Logic :

①
Value of the number ATE

= 100 × A + 10 × T + 1 × E
= 100A + 10T + E

Value of 3 ATE = 300A + 30T + 3E

Value of the number NINE
= 1000 × N + 100 × I + 10 × N + 1 × E
= 1010N + 100I + E

So,

Value of 3 ATE = Value of NINE

300A + 30T + 3E = 1010N + 100I + E

300A + 30T + 2E = 1010N + 100I


Divide both sides by 2 ,

150A + 15T + E = 505N + 50I

E = 505N + 50I - 15T - 150A

E = 5(101N + 10I - 3T - 30A)


This tells us that E is a multiple of 5 since there is factor 5 on the right side.

The only digit that is a multiple of 5, is 5 itself.

So, E = 5


②
150A + 15T + 5 = 505N + 50I

Divide both sides by 5,

30A + 3T + 1 = 101N + 10I

(Recall that this equation came from reducing the original equation above)


3T + 1 = 101N + 10I - 30A

3T + 1 - N = 100N + 10I - 30A

3T + 1 - N = 10(10N + I - 3A)

This tells us that 3T + 1 - N is a multiple of 10.


③
Now T and N are single non-zero digits. The highest possible is 9, lowest possible is is 1.


1 ≤ T ≤ 9
3 ≤ 3T ≤ 27
4 ≤ 3T + 1 ≤ 28
4 - N ≤ 3T + 1 - N ≤ 28 - N

N is positive so 28 - N is smaller than 28.


Since it's a multiple of 10,
3T + 1 - N can only be 10 or 20.

④
But let's consider this :

Since we know 3ATE is the addition of 3 identical three-digit numbers, the total is less than 3000 (maximum of ATE is only 985 since all 3 digits are different and we know E = 5)

So N = 1 or 2

If N = 1, then 3T + 1 - 1 = 10 or 3T + 1 - 1 = 20
3T = 10 or 3T = 20
T = 10/3 = 3⅓ or T = 20/3 = 6⅔
(Rejected since T is an integer)

So N = 2


Then,

3T + 1 - 2 = 10 or 3T + 1 - 2 = 20
3T = 11 or 3T = 21
T = 11/3 = 2⅔ or T = 7
(Rejected as T is an integer)

So T = 7


Substituting E = 5, N = 2, T = 7 into 3T + 1 - N = 10(10N + I - 3A),

20 = 10(2(10) + I - 3A)
2 = 20 + I - 3A
I = 3A - 18
I = 3(A - 6)

So I is a multiple of 3. I = 3, 6, or 9

3 = 3A - 18 or 6 = 3A - 18 or 9 = 3A - 18

3A = 21 or 3A = 24 or 3A = 27

A = 7
(rejected as T = 7, and they are all different digits)

Or A = 8

Or A = 9
(rejected as this would mean A = I, but they all have to be different digits)


So E = 5, N = 2, T = 7, A = 8 and I = 6

The answer is correct. Why the downvote?