Solve The Combined Operations With Rational Numbers. A Store Has 220 Jars Of Milk, Each Sold At $2.60. In The Morning, 2/5 Of The Total Are Sold, And In The Afternoon, 3/4 Of The Remainder Are Sold. How Many Jars Are Sold In The Afternoon?
Introduction to Combined Operations with Rational Numbers
In the realm of mathematics, solving combined operations with rational numbers is a fundamental skill. Rational numbers, which encompass fractions, decimals, and integers, are the building blocks of many mathematical concepts. Mastering the art of combining operations such as addition, subtraction, multiplication, and division with these numbers is crucial for problem-solving in various real-world scenarios. This article delves into a practical problem involving combined operations with rational numbers, illustrating how these mathematical principles can be applied to everyday situations. We will explore the steps involved in solving the problem, emphasizing the importance of order of operations and the properties of rational numbers. By understanding these concepts, you will be better equipped to tackle similar problems and appreciate the versatility of mathematics in practical contexts.
When it comes to combined operations involving rational numbers, understanding the order of operations is paramount. This concept, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed to arrive at the correct answer. Failing to adhere to this order can lead to significant errors, especially when dealing with complex expressions. In our problem involving the sale of milk jars, we encounter a combination of multiplication, subtraction, and division, each representing different aspects of the transaction. By carefully applying the order of operations, we can systematically break down the problem into smaller, manageable steps, ultimately leading to the solution. This approach not only ensures accuracy but also promotes a deeper understanding of the underlying mathematical principles.
The properties of rational numbers play a crucial role in simplifying combined operations. These properties, such as the commutative, associative, and distributive laws, allow us to rearrange and manipulate expressions to make calculations easier. For example, the distributive property allows us to multiply a single term by a group of terms within parentheses, while the commutative and associative properties enable us to change the order and grouping of terms without affecting the result. In the context of our milk jar problem, these properties can be applied to simplify fractions, combine like terms, and ultimately arrive at the final answer more efficiently. By understanding and applying these properties, we can transform complex expressions into simpler forms, making the problem-solving process more manageable and intuitive. This not only enhances our computational skills but also fosters a deeper appreciation for the elegance and power of mathematical principles.
Problem Statement: The Milk Jar Sales Scenario
Our problem revolves around a hypothetical establishment that stocks and sells milk jars. Imagine a scenario where a store has 220 jars of milk, each priced at $2.60. The store's sales pattern is divided into two parts: the morning and the afternoon. During the morning, the store manages to sell 2/5 of the total stock. As the day progresses, in the afternoon, the store sells 3/4 of the remaining jars. The core question we aim to answer is: How many jars of milk are sold during the afternoon? This problem is a perfect example of how combined operations with rational numbers are used in real-life scenarios. It involves understanding fractions, calculating parts of a whole, and applying these concepts to determine the quantity of items sold. To solve this, we will need to carefully analyze each step, apply the correct mathematical operations, and interpret the results in the context of the problem.
This problem is more than just a mathematical exercise; it reflects the kind of calculations that businesses perform daily to manage inventory, track sales, and make informed decisions. Understanding how to solve such problems is valuable not only for students learning mathematics but also for anyone involved in business, finance, or retail. The ability to accurately calculate quantities, proportions, and remainders is essential for effective management and decision-making in these fields. By working through this problem, we gain insights into the practical applications of mathematics and its relevance to various aspects of our lives. This reinforces the importance of mathematical literacy and its role in empowering us to navigate real-world challenges.
The problem's complexity lies in its multi-step nature. To arrive at the final answer, we cannot simply apply a single calculation. Instead, we must break the problem down into smaller, more manageable steps. First, we need to determine the number of jars sold in the morning. This involves calculating a fraction of the total number of jars. Next, we need to calculate the number of jars remaining after the morning sales. This step requires subtracting the number of jars sold from the initial stock. Finally, we need to calculate the number of jars sold in the afternoon, which involves finding a fraction of the remaining jars. Each of these steps relies on the fundamental principles of arithmetic and the properties of rational numbers. By carefully executing each step and keeping track of the results, we can successfully solve the problem and gain a deeper understanding of the underlying mathematical concepts.
Step-by-Step Solution
To solve this problem, we will break it down into a series of steps, each involving a specific mathematical operation. This step-by-step approach will help us to understand the logic behind each calculation and ensure that we arrive at the correct answer. The key to success in this type of problem is to carefully follow the order of operations and to correctly interpret the information provided in the problem statement.
