The area of the flower bed is 30*24 = 720 square yards.


1/4 of it is   = 180 square yards.


Let x be the uniform width of the walkway around the flower bed.


Then the outer dimensions of the larger rectangle are (30+2x) and (24+2x) yards,

so the area of the larger rectangle is  (30+2x)*(24+2x).


From the condition, we are given that


    (30+2x)*(24+2x) - 30*24 = 180   (it is exactly the area of the path !)


Simplify 


    30*24 + 48x + 60x + 4x^2 - 30*24 = 180,

    4x^2 + 108x - 180 = 0,

    x^2  +  27x  - 45 = 0.


     =  =  = .


Of the two roots, choose the positive x =  = 1.57, 

since only positive root makes sense.


ANSWER.  The walkway's uniform width is  1.57 yards.


CHECK.   (30+2*1.57)*(24+2*1.57) - 30*24 = 179.4 square yards.

          Taking rounding into account, it is good solution.

    From my solution, you need to learn TWO things:

        1)  The outer dimensions of the larger rectangle are 2x enlarged dimensions of the base rectangle, and

        2)  Learn how to create the base equation to find the unknown "x".

     The rest is just a technique.