Question 1075905
The equation {{{1/x+1/y = 1/12 }}} can be re-arranged:
{{{ (x+y)/xy = 1/12 }}}
{{{ 12(x+y) = xy }}}    (1)


If we think of xy as the area of a rectangle, then the right hand side of (1) is the area, and the left hand side is 6 times the perimeter.   The maximum area for minimal perimeter is when the shape is a square (x=y), so let's try x=y:   
          {{{ 12(2x) = x^2 }}}
          {{{  24x = x^2 }}}
          {{{ x^2 - 24x = 0 }}}
          {{{ x(x-24) = 0 }}}   
          Discard x=0 because x and y are positive integers.
          {{{  x=24 }}} —> {{{ y=24 }}}

So {{{ highlight(x+y = 48) }}}  is the minimum value of x+y.