A rectangular photograph is surrounded by a border that is 1 inch wide on each side. The total area of the photograph and the border is M square inches. If the border had been 2 inches wide on each side, the total area would have been (M + 52) square inches. What is the perimeter of the photograph, in inches?

A. 34
B. 36
C. 38
D. 40
E. 42

D.

First, we know that M is the area of the picture AND the border, and that the border is 1 in wide. If you have a border around a picture that is 1" wide, it doesn't add just 1" to the width and height, but actually 2" to the width and 2" to the height because you have an extra inch on each side for the border.

So, if we want to let x = width of the photograph (no border) and y = height of the photograph (no border), then we know that M = (border width + photograph width) * (border + photograph height).

A border adds twice its width to the width of the photograph. A 4" picture will have a 1" border on the right and left, so it adds 2", for a new width of 6". (Hypothetical example). For a 2" wide border, the added dimensions will be twice the width of the border, or 4" (2" for each left and right side).

(x+4)(y+4) = M + 52

But we know that M is the area of the photograph plus a 1" border. This is represented alegbraically like:

(x+2)(y+2) = M

So substitute in:

(x+4)(y+4) = (x+2)(y+2) + 52, Now lets reduce as far as we can. First do FOIL.

xy + 4x + 4y + 16 = xy + 2x +2y +4 +52

The xy on each side cancels each other out.

4x + 4y + 16 = 2x + 2y + 56
subtract the following values from each side: 2x, 2y and 16

2x + 2y = 40.

We can reduce further if you want to, but it isn't necessary.

If you have a rectangle with height (x) and width (y) and you want to find the perimeter, then you can do x + x + y + y = perimter. Also known as 2x + 2y = Perimeter.

This shows us that the perimter should be 40.


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Agreed.

Outer rectangle area when width is two inches on either side - Outer rectangle area when the width is 1 inch
= M + 52 - M.

2 (l+b) = 40 units.
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I did not understand how 1 inch border is counted as 2. Can somebody please explain?

Here we go----

Let Length and Breadth of photograph be L and B respectively.
Perimeter is given by 2*(L+b) -----(1)

According to the question:

(L+2)(B+2) = m ----(2)

and

(L+4)(B+4) = m + 52 ---------> (L+4)(B+4) - 52 = m ------(3)

Equating (2) and (3)

(L+2)(B+2) = (L+4)(B+4) - 52

LB + 2L + 2B + 4 = LB + 4L + 4B + 16 -52

Simplify
2L + 2B = 40 ------> 2(L+B) = 40 (Check eq (1))

Answer is D

Please see attached visual.

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LB=M
(L+2)(B+2) = M+52
2(L+B) = 48
Perimeter of pic = 2x (L-2+B-2) = 2(L+B)-8 = 48-8 =40

Answer is D
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