Calculating Paint Coverage How To Find Unit Rate Step By Step

Hey guys! Let's dive into a cool math problem that involves calculating how much someone can paint with a certain amount of paint. This is a classic example of finding a unit rate, which basically tells us how much of one thing we can do with just one unit of another thing. In our case, we want to figure out how many walls Martin can paint with one gallon of paint. This kind of problem is super useful in real life, whether you’re planning a home improvement project or just trying to understand rates and proportions better.

So, the question is: Martin uses $\frac{5}{8}$ of a gallon of paint to cover $\frac{4}{5}$ of a wall. What is the unit rate at which Martin paints in walls per gallon? This sounds a bit tricky at first, but don’t worry, we'll break it down step by step. We need to figure out how much wall Martin can paint with one whole gallon of paint. That's what finding the unit rate is all about – scaling things to a single unit. Think of it like figuring out the price per apple if you know the price for a bag of apples. It's the same concept, just with paint and walls!

To solve this, we'll use a little bit of fraction division. The key idea here is that if we know how much of a wall is painted by a fraction of a gallon, we can divide the amount of wall painted by the fraction of a gallon used. This will give us the amount of wall painted per one gallon. It's like saying, "If this much paint covers this much wall, then one gallon covers how much wall?" The answer is the unit rate we are looking for. Understanding this concept is crucial because unit rates pop up everywhere – from calculating gas mileage in your car (miles per gallon) to figuring out the cost of groceries (price per pound). So, mastering this skill will help you in all sorts of situations.

Okay, so let’s get down to the nitty-gritty of how to solve this problem. Remember, we know that Martin used $\frac{5}{8}$ of a gallon to paint $\frac{4}{5}$ of a wall. The first thing we need to do is set up a proportion. A proportion is just a way of saying that two ratios are equal. In our case, the ratio we know is the amount of wall painted to the amount of paint used. We can write this as a fraction: $\frac{\frac{4}{5} \text{ wall}}{\frac{5}{8} \text{ gallon}}$. This looks a little messy, but don't worry, we'll clean it up soon!

What we want to find is the number of walls painted per one gallon. This is our unit rate. We can represent this as a fraction too: $\frac{x \text{ walls}}{1 \text{ gallon}}$, where x is the unknown number of walls we're trying to find. Now, we can set these two ratios equal to each other to form our proportion: $\frac{\frac{4}{5}}{\frac{5}{8}} = \frac{x}{1}$. See how we've lined everything up? Walls on top, gallons on the bottom. This is important to keep the units straight and make sure our calculations make sense. Setting up the problem correctly is half the battle, so make sure you understand this step completely before moving on. Think of it like building a house – you need a solid foundation before you can start putting up the walls!

Now, you might be wondering, "Why are we setting up a proportion like this?" Well, the beauty of proportions is that they allow us to compare two related quantities. In this case, we're comparing the amount of wall painted to the amount of paint used. By setting up a proportion, we're essentially saying that the rate at which Martin paints is constant. In other words, he paints the same amount of wall for every gallon of paint he uses. This assumption is what allows us to use proportions to solve the problem. If Martin's painting rate wasn't consistent, we wouldn't be able to use this method. So, keep in mind that proportions are most useful when dealing with situations where quantities are changing at a constant rate.

Alright, we've set up our proportion: $\frac{\frac{4}{5}}{\frac{5}{8}} = \frac{x}{1}$. Now comes the fun part: solving for x, which is the unit rate we're after. Since anything divided by 1 is just itself, we can simplify the right side of the equation to just x. So, we really need to figure out what $\frac{\frac{4}{5}}{\frac{5}{8}}$ equals. And that, my friends, means dividing fractions!

Remember the golden rule of dividing fractions? It’s "Keep, Change, Flip." This is a handy way to remember the steps involved. We keep the first fraction ($\frac{4}{5}$), change the division sign to a multiplication sign, and flip the second fraction (rac{5}{8} becomes $\frac{8}{5}$). So, our equation now looks like this: $\frac{4}{5} \times \frac{8}{5} = x$. See how we turned a potentially scary division problem into a much more manageable multiplication problem? This little trick makes fraction division way less intimidating.

Now, multiplying fractions is pretty straightforward. You just multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, we have $(4 \times 8)$ on top and $(5 \times 5)$ on the bottom. This gives us $\frac{32}{25} = x$. Boom! We've found our unit rate! But wait, it's an improper fraction (the numerator is bigger than the denominator). While $\frac{32}{25}$ is perfectly correct, it's often helpful to convert it to a mixed number so we can get a better sense of the quantity. This will help us visualize how much wall Martin can actually paint with one gallon.

So, we've got our unit rate as $\frac{32}{25}$ walls per gallon. To make this easier to understand, let's turn it into a mixed number. Remember, a mixed number is just a whole number plus a fraction. To do this, we ask ourselves, "How many times does 25 go into 32?" The answer is 1 time, with a remainder of 7. So, we have 1 whole and $\frac{7}{25}$ left over. This means $\frac{32}{25}$ is the same as $1 \frac{7}{25}$.

Therefore, our final answer is that Martin can paint $1 \frac{7}{25}$ walls per gallon of paint. Now, let's think about what this actually means. It means that with one gallon of paint, Martin can paint one entire wall and a little bit more – specifically, $\frac{7}{25}$ of another wall. This gives us a much clearer picture than just saying $\frac{32}{25}$ walls. Seeing the "1" in front makes it easy to grasp that he can definitely complete one wall.

This is why converting to a mixed number is often a good idea when dealing with improper fractions, especially in real-world problems. It helps us put the answer in context and understand its practical implications. Imagine telling someone, "Martin can paint $\frac{32}{25}$ walls per gallon." It's not as immediately clear as saying, "Martin can paint one and a bit more than a quarter of another wall per gallon." The mixed number just makes the information more accessible and relatable.

So, to recap, we started with a problem about Martin using $\frac{5}{8}$ of a gallon of paint to cover $\frac{4}{5}$ of a wall. We wanted to find the unit rate – how many walls Martin can paint with one gallon. We set up a proportion, divided fractions, and converted an improper fraction to a mixed number. And finally, we arrived at our answer: Martin can paint $1 \frac{7}{25}$ walls per gallon of paint.

Therefore, the correct answer is A. $1 \frac{7}{25}$ walls per gallon.

I hope this step-by-step explanation has made the process crystal clear for you guys! Remember, the key to solving these kinds of problems is to break them down into smaller, manageable steps. Don't be intimidated by the fractions – just take your time, follow the rules, and you'll get there. And remember, understanding unit rates is a super valuable skill that will help you in all sorts of situations, both in math class and in the real world. So keep practicing, and you'll become a pro in no time!