perplexus.info :: Shapes : Cinq Confusion

Cinq Confusion (Posted on 2014-07-10)

A farmer has two rectangular fields with a peculiar relationship. Both fields have integer dimensions. The corn field has five times the area of the hay field but the hay field has five times the perimeter of the corn field.

It is also known that the dimensions of the corn field differ by 1, and the shorter dimension of the hay field is less than 100.

Find the dimensions of the two fields.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Analytic attempt

Comment 3 of 3 |

Let the corn field be C by C+1 and the hay field be H by W with H>100>W.

Then I can create two equations: 5*(2C+1)=H+W and C*(C+1)=5*H*W.

Next I solved the first equation for C and substituted into the second. After some simplification I got H^2 - 498HW + W^2 = 25.

At this point I made an assumption: I decided to let H and W be multiples of 5 to match the 25 on the right side of the equation. Then the smallest candidate for W is 5.

If W=5 then H^2 - 2490*H=0, or H=2490. Then 5*(2C+1)=2495, or C=249.

These fit the problem nicely. The corn field is 249 by 250 and the hay field s 5 by 2490.

Posted by Brian Smith on 2024-01-25 13:15:19

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