Calculate Pretzel Price Per Pound Solving Division Problems

In this article, we'll break down a common mathematical problem encountered in everyday life: calculating the unit price of an item. Specifically, we'll be addressing the question: Yen paid $10.50 for 2.5 pounds of pretzels. What price did she pay for each pound of pretzels? We'll not only solve the problem but also delve into the step-by-step process of division, ensuring a clear understanding of the underlying concepts. Furthermore, we will analyze the subsequent step in the long division process to enhance your comprehension of mathematical operations.

Understanding the Problem: Price per Pound

The core of this problem lies in determining the price of a single pound of pretzels. We know the total cost ($10.50) and the total weight (2.5 pounds). To find the price per pound, we need to divide the total cost by the total weight. This concept is crucial in everyday shopping, allowing us to compare prices and make informed decisions. Understanding price per pound also allows for effective budgeting and cost management. This principle is applicable across various scenarios, from grocery shopping to evaluating bulk purchases. It enables consumers to accurately assess the true cost of goods and services, leading to smarter spending habits. In essence, mastering the calculation of price per pound empowers individuals to make financially sound choices in their daily lives.

The Division Process: Unveiling the Solution

To find the price per pound, we perform the division: $10.50 ÷ 2.5. However, dividing by a decimal can be tricky. To simplify the process, we can multiply both the dividend ($10.50) and the divisor (2.5) by 10. This shifts the decimal point one place to the right, transforming the problem into 105 ÷ 25. This transformation maintains the ratio and makes the division easier to manage. Multiplying both numbers by the same factor does not change the result of the division, which is a fundamental principle of mathematics. This technique is widely used in various calculations to avoid dealing with decimals directly, making the process less prone to errors. By converting the problem to whole numbers, we streamline the long division process and enhance our ability to arrive at an accurate solution efficiently.

Now, we proceed with long division. We set up the problem with 105 as the dividend and 25 as the divisor. First, we see how many times 25 goes into 10. Since 25 is larger than 10, it goes in 0 times. Next, we consider 105. How many times does 25 go into 105? It goes in 4 times (4 x 25 = 100). So, we write '4' above the '5' in 105. This step involves estimating how many times the divisor fits into a portion of the dividend, which is a crucial skill in long division. Accurate estimation speeds up the process and reduces the likelihood of errors. This is an iterative process where we continually assess the quotient and adjust it as needed until we have fully divided the dividend. The ability to correctly estimate the quotient is a testament to one's understanding of multiplication and division relationships.

After multiplying 25 by 4 and placing the result (100) under 105, the next crucial step in the long division process is B. subtract 100 from 105.

The Next Step: Subtraction is Key

Subtracting 100 from 105 gives us a remainder of 5. This remainder represents the portion of the dividend that hasn't been fully divided yet. The subtraction step is a cornerstone of long division, as it allows us to determine the remaining amount to be divided in subsequent steps. Without this subtraction, we cannot accurately track the progress of the division and may end up with an incorrect quotient. This step essentially isolates the portion of the dividend that needs further processing, ensuring that we account for every part of the original number. The remainder obtained from this subtraction plays a vital role in the subsequent steps of the long division process.

Once we have the remainder, we bring down the next digit (which, in this case, we'll need to add a '0' since we're dealing with decimals). This process of bringing down the next digit allows us to continue the division process with a larger number. It's a systematic way of incorporating the remaining portion of the dividend into the subsequent division steps. Bringing down the next digit essentially extends the dividend, allowing for a more precise calculation of the quotient. This step ensures that we consider all parts of the dividend and arrive at a complete and accurate answer. The ability to correctly bring down the next digit is crucial for maintaining the flow and accuracy of long division.

We then repeat the process: how many times does 25 go into 50? It goes in 2 times (2 x 25 = 50). We write '2' next to the '4' in our quotient. Multiplying the divisor by this new digit and subtracting the result from the current remainder helps us refine our quotient further. This iterative process is the heart of long division, where we progressively narrow down the quotient until we reach a point where the remainder is zero or sufficiently small. This iterative refinement ensures that we obtain the most accurate quotient possible.

Completing the Calculation: The Final Answer

Subtracting 50 from 50 leaves us with a remainder of 0. This signifies that the division is complete. The quotient we have, 4.2, represents the price per pound in terms of '105 ÷ 25'. Now, we need to correctly place the decimal point. Since we multiplied both the dividend and divisor by 10 earlier, the decimal point was effectively shifted one place to the right in both numbers. However, the quotient 4.2 obtained from 105 ÷ 25 directly represents the solution to the original problem, 10.50 ÷ 2.5. Therefore, the final answer is $4.20 per pound. This final step demonstrates the importance of understanding the effect of prior transformations on the result, ensuring that the answer is correctly interpreted in the context of the original problem.

Therefore, Yen paid $4.20 for each pound of pretzels. This price per pound calculation allows consumers to easily compare the cost-effectiveness of different products and make informed purchasing decisions. Understanding unit pricing, like price per pound, is a fundamental skill for managing personal finances and making savvy choices in the marketplace. By mastering this skill, individuals can effectively assess value, optimize their spending, and achieve their financial goals.

Why Other Options are Incorrect

Let's briefly discuss why the other options are incorrect:

• A. Multiply 25 by 4: While we did multiply 25 by 4 during the process, this was to determine how many times 25 goes into 105. It's not the next step after performing the multiplication.
• C. Bring down the next number: We bring down the next number after subtracting. Subtraction is the immediate next step.

Key Takeaways: Mastering Long Division

This problem highlights the importance of understanding the steps involved in long division. It's not just about getting the right answer; it's about understanding the process that leads to the solution. Mastering long division is crucial for various mathematical concepts and real-world applications. This foundational skill is essential for understanding more complex mathematical operations, such as algebra and calculus. Moreover, long division is a valuable tool in practical scenarios, including budgeting, financial analysis, and even everyday tasks like dividing recipes. By developing a strong understanding of long division, individuals can confidently tackle a wide range of mathematical challenges in both academic and real-world settings.

By carefully following each step – dividing, multiplying, subtracting, and bringing down – you can confidently solve similar problems. Remember, practice makes perfect! The more you practice long division, the more fluent and accurate you'll become. Regular practice will not only solidify your understanding of the process but also enhance your ability to perform calculations quickly and efficiently. Consistency is key to mastering any mathematical skill, and long division is no exception. Dedicate time to practicing various long division problems, and you'll find your confidence and proficiency growing steadily.

This exercise demonstrates how basic mathematical skills, like division, are used in everyday situations. Understanding these concepts allows us to make informed decisions and solve problems effectively. By mastering the fundamentals of mathematics, we equip ourselves with valuable tools for navigating the complexities of the world around us. Mathematical literacy is essential for informed citizenship, enabling individuals to critically analyze data, make sound judgments, and participate fully in society. Moreover, a strong foundation in mathematics opens doors to a wide range of career opportunities in fields such as science, technology, engineering, and mathematics (STEM), as well as finance, business, and many others. Embracing mathematical learning is an investment in one's future and a key to unlocking countless opportunities.