Paint Calculation: How Many Gallons Needed?

Hey guys, gather 'round! Today we're diving into a real-world scenario that involves a bit of math magic. Imagine Tom and Sami are gearing up for two painting jobs. They've got a limited shopping trip – only one stop allowed at the paint store! This means they need to be super smart about how much paint they grab. The first job is a moderate one, requiring 1 rac{4}{5} gallons of paint. Think of a couple of rooms, maybe a hallway, or a decent-sized feature wall. It’s not a massive undertaking, but it definitely needs its own supply. Now, the second job? That’s a different beast altogether. It’s a much bigger project, calling for a hefty 12 rac{1}{3} gallons of paint. This could be an entire house, a large commercial space, or maybe a really ambitious outdoor deck project. The key challenge here is that they can only make one trip to the store. This isn't a 'pop back for more later' situation, guys. They've got to get it right the first time. So, the burning question is: How many gallon cans of paint should they buy? This isn't just about adding two numbers; it's about understanding how paint is sold – in whole cans! You can't buy fractions of a gallon. This means we'll have to round up to make sure they have enough paint to complete both jobs without running out halfway through. Let's break down the math and figure out the smartest way for Tom and Sami to tackle this shopping trip and ensure their painting projects are a smashing success. We need to consider the total amount of paint needed and then figure out how many full gallon cans that translates to. This is a classic problem that pops up in everyday life, from DIY home renovations to managing supplies for larger projects. It teaches us the importance of planning and accurate estimation. So, let's put on our thinking caps and get ready to solve this paint puzzle!

Adding Up the Paint Needs

Alright, let's get down to business, folks! The total amount of paint Tom and Sami need is the sum of the paint required for both jobs. We have the first job needing 1 rac{4}{5} gallons, and the second job demanding a whopping 12 rac{1}{3} gallons. To find the total, we need to add these two mixed numbers together. Now, I know adding fractions can sometimes feel a bit tricky, but it's totally manageable if we follow the steps. First off, we need to find a common denominator for the fractions rac{4}{5} and rac{1}{3}. The least common multiple of 5 and 3 is 15. So, we'll convert both fractions to have a denominator of 15. For rac{4}{5}, we multiply both the numerator and the denominator by 3: rac{4 imes 3}{5 imes 3} = rac{12}{15}. For rac{1}{3}, we multiply both the numerator and the denominator by 5: rac{1 imes 5}{3 imes 5} = rac{5}{15}.

Now, our mixed numbers look like this: 1 rac{12}{15} and 12 rac{5}{15}. With the fractions now having the same denominator, we can add them easily. We add the whole numbers together: $1 + 12 = 13$. And we add the fractions together: rac{12}{15} + rac{5}{15} = rac{12+5}{15} = rac{17}{15}.

So, the total paint needed is 13 rac{17}{15} gallons. But hold on a sec! The fraction rac{17}{15} is an improper fraction because the numerator (17) is larger than the denominator (15). This means it represents more than one whole. We can convert this improper fraction into a mixed number. rac{17}{15} is equal to $1$ whole and rac{2}{15} remaining (since $17 ext{ divided by } 15 ext{ is } 1 ext{ with a remainder of } 2$).

Now we add this '1' back to our whole number sum from before. So, 13 + 1 rac{2}{15} = 14 rac{2}{15} gallons. This is the exact total amount of paint Tom and Sami need for both jobs. Phew! We've done the heavy lifting with the fraction addition. This 14 rac{2}{15} gallons is the precise volume of paint required. It's crucial to get this number right because it forms the basis for our final purchasing decision. Remember, they can't buy fractions of a gallon, so this precise total is what we'll use to figure out how many cans they need to pick up from the store on their single trip. This step is all about accuracy; we've meticulously added the requirements of both projects to get a clear picture of the total paint volume.

