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Larger area / larger perimeter (Posted on 2017-07-03) Difficulty: 2 of 5
Find the smallest pair of integer sided rectangles that fit the following criteria:

The first has three times the area of the second and
the second has three times the perimeter of the first.

No Solution Yet Submitted by Jer    
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Solution Analytic Solution Comment 3 of 3 |
Finding a good approach was harder than I expected.  I eventually came up with this:

Let x be a side of the large perimeter rectangle.
Let y be a side of the large area rectangle.
Let s be the semiperimeter if the large area rectangle.

Then the other side of the large perimeter rectangle is 3s-x and the other side of the large area rectangle is s-y.

Then the areas create the equation y*(s-y) = 3*x*(3s-x).  Solving for s yields s = (3x^2-y^2) / (9x-y).

Then it is just a matter of trying small x,y pairs.  The smallest result occured for x=1 and y=15 yielding rectangles of 1x110 and 15x22.

  Posted by Brian Smith on 2017-07-03 23:35:42
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