There is a matte with a picture centered. The picture is twice as long as it is wide. Around the picture is a 2 inch border. The entire matte with picture has an area of 5616 sq. inches.

What are the dimensions of the picture?

Here's the set up:

Let x be the width. (2x+4)(x+4)=5616

The answer is 100 in. x 50 in.

Now do the same problem with one change. The total area of the matte and picture is 5616 sq. inches. The border is now 1 inch instead of 2. Intuitively, the picture should be 102 in. x 52 in.

Here's the set up:

Let x be the width.

(2x+2)(x+2)=5616

But when you do the mathematical computation, you get a different answer (non integer). What the heck is happening?

102 x 52 in cannot be correct. The statement of the problem says that the length of the picture is twice the width of the picture. 51.493 x 102.986 for the picture gives the correct aspect ratio. The matte is then 53.493 x 104.986, which gives the correct area (after accounting for the approximation to 3 decimal places).

Your intuition has steered you wrong. You were assuming that the dimensions of the matte would stay the same in the modified problem, but that led to a “solution” that did not satisfy the conditions stated in the problem.

Thank You! I knew I had to be doing something wrong.

Once again ... THANKS!