The Importance of Buying Whole Cans

Now, here's where the practical reality kicks in, guys! We've calculated that Tom and Sami need exactly 14 rac{2}{15} gallons of paint. That's the ideal amount. However, paint isn't sold in fractions of a gallon. You can't walk into the store and ask for rac{2}{15} of a gallon can. Stores sell paint in standardized container sizes, most commonly in 1-gallon cans. This means that even though they only need a tiny bit more than 14 gallons, they have to buy enough whole cans to cover that amount. If they were to buy just 14 cans, they would have exactly 14 gallons. But they need 14 rac{2}{15} gallons. That leaves them short by rac{2}{15} of a gallon. Imagine running out of paint when you're just about to finish the last coat on a wall – that would be a total nightmare, right? Especially since they can only stop at the store once. This is precisely why we need to round up to the next whole number. We must ensure they have at least 14 rac{2}{15} gallons. So, if they need 14 rac{2}{15} gallons, and they can only buy in 1-gallon increments, they need to buy enough cans to exceed or meet this total. Buying 14 cans gives them 14 gallons, which isn't enough. Therefore, they must buy the next whole number of cans, which is 15 gallons. This guarantees they have enough paint, with a little bit left over, just in case. This leftover paint can be super handy for touch-ups later on, too! The principle here is critical for any purchasing decision where items are sold in discrete units. Whether it's paint cans, boxes of tiles, or bags of concrete, you always round up to ensure you have sufficient quantity to complete the job. It's about being prepared and avoiding costly return trips or project delays. This step highlights the difference between a theoretical calculation and a practical application, emphasizing that real-world constraints often require adjusting our plans. So, always remember to account for how items are packaged and sold when you're budgeting or planning materials for a project.

Determining the Final Answer

So, we've crunched the numbers, figured out the exact paint needed, and understood the real-world constraint of buying paint in whole gallon cans. We determined that Tom and Sami need a total of 14 rac{2}{15} gallons of paint to complete both their projects. Since they can only visit the store once, they must purchase enough whole gallon cans to meet or exceed this requirement. If they buy 14 cans, they will have exactly 14 gallons, which is less than the 14 rac{2}{15} gallons they need. This would leave them short and unable to finish the jobs. Therefore, they must round up to the next whole number of cans. The next whole number after 14 is 15. By purchasing 15 gallon cans, they will have a total of 15 gallons of paint. This is more than the 14 rac{2}{15} gallons required, ensuring they have enough paint to finish both jobs without any issues. They'll even have a small surplus, which is always a good thing for potential touch-ups down the line! Looking back at the options provided:

A. 11 cans B. 13 cans C. 14 cans D. Discussion category : mathematics

Option C, 14 cans, would leave them short. Options A and B are clearly insufficient as they are less than the total calculated amount. Our calculation shows they need to buy 15 cans. It seems there might be a slight discrepancy with the provided options, as 15 is not listed. However, based on standard mathematical and practical problem-solving, the correct approach leads to needing 15 cans. Let's re-evaluate the question and options to ensure we haven't missed anything. The question asks, "How many gallon cans of paint should they buy?" We calculated the exact need as 14 rac{2}{15} gallons. To meet this, they must buy 15 cans. If we must choose from the given options, and assuming there might be a typo or a need to select the closest answer that ensures enough paint, we'd still lean towards needing more than 14. However, in a typical math problem, the answer would be 15. Let's consider if the question implies something else, like estimating. But the phrasing is direct. Given the standard way these problems are structured, and the necessity of having enough paint, the answer derived is 15 cans. If we were forced to pick the 'least wrong' from the options, it would highlight a flaw in the question's provided choices. However, the process is: Calculate total needed (14 rac{2}{15} gallons) -> Determine how many 1-gallon units are needed -> Round UP to the nearest whole unit. This gives 15 cans. Let's assume, for the sake of proceeding with the given options, that maybe there's a misunderstanding or a typo in the options and focus on the correct mathematical reasoning. The correct mathematical answer is 15 cans. If we had to select one of the options provided, it points to an issue with the question's choices rather than the calculation. However, if this were a multiple-choice test and 15 was not an option, it might indicate a need to re-read the question very carefully for subtle nuances, or it might simply be a poorly constructed question. For clarity and correctness, the answer is 15 cans. It's vital in these types of problems to always round up to ensure sufficiency.