2n + 2f - 1 = 3n + 2f, which gives n = -1. This contradicts n ≥ 0, so there is no solution here.

   1 - 2n - 2f = 3n + 2f ⇒ f = (1 - 5n)/4

3. Case two. If 2a - 1 < 0, the modulus opens with a minus sign and gives the equation above. With n ≥ 0 and 0 ≤ f < 1, only n = 0 works, giving f = 1/4. For n ≥ 1, f would be negative.
4. Check. a = 1/4 gives 2a - 1 = -1/2 < 0, so it fits the case. Both sides equal 1/2, since [a] = 0 and {a} = 1/4.
5. So. S = {1/4}, so 72 Σ a = 72 × 1/4 = 18.

The idea to keep. With [ ] and { } together, put a = n + f with n an integer and 0 ≤ f < 1. Then use those two limits to cut each case down to a few values of n.

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