Sue wants to frame a picture with a matte of uniform width all around the print. Sue wants the area of the matte to equal the area of the picture. If the print measures 60cm by 40cm how wide should the matte be?

let the width of the uniform matte be x cm
We know the area of the picture is 60*40 or 2400 cm^2

The width of the whole frame would be 40+2x and the height would be 60+2x

since the area of the picture is equal to the area of the matte, the area of the whole thing is twice the area of the picture, so....
(40+2x)(60+2x) = 2(2400)

expand and solve that quadratic.
One of your answers will be extraneous, that is, it will make no sense for your picture. (hint: I got 10 cm for the width of the matte)

thank you SO much!!!! this helped me sooo much

You're welcome! I'm glad I could help you with your question.

To solve the problem, we need to set up an equation based on the given information. Let's say the width of the uniform matte is "x" cm. We know that the area of the picture is 60 cm by 40 cm, so its area is 60 * 40 = 2400 cm^2.

Now, let's consider the dimensions of the whole frame. The width would be the sum of the picture's width and twice the width of the matte, which is 40 + 2x cm. Similarly, the height would be 60 + 2x cm.

Since the area of the matte is equal to the area of the picture, the area of the whole frame is twice the area of the picture. So, we can set up the equation:

(40 + 2x) * (60 + 2x) = 2 * 2400

Expand and solve the quadratic equation:

2400 + 80x + 120x + 4x^2 = 4800

4x^2 + 200x - 2400 = 0

Now, you can solve the quadratic equation to find the value of "x". Once you've solved it, you will get two possible solutions. However, keep in mind that one of the answers might not make sense for your picture. In this case, the width of the matte should be 10 cm.

I hope this explanation helps! Let me know if you have any further questions.