Question 257447
Let x = the width of the picture.
Then, since the length is 4/3 of the width, the length of the picture is {{{(4/3)x = 4x/3}}}<br>
Let's look at a picture of the situation:
{{{drawing(400, 400, 0, 12, 0, 12, line(1, 1, 1, 11), line(9, 1, 9, 11), line(3, 3, 3, 9), line(7, 3, 7, 9), line(1, 1, 9, 1), line(1, 3, 9, 3), line(1, 9, 9, 9), line(1, 11, 9, 11), locate(9.1, 10, 2), locate(8, 9, 2), locate(2, 9, 2), locate(5, 8.7, x), locate(6, 6.2, 4x/3), locate(9.1, 2.5, 2))}}}
The inner rectangle is the picture.<br>
The four outer rectangles make up the frame. The two larger rectangles, top and bottom, have a length of 2 + x + 2 or x+4 and a width of 2. The area of each is (x+4)2 = 2x+8.<br>
The two smaller rectangles on the sides have a length of 4x/3 and a width of 2. The area of each is 4x/3*2 = 8x/3.<br>
The combined areas of the four rectangles is 2x+8 + 2x+8 + 8x/3 + 8x/3 = 4x+16+16x/3. We are told that the area of the frame is 100 square centimeters. So
{{{4x+16+16x/3 = 100}}}
We can use Algebra to solve this. We'll start by eliminating the fraction by multiplying both sides by 3:
{{{12x + 48 + 16x = 300}}}
which simplifies to:
{{{28x + 48 = 300}}}
Subtracting 48 from each side we get:
{{{28x = 252}}}
Dividing both sides by 28 we get:
{{{x = 9}}}
Since x is the width of the picture the width is 9 centimeters. The length picture is 4x/3 = 4(9)/3 = 12 centimeters.