Which Expression Represents The Width Of The Framed Picture

Which Expression Represents the Width of the Framed Picture? A Deep Dive into Mathematical Problem Solving

Finding the correct expression to represent the width of a framed picture might seem straightforward, but it's a great example of how seemingly simple problems can require careful consideration of variables and their relationships. This article will explore various scenarios, demonstrating how to derive the correct expression for the width of a framed picture under different conditions. We'll also delve into the underlying mathematical concepts and problem-solving strategies, equipping you with the skills to tackle similar problems with confidence.

Understanding the Problem: Defining Variables and Relationships

Before we jump into different scenarios, let's establish a foundational understanding of the variables involved. We'll primarily be focusing on three key variables:

• w (Picture Width): The width of the unframed picture itself. This is often the unknown we're trying to find.
• f (Frame Width): The width of the frame on one side. It's crucial to remember that a frame adds to the width on both sides of the picture.
• W (Total Width): The total width of the framed picture, including the picture and the frame. This is frequently the known value given in a problem.

The core relationship between these variables is the addition of the frame's width to the picture's width to obtain the total width. However, the specific expression depends on the information provided in the problem.

Scenario 1: The Total Width and Frame Width are Known

This is the simplest scenario. We know the total width (W) of the framed picture and the frame's width (f). The expression for the picture's width (w) is derived by subtracting the frame's width from both sides of the picture from the total width:

w = W - 2f

Example:

If the total width of the framed picture (W) is 24 inches and the frame's width on one side (f) is 2 inches, then the picture's width (w) is:

w = 24 - 2 * 2 = 24 - 4 = 20 inches.

This formula directly reflects the fact that the frame adds f inches to the left and f inches to the right of the picture.

Scenario 2: The Total Width and Picture Width are Known

This scenario flips the previous one. We know the total width (W) and the picture width (w). We want to find the frame width (f). The expression is derived by rearranging the fundamental equation:

f = (W - w) / 2

Example:

If the total width of the framed picture (W) is 30 inches and the picture width (w) is 26 inches, then the frame width (f) is:

f = (30 - 26) / 2 = 4 / 2 = 2 inches.

This makes intuitive sense: the total width minus the picture width gives us the combined width of the frames on both sides, which, when divided by two, gives the frame width on one side.

Scenario 3: Only the Picture Width and Frame Width are Known

In this case, we know the picture width (w) and the frame width (f). We want to find the total width (W). This is the most straightforward calculation:

W = w + 2f

Example:

If the picture width (w) is 18 inches and the frame width (f) is 1 inch, then the total width (W) is:

W = 18 + 2 * 1 = 18 + 2 = 20 inches.

This simply adds the frame's contribution to both sides of the picture to the picture's width.

Scenario 4: Dealing with More Complex Frames

Let's consider scenarios with more complex frame structures. For example, what if the frame has different widths on different sides?

Let's say the frame has width f1 on the left and f2 on the right. In this case, the total width (W) is represented by:

W = w + f1 + f2

And conversely, if we know W, f1, and f2, we can solve for the picture width:

w = W - f1 - f2

Scenario 5: Incorporating Units of Measurement

It's critically important to maintain consistency in units of measurement. If the picture width is given in centimeters and the frame width is given in inches, you must convert one to match the other before performing any calculations. Always explicitly state the units in your final answer.

Scenario 6: Solving Word Problems Involving the Width of a Framed Picture

Word problems often present the information in a less direct way. It's vital to carefully read the problem and identify the relevant variables. Let's look at an example:

"A picture measuring 10 inches wide is framed with a 1.5-inch-wide frame. What is the total width of the framed picture?"

Here, w = 10 inches and f = 1.5 inches. We use the formula:

W = w + 2f = 10 + 2 * 1.5 = 10 + 3 = 13 inches.

The total width of the framed picture is 13 inches.

Advanced Concepts and Problem-Solving Strategies

While the above scenarios cover the basics, more complex problems might involve:

• Multiple frames: Imagine a picture with two frames, each with a different width. The total width would be the sum of the picture width and the widths of both frames on each side.
• Proportions and ratios: Problems might involve proportions, where the ratio of the picture width to the frame width is given. You'd then use algebraic techniques to solve for the unknowns.
• Geometric shapes: The picture might not be rectangular, leading to the need to use geometric principles to find the width or other dimensions.

Mastering the Art of Problem Solving: A Step-by-Step Guide

1. Identify the knowns and unknowns: Carefully read the problem and list what you already know and what you need to find.

2. Define variables: Assign appropriate variables to each quantity (e.g., w, f, W).

3. Draw a diagram: A visual representation can greatly simplify the problem. Draw the picture and the frame, labeling the dimensions.

4. Write down the relevant formula: Based on the problem's scenario, choose the appropriate formula from the ones discussed above or derive a new one.

5. Substitute and solve: Plug in the known values into the formula and solve for the unknown.

6. Check your answer: Make sure your answer is reasonable and consistent with the information given in the problem. Does it make sense in the context of the word problem?

7. State your answer clearly: Include the appropriate units of measurement in your final answer.

By following this systematic approach, you can confidently tackle any problem involving the width of a framed picture, no matter how complex it may appear. Remember, the key is to carefully analyze the problem, identify the relationships between variables, and apply the appropriate mathematical principles.