**From New Scientist #2152, 19th September 1998** [link]

A small building site is offered for sale, divided into three plots, each at the same price per acre.

The plots are all rectangles of different sizes but each is the same shape as the overall site — that is, the ratio of the sides is the same for each, although two of the rectangles are rotated through 90° relative to the other two.

If the asking price of the largest plot is £20,000 more than that of the smallest, how much is the middle-sized plot?

[enigma997]

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If the cost of the medium sided plot is

k, and it has long side lengthaand short size lengthf.athen we can write:(equating area and cost).

The smallest plot, has long side

f.a(the same as the shortest side of the medium plot) and so its shortest side isf².a.And its area (and hence cost) is:

The largest plot has longest side equal to

a(1 + f²)(the sum of the shortest side of the smallest plot and the longest side of the medium plot). And so the shortest side isf.a(1 + f²), giving an area (and cost) of:And the difference between the costs is £20,000, so:

writing:

x = f², we have:So, if we knew

x, we could calculatek.If we look at the long side of the site we know it is the sum of the short side of the largest plot and the short side of the medium plot, which is:

And

fis the ratio of the short side to the long side of any of the rectangles and the site. Applying this to the site we get:substituting

xforf²:So:

Solution:The cost of the middle sized plot is £10,000.We can use the equation

x² + x = 1to calculatexand hencef:Here is a simple program to determine numerical values for

x,fand the cost of the plots:Run:[ @repl.it ]