Step 1: Calculate the Number of Jars Sold in the Morning
The first step is to determine the number of milk jars sold during the morning. The problem states that the store sold 2/5 of the total stock of 220 jars. To find this number, we need to multiply the fraction 2/5 by the total number of jars, which is 220. This can be written as:
(2/5) * 220
To perform this multiplication, we can first multiply the numerator (2) by the whole number (220), and then divide the result by the denominator (5). This gives us:
(2 * 220) / 5 = 440 / 5 = 88
Therefore, the store sold 88 jars of milk in the morning. This calculation is a fundamental application of fractions and multiplication, demonstrating how we can find a part of a whole. Understanding this concept is crucial for solving many real-world problems involving proportions and quantities.
Step 2: Calculate the Number of Jars Remaining After the Morning Sales
After selling 88 jars in the morning, we need to determine how many jars are left in the store's inventory. To do this, we subtract the number of jars sold from the total initial stock. The initial stock was 220 jars, and we sold 88 jars, so the remaining number of jars is:
220 - 88 = 132
This means that there are 132 jars of milk remaining after the morning sales. This simple subtraction is a key step in the problem, as it sets the stage for the next calculation. The number of remaining jars is the basis for calculating the number of jars sold in the afternoon. This step highlights the importance of understanding how subtraction works in the context of real-world problems and how it helps us keep track of quantities.
Step 3: Calculate the Number of Jars Sold in the Afternoon
Now, we need to calculate the number of jars sold in the afternoon. The problem states that the store sold 3/4 of the remaining jars in the afternoon. We know from the previous step that there were 132 jars remaining. To find 3/4 of 132, we multiply the fraction 3/4 by 132:
(3/4) * 132
Similar to the first step, we can multiply the numerator (3) by the whole number (132) and then divide the result by the denominator (4). This gives us:
(3 * 132) / 4 = 396 / 4 = 99
Therefore, the store sold 99 jars of milk in the afternoon. This calculation again demonstrates the application of fractions and multiplication, but this time it is applied to a quantity that has already been reduced. This step reinforces the importance of carefully reading the problem statement and understanding which quantity is being referred to in each part of the problem.
Final Answer: Jars Sold in the Afternoon
After completing the step-by-step calculations, we arrive at the final answer to our problem. We have determined that the store sold 99 jars of milk during the afternoon. This answer is the result of a series of calculations involving fractions, multiplication, and subtraction, all applied in the correct order and with careful attention to the details of the problem. This final answer not only provides a numerical solution but also demonstrates the power of mathematical reasoning and problem-solving skills.
This problem illustrates how combined operations with rational numbers can be used to solve practical problems in everyday life. By breaking down the problem into smaller steps, we were able to systematically work through the calculations and arrive at the correct answer. This approach is applicable to a wide range of problems, from managing finances to planning events. The ability to apply mathematical concepts to real-world scenarios is a valuable skill that can help us make informed decisions and solve complex problems.
The process of solving this problem also highlights the importance of accuracy and attention to detail. Each step in the calculation depends on the results of the previous steps, so any errors made along the way can lead to an incorrect final answer. By carefully checking our work and ensuring that we have correctly applied the mathematical operations, we can increase our confidence in the accuracy of our results. This attention to detail is a valuable habit that can benefit us in many areas of life, not just in mathematics.
Conclusion: The Power of Combined Operations
In conclusion, solving combined operations with rational numbers is a critical skill that extends beyond the classroom and into real-world applications. The milk jar sales scenario provided a practical context for understanding how these mathematical principles work in tandem to solve everyday problems. By breaking down the problem into manageable steps, we successfully calculated the number of jars sold in the afternoon, demonstrating the power and versatility of combined operations. This exercise not only reinforces our understanding of fractions, multiplication, subtraction, and the order of operations but also highlights the importance of mathematical literacy in navigating real-life situations.
Furthermore, the problem-solving process emphasized the significance of careful reading, attention to detail, and systematic thinking. Each step in the solution built upon the previous one, underscoring the need for accuracy and precision in calculations. The ability to break down a complex problem into smaller, more manageable parts is a valuable skill that can be applied to various challenges in life, whether they involve mathematics, finance, or any other field. This problem served as a reminder that mathematics is not just an abstract subject but a powerful tool for understanding and interacting with the world around us.
The insights gained from this exercise can be applied to a wide range of scenarios, from managing personal finances to making informed business decisions. Understanding how to calculate proportions, percentages, and remainders is essential for effective budgeting, investing, and planning. Moreover, the ability to analyze data, identify patterns, and make predictions based on mathematical models is becoming increasingly valuable in today's data-driven world. By mastering the fundamentals of combined operations with rational numbers, we equip ourselves with the tools and skills necessary to succeed in a variety of endeavors. This reinforces the importance of continuous learning and the pursuit of mathematical knowledge as a means of empowering ourselves and contributing to society.