Olive Oil Sales: Maximizing Profit With Minimum Bottles

Hey guys! Let's break down this interesting math problem about an olive oil merchant trying to maximize his profit. We'll take it step by step to make sure everyone understands. So, grab your thinking caps, and let's dive in!

Understanding the Olive Oil Purchase

First, let's figure out how much the merchant spent on the olive oil. Our merchant bought two batches of olive oil: 42 liters at 40 TL per liter and 63 liters at 50 TL per liter. To calculate the total cost, we need to figure out the cost of each batch and then add them together. For the first batch, we multiply the quantity (42 liters) by the price per liter (40 TL). This gives us 42 * 40 = 1680 TL. So, the first batch cost 1680 TL. Next, we do the same for the second batch. We multiply the quantity (63 liters) by the price per liter (50 TL). This gives us 63 * 50 = 3150 TL. Therefore, the second batch cost 3150 TL. Now, to find the total cost, we add the cost of the two batches: 1680 TL + 3150 TL = 4830 TL. So, the merchant spent a total of 4830 TL on olive oil. This initial calculation is crucial because it forms the baseline against which we will measure the merchant's profit. Knowing the total cost helps us understand how much revenue the merchant needs to generate to break even and, more importantly, to make a profit. Without this foundation, we cannot accurately assess the profitability of the merchant's venture. Therefore, accurately calculating the total cost of the purchase is the first and one of the most important steps in solving this problem.

Determining the Bottle Size

The next key step in solving this problem involves figuring out the size of the bottles the merchant will use. The merchant wants to use the fewest number of bottles possible while ensuring each bottle is filled identically. This means we need to find the greatest common divisor (GCD) of the two quantities of olive oil: 42 liters and 63 liters. The GCD is the largest number that divides both 42 and 63 without leaving a remainder. There are a couple of ways we can find the GCD. One method is to list the factors of each number and find the largest factor they have in common. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The factors of 63 are 1, 3, 7, 9, 21, and 63. Looking at these lists, we can see that the largest factor they have in common is 21. Alternatively, we can use the Euclidean algorithm, which is a more efficient method for larger numbers. The Euclidean algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD. So, we divide 63 by 42, which gives us a quotient of 1 and a remainder of 21. Then, we divide 42 by 21, which gives us a quotient of 2 and a remainder of 0. Since the remainder is 0, the GCD is 21. This means the largest bottle size the merchant can use is 21 liters. Using 21-liter bottles ensures that the merchant uses the fewest possible bottles, which aligns with the problem's requirement. Furthermore, understanding the GCD is crucial in various real-world applications, such as dividing resources equally, simplifying fractions, and even in cryptography. So, this step not only helps us solve the olive oil problem but also reinforces an important mathematical concept.

Calculating the Number of Bottles

Now that we know the size of each bottle (21 liters), we can figure out how many bottles the merchant will need. To do this, we simply divide the quantity of each type of olive oil by the bottle size. For the 42 liters of olive oil, we divide 42 by 21, which gives us 2 bottles. So, the merchant will need 2 bottles for the first batch of olive oil. For the 63 liters of olive oil, we divide 63 by 21, which gives us 3 bottles. Thus, the merchant will need 3 bottles for the second batch of olive oil. To find the total number of bottles, we add the number of bottles needed for each batch: 2 bottles + 3 bottles = 5 bottles. Therefore, the merchant will use a total of 5 bottles. This calculation is a straightforward application of division, but it’s an essential step in determining the overall profitability of the venture. The number of bottles directly impacts the revenue generated, as each bottle will be sold for 500 TL. Knowing the total number of bottles allows us to calculate the total revenue, which is a key component in determining the profit. Moreover, this step highlights the importance of efficient packaging and distribution in business. By using the largest possible bottle size, the merchant minimizes the number of bottles needed, which can save on packaging costs and simplify logistics. So, while the calculation itself is simple, its implications for the business's efficiency and profitability are significant. Understanding how to optimize packaging and distribution is a valuable skill in any business context.

Determining the Total Revenue

Okay, let's talk money! We need to calculate how much the merchant will make by selling these olive oil bottles. We know the merchant is selling each bottle for 500 TL, and we've already figured out that he has a total of 5 bottles. So, to find the total revenue, we multiply the number of bottles by the selling price per bottle. This means we do 5 bottles * 500 TL/bottle = 2500 TL. So, the merchant will make a total of 2500 TL from selling all the olive oil. This is a crucial number because it represents the total income generated from the sales. However, it's important to remember that this is not the profit yet. To calculate the profit, we need to subtract the total cost of the olive oil from this revenue. But for now, we have a clear picture of how much money is coming in. Understanding how to calculate revenue is fundamental in business and finance. It’s the first step in assessing the financial viability of any venture. Whether you’re selling olive oil, lemonade, or software, knowing your potential revenue helps you make informed decisions about pricing, marketing, and overall business strategy. So, this simple calculation is a powerful tool for anyone looking to start or run a business. Moreover, it's a practical application of basic multiplication that can be used in everyday situations, such as calculating the total cost of items you’re buying or estimating your earnings from a part-time job.

Calculating the Profit

Alright, time to get down to the real deal – figuring out the merchant's profit! Profit is what's left after we subtract the total costs from the total revenue. We already know that the merchant spent 4830 TL on olive oil, and he's going to make 2500 TL from selling it. So, to find the profit, we subtract the cost from the revenue: 2500 TL (revenue) - 4830 TL (cost) = -2330 TL. Wait a minute... that's a negative number! This means the merchant didn't actually make a profit; he incurred a loss of 2330 TL. This is a crucial finding because it tells us that the merchant's business strategy, at least in this scenario, is not financially viable. Understanding how to calculate profit is essential in any business context. It’s the bottom line that determines whether a business is successful or not. A negative profit, or a loss, indicates that the business is spending more money than it's making, which is not sustainable in the long run. In this case, the merchant's selling price of 500 TL per bottle was not high enough to cover the cost of the olive oil. This could be due to a variety of factors, such as the high purchase price of the olive oil, inefficient packaging, or a misjudgment of the market demand. So, while the calculation itself is simple subtraction, the implications for the merchant's business are significant. It highlights the importance of carefully considering costs, pricing, and market conditions when making business decisions. A thorough understanding of these factors can help entrepreneurs avoid losses and build profitable businesses.

Final Thoughts

So, in this scenario, our olive oil merchant didn't make a profit; he actually lost money. This highlights the importance of careful planning and pricing in any business venture. Always make sure your selling price covers your costs! We walked through each step, from calculating the initial cost to figuring out the bottle size and ultimately determining the profit (or in this case, the loss). I hope this breakdown was helpful and that you guys feel more confident tackling similar problems in the future! Remember, math is all about breaking things down into smaller, manageable steps. Keep practicing, and you'll become a pro in no time! This problem demonstrates a practical application of several mathematical concepts, such as the greatest common divisor, division, multiplication, and subtraction. These concepts are not only useful in business contexts but also in everyday life. Whether you’re calculating the cost of groceries, splitting a bill with friends, or managing your personal finances, the ability to perform these calculations accurately is essential. Moreover, this problem illustrates the importance of critical thinking and problem-solving skills. It requires us to analyze the information provided, identify the key steps needed to solve the problem, and execute those steps in a logical and systematic manner. These skills are valuable not only in mathematics but also in various other fields and aspects of life. So, by working through this problem, we’ve not only enhanced our mathematical abilities but also strengthened our critical thinking and problem-solving skills, which are crucial for success in any endeavor.