Calculating The Blue Area On Marcus's Wall Math Problem Solved
Hey guys! Let's dive into a fun math problem about painting a wall. Marcus has a wall in his bedroom that he wants to paint, and we need to figure out how much of it will be blue. This involves a bit of fractions and area calculations, but don't worry, we'll break it down step by step so it's super easy to understand. So, grab your thinking caps, and let's get started!
Understanding the Wall Dimensions
First, let's talk about the wall itself. Marcus's wall isn't just any wall; it has specific dimensions. It's 8 1/3 feet high and 16 1/5 feet long. Now, these aren't your typical whole numbers, are they? We're dealing with mixed fractions here, which means we have a whole number part and a fractional part. To make things easier to work with, especially when we start multiplying, we need to convert these mixed fractions into improper fractions. An improper fraction is where the numerator (the top number) is larger than or equal to the denominator (the bottom number).
So, how do we do that? For the height, which is 8 1/3 feet, we multiply the whole number (8) by the denominator (3), which gives us 24. Then, we add the numerator (1) to that result, which gives us 25. We keep the same denominator (3), so the improper fraction for the height is 25/3 feet. See? Not so scary, right? We're just changing the way it looks without changing its actual value. It’s like dressing up the fraction in a new outfit!
Let's do the same thing for the length, which is 16 1/5 feet. We multiply the whole number (16) by the denominator (5), which gives us 80. Then, we add the numerator (1), which gives us 81. Keep the denominator (5), and we get 81/5 feet. So, now we know the wall is 25/3 feet high and 81/5 feet long. Converting to improper fractions was a crucial first step because it makes the multiplication process much smoother.
Calculating the Total Area of the Wall
Now that we have the dimensions in a more usable form, let's figure out the total area of the wall. Remember, area is the amount of surface something covers, and for a rectangle (which we're assuming Marcus's wall is), we calculate it by multiplying the length by the width (or in this case, the height). So, we need to multiply 25/3 feet by 81/5 feet. When multiplying fractions, it’s super straightforward: you just multiply the numerators together and the denominators together.
So, we multiply 25 by 81, which equals 2025. Then, we multiply 3 by 5, which equals 15. That gives us a fraction of 2025/15 square feet. But wait, we're not quite done yet! This fraction looks a bit intimidating, doesn't it? It's a large improper fraction, and we can simplify it to make it easier to understand. To simplify, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that divides evenly into both numbers. In this case, the GCD of 2025 and 15 is 15 itself.
When we divide 2025 by 15, we get 135. And when we divide 15 by 15, we get 1. So, the simplified fraction is 135/1, which is just 135. This means the total area of Marcus's wall is 135 square feet. That's a pretty big wall! It’s important to remember the units here – we’re talking about square feet because we’re measuring area, which is a two-dimensional space.
Determining the Blue Area
Okay, we know the total area of the wall, but Marcus isn't painting the whole wall blue. He's only painting 1/3 of it blue. This is where fractions come into play again, but this time we're finding a fraction of a whole number. To find 1/3 of 135 square feet, we simply multiply 135 by 1/3. Remember, multiplying by a fraction is the same as dividing by the denominator of the fraction. So, we're essentially dividing 135 by 3.
When we divide 135 by 3, we get 45. So, 1/3 of the wall's area is 45 square feet. This means Marcus will be painting 45 square feet of his wall blue. Isn't it cool how we can use fractions to figure out parts of a whole? This is a super useful skill in everyday life, from cooking to home improvement projects like this one. We’ve successfully calculated the blue area by finding a fraction of the total area.
Final Calculation and Answer
So, to recap, we started with the mixed fraction dimensions of the wall, converted them to improper fractions, calculated the total area of the wall, and then found 1/3 of that area to determine the blue portion. Our final answer is that Marcus will be painting 45 square feet of his wall blue. We’ve gone through all the steps, and we’ve arrived at a clear, concise answer. Isn't math awesome when you break it down into manageable steps? Each step builds upon the previous one, leading us to the final solution. This problem highlights the practical application of fractions and area calculations in real-life scenarios.
Conclusion: The Blue Hue
So there you have it! Marcus will have 45 square feet of blue on his bedroom wall. We tackled mixed fractions, improper fractions, area calculations, and fractions of a whole number. Hopefully, you found this explanation helpful and maybe even a little bit fun. Remember, math isn't just about numbers; it's about problem-solving and understanding the world around us. Now, you're all equipped to calculate areas and fractions like pros. Maybe you can even help your friends with their math homework now! Keep practicing, and you'll be amazed at what you can achieve. And hey, who knows? Maybe you'll even inspire someone to paint their room blue!
