Rectangle Resize: Calculating The New Width

Have you ever wondered how changing the dimensions of a shape affects its other measurements? Let's dive into a common problem involving rectangles and scaling. Picture this: Shawna is working with a rectangle that's initially 24 inches wide and 12 inches tall. She decides to reduce the height to just 2 inches. The big question is, what does this do to the width? How do we figure out the new width of this scaled-down rectangle?

Understanding the Problem

When dealing with geometric scaling problems, it's super important to understand the relationships between the original and the scaled figures. In our case, we have a rectangle, and we're shrinking it while maintaining its proportions. This means we're not just chopping off a piece; we're reducing it proportionally. The key here is that the ratio of width to height remains constant. Think of it like resizing a photo on your computer; you want to make it smaller without distorting the image.

Let's define our terms:

• Original width = 24 inches
• Original height = 12 inches
• New height = 2 inches
• New width = ? (This is what we want to find!)

The relationship we need to remember is:

(Original Width) / (Original Height) = (New Width) / (New Height)

This proportion tells us that whatever factor we're using to shrink the height, we must use the same factor to shrink the width. If we don't, our rectangle will turn into some weird, distorted shape, which is not what we want!

Setting Up the Proportion

Okay, guys, now that we understand the concept, let's put some numbers into action. We know the original width and height, and we know the new height. We're trying to find the new width. Let's set up the proportion using the formula we just talked about:

24 / 12 = New Width / 2

This equation is the heart of our problem. It tells us that the ratio of 24 to 12 (original dimensions) is equal to the ratio of the new width to 2 (new height). Our next step is to solve for the "New Width."

Solving for the New Width

To solve for the "New Width," we need to isolate it on one side of the equation. A common way to do this is by cross-multiplying. Here's how it works:

24 / 12 = New Width / 2

Cross-multiply: 24 * 2 = 12 * New Width

This simplifies to:

48 = 12 * New Width

Now, to get "New Width" by itself, we divide both sides of the equation by 12:

48 / 12 = New Width

This gives us:

4 = New Width

So, the new width is 4 inches! Easy peasy, right?

Checking Our Answer

It's always a good idea to check our work to make sure our answer makes sense. Let's go back to our original proportion:

24 / 12 = New Width / 2

We found that the "New Width" is 4 inches. So, let's plug that back into the equation:

24 / 12 = 4 / 2

Simplify both sides:

2 = 2

Since both sides of the equation are equal, our answer is correct! This confirms that the new width of the rectangle is indeed 4 inches when the height is reduced to 2 inches.

Real-World Applications

You might be thinking, "Okay, that's cool, but when will I ever use this?" Well, understanding scaling and proportions is super useful in many real-world scenarios. Here are a few examples:

• Photography: When you resize a photo, you're essentially scaling it. Knowing how to maintain proportions prevents your pictures from looking stretched or squished.
• Architecture: Architects use scaling to create blueprints of buildings. They need to ensure that the proportions are accurate so that the final structure is sound and aesthetically pleasing.
• Graphic Design: Designers use scaling all the time when creating logos, websites, and other visual materials. Maintaining the correct proportions is crucial for a professional look.
• Cooking: Ever tried doubling a recipe? That's scaling! You need to adjust all the ingredients proportionally to ensure the dish turns out right. So next time you're baking a cake, remember Shawna and her rectangle!

Common Mistakes to Avoid

When working with scaling problems, there are a few common mistakes that people often make. Here are some tips to help you avoid them:

• Forgetting to Maintain Proportions: The biggest mistake is not keeping the ratios consistent. If you change one dimension, you must adjust the other dimensions proportionally.
• Incorrectly Setting Up the Proportion: Make sure you set up the proportion correctly. Double-check that you have the original width and height on one side and the new width and height on the other side.
• Math Errors: Simple math errors can throw off your entire calculation. Take your time and double-check your work, especially when cross-multiplying and dividing.
• Not Checking Your Answer: Always check your answer to make sure it makes sense in the context of the problem. This can help you catch any errors you might have made.

Practice Problems

Want to test your understanding? Try these practice problems:

1. A rectangle is originally 15 inches wide and 5 inches tall. If the height is reduced to 1 inch, what is the new width?
2. A poster is 36 inches wide and 24 inches tall. If you want to reduce the width to 12 inches, what will the new height be?
3. A garden is 20 feet long and 10 feet wide. If you want to double the length, what will the new width be to maintain the same proportions?

See if you can solve these problems using the methods we discussed. Good luck!

Conclusion

So, there you have it! By understanding the principles of scaling and proportions, we were able to help Shawna find the new width of her rectangle. Remember, the key is to maintain the ratio between the dimensions. Whether you're resizing photos, designing buildings, or doubling recipes, these concepts will come in handy. Keep practicing, and you'll become a scaling pro in no time!

And remember, math isn't just about numbers; it's about understanding the relationships between things. Once you grasp those relationships, you can solve all sorts of interesting problems.