Question

A small garden measures 11 ft by 7 ft. A uniform border around the garden increases the total area to 140 ft2. What is the width of the border?

Answer 1

There are many ways to work a problem like this. You can write an equation for the area of the border and solve for its width; you can graph an equation for the border and find its solutions; or you can use a little number sense.

The difference in length and width of the garden area is 4 ft. If the border is uniform, the difference in length and width of the overall area will remain 4 ft. Thus you can look for factors of 140 (the area) that differ by 4.
.. 140 = 1*140 = 2*70 = 4*35 = 5*28 = 7*20 = 10*14
The last two factors listed differ by 4, so we take these as the overall dimensions of the bordered area.

Without the border, the garden area width is 7 ft. With a border on both sides, the garden area width is 10 ft. Then twice the border width is
.. 10 ft -7 ft = 3 ft . . . . twice the border width

The border width is 1.5 ft.

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The attached graph shows the solution to
.. (7 +2x)*(11 +2x) -140 = 0
The overall width with a border of width x will be 7 +2x. The overall length with a border of width x will be 11 +2x. The product of these dimensions is the overall area, 140 t^2. When we subtract that area from the product of dimensions, we want the result to be zero. The graphing program easily tells us the value of x that makes the result zero: 1.5 ft.