Question 120184
Let's say that the width of the border is x.  Then the overall width of the garden and its border on both sides must be {{{4+2x}}}.  Likewise, the overall length must be {{{6+2x}}}.  See the diagram.


{{{drawing(400,400,-10,10,-10,10,
line(-4,6,4,6),
line(-4,6,-4,-6),
line(4,6,4,-6),
line(-4,-6,4,-6),
green(line(-6,8,6,8),
line(-6,8,-6,-8),
line(6,8,6,-8),
line(-6,-8,6,-8)),
red(locate(-1.5,-5,4*meters),
locate(-3.5,0,6*meters),
locate(-.4,-6.6,x),
locate(-5.4,0,x))
)}}}


So the dimensions of the outer edge of the border are {{{4+2x}}} by {{{6+2x}}}.  Since the area of the garden is {{{4*6=24m^2}}}, and the area of the border has to be equal to that, the overall area of the garden and the border must be 2 times the area of the garden or {{{48m^2}}}.  But the overall area is also given by its length times its width, so we can write:


{{{(4+2x)(6+2x)=48}}}


Use FOIL:


{{{24+20x+4x^2=48}}}


Put into standard form:


{{{4x^2+20x-24=0}}}


Divide by 4:


{{{x^2+5x-6=0}}}


Factors of -6 that add to +5 are 6 and -1, so:


{{{(x-1)(x+6)=0}}}


{{{(x-1)(x+6)=0}}} if and only if {{{x=1}}} or {{{x=-6}}}, but you can exclude {{{x=-6}}} because we are looking for a length measurement.


Therefore {{{x=1m}}}


Check:

The outside dimensions of the border must be {{{4+2*1=6}}} by {{{6+2*1=8}}} and the overall area must be {{{6*8=48m^2}}}.  Subtract the area of the original garden, {{{24m^2}}}, and the area of the border is {{{48-24=24m^2}}}.  All conditions of the problem are met:  The border is uniform, that is it is 1 meter wide on all sides, and the area of the border is equal to the area of the garden.  Answer checks.


Hope that helps,
John