The shaded region in the figure represents a rectangular frame with length 18 and width 15. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the length and width of the frame, what is the length of the picture?
The official answer is as follows:
Let l and w be the length and width of the picture, so:
1) lw = 270 - lw
2) l/w = 18/15
From these two equations, we can solve to get the answer of 9(2)^1/2
My question is this - for equation #1, I wrote it as:
1) lw = (18-l)(15-w)
The reasoning here is that the area of the picture (lw) is equal to the area of the shaded region or the frame which should be (18-l)(15-w).
I then use the same equation 2 to solve for l.
I know there is a flaw in my reasoning here, but could someone please help point out what it is?...I can't seem to get my head around what I'm doing wrong.
Thanks!
[/img]
The official answer is as follows:
Let l and w be the length and width of the picture, so:
1) lw = 270 - lw
2) l/w = 18/15
From these two equations, we can solve to get the answer of 9(2)^1/2
My question is this - for equation #1, I wrote it as:
1) lw = (18-l)(15-w)
The reasoning here is that the area of the picture (lw) is equal to the area of the shaded region or the frame which should be (18-l)(15-w).
I then use the same equation 2 to solve for l.
I know there is a flaw in my reasoning here, but could someone please help point out what it is?...I can't seem to get my head around what I'm doing wrong.
Thanks!
[/img]
