Question 763030
Though this looks complicated, it isn't and can be solved through linear equations.

Step 1:
Let the digit in 10's place be x and digit in unit's place be y. 
Then digit's in 100's place = 19 - (x+y) = 19 - x - y (since sum of the 3 digits is 19)


Step 2:
Value of the original number = {{{100*(19-x-y) + 10*x + y}}} = {{{-90*x -99*y + 1900}}}


Step 3:
If 10's and unit's places are interchanged, 10's place becomes y and units place is x. 100's place is still 19-x-y.

The value of the new number is
{{{100*(19-x-y) + 10*y + x}}} = {{{-90*y - 99*x +1900}}}

Given that this is less than the original number by 27, we get the equation

{{{-90*x -99*y + 1900 - 27 = -90*y - 99*x +1900}}}

Simplifying, we get {{{9*x - 9*y = 27}}} or {{{x - y = 3}}} Eqn (1)


Step 4:
If 100's and 10's places are interchanged, x comes to the 100's place and (19-x-y) comes to the 10's place. Units place is stilly.

The value of this new number is
{{{100*x + 10*(19-x-y) + y}}} = {{{-90*x - 9*x +190}}}

Given that this is 180 more than the original number, we get the equation

{{{-90*x - 9*x +190 = -90*x -99*y + 1900 + 180}}}

Simplifying, we get: {{{180*x + 90*y = 1890}}} or {{{2*x + y = 21}}} Eqn (2)


Step 5:
We can solve Eqn(1) and (2) to get x and y
Add eqns(2) and (1) to get {{{3*x = 21 + 3 = 24}}} or {{{x = 8}}}

Since x - y = 3, {{{y = 8 - 3 = 5}}}

Hence 100'th place = {{{19 - x - y = 19 - 8 - 5 = 6}}}

The original number = {{{highlight(685)}}}

Check your answer:
Swapping 10's and units, we get 658, and 685 - 658 = 27
Swapping 100's and tens, we get 865, and 865 = 685 + 180.

Got it?
:)