Hello there! Could You please help me with this Geometry word problem?

Here's the problem: Helen painted a picture that was 10 inches longer than it was wide. When she framed the picture, the outside dimensions (that is, the length and the width) were each two inches longer. If the area of the picture with the frame was 40 sq inches more than the area of the picture without its frame, what was the length of the original painting?

Any help you could provide would be greatly appreciated!

length = w+10

(w+2)(w+10+2) = w(w+10)+40
w = 4

check:
picture: 4x14 = 56
total : 6*16 = 96 = 56+40

Hello! I would be happy to help you with this geometry word problem. Let's break it down step by step.

Step 1: Assign variables
Let's use variables to represent the unknown quantities in the problem. We'll let "w" represent the width of the original painting.

Step 2: Express the length in terms of the width
According to the problem, the length of the painting is 10 inches longer than its width. Therefore, the length can be expressed as (w + 10).

Step 3: Calculate the area of the painting without the frame
The area of a rectangle is given by the formula: Area = length * width. In this case, the area without the frame is (w * (w + 10)).

Step 4: Calculate the outside dimensions (with the frame)
According to the problem, the outside dimensions are each two inches longer than the original painting. Therefore, the outside length would be ((w + 10) + 2) and the outside width would be (w + 2).

Step 5: Calculate the area of the painting with the frame
Similar to step 3, the area of the painting with the frame is given by the formula: Area = length * width. In this case, the area with the frame is ((w + 10 + 2) * (w + 2)).

Step 6: Set up the equation
The problem states that the area of the picture with the frame was 40 square inches more than the area of the picture without its frame. So, we can set up the equation:

((w + 10 + 2) * (w + 2)) = (w * (w + 10)) + 40

Step 7: Solve the equation
Simplify the equation above and solve for w:

(w + 12) * (w + 2) = w^2 + 10w + 40
w^2 + 14w + 24 = w^2 + 10w + 40
4w + 24 = 40
4w = 16
w = 4

Step 8: Find the length of the original painting
The length of the original painting is 10 inches longer than its width, which we have found to be 4 inches. So, the length would be (4 + 10) = 14 inches.

Therefore, the original painting had a length of 14 inches.

I hope this step-by-step explanation helps you understand how to approach and solve geometry word problems like this one! Let me know if there's anything else I can assist you with.