A number consists of 3 digits whose sum is 18 and the middle digit is equal to the sum of the other two. If the number increases by 297 when its digits are reversed, then what is the number?

Correct option is D

​Given:
Let the digits be a, b, c (hundreds, tens, units).

​$$a + b + c = 18, \, \\ \ \\ b = a + c$$

Original number:

100a + 10b + c

Reversed number:

100c + 10b + a

​​Solution:

​Difference:

​$$297 = 100c + 10b + a - (100a + 10b + c)$$​

$$297 = 99c - 99a \\ \ \\ \implies c - a = 3$$ ....(1) 

We know, 

a + b + c = 18, 

Putting (b = a + c) into this;

a + (a + c) + c = 18

$$2a + 2c = 18\\ \ \\ \implies a + c = 9$$ ....(2)

From equ(1) and (2).

a = 3, c = 6,

Then

b = a + c = 9 

Thus , Number:

100a+10b+c = 100(3)+10(9)+6 = 396