Let the width of the gravel border be x.

Then the rectangle's sides after the gravel is:

Length = 10 + x

Width = 4 + x

The area of the rectangle + the border = (x+10)*(x+4)

==> The area of the border = (x+10)(x+4) - the area of the rectangle

==> The area of the border = (x+10)(x+4) - 10*4

==> The area of the border = x^2 + 14x + 40 - 40 = x^2 + 14x

==> But we are given that the area of the gravel border = 6 times the area of the garden

==> x^2 + 14x = 6*40

==> x^2 + 14x = 240

==> x^2 + 14x -240 = 0

Now we will factor and solve for x.

==> (x- 10)(x+24) = 0

==> x = 10

==> x = -24 ( Not valid )

Then the width of the border is x = 10

Then the length of the garden now = 10+10 = 20

The width of the garden with the border = 14

Then the perimeter = 2*20 + 2*14 = 40 + 28 = 68

**Then, the perimeter of the gravel border = 68 ft**

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