How To Calculate (18 * 5 - 12 * 5) / 0.3 A Step-by-Step Guide
In this article, we will walk through the process of solving the mathematical expression (18 * 5 - 12 * 5) / 0.3. This problem involves basic arithmetic operations such as multiplication, subtraction, and division. By breaking down the expression step-by-step, we can arrive at the correct answer and understand the order of operations.
Understanding the Order of Operations
Before we begin, it's important to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform mathematical operations:
- Parentheses: Operations inside parentheses are performed first.
- Exponents: Exponents are evaluated next.
- Multiplication and Division: These operations are performed from left to right.
- Addition and Subtraction: These operations are performed from left to right.
Following this order ensures that we solve the expression correctly.
Step 1: Perform the Multiplications
Our expression is (18 * 5 - 12 * 5) / 0.3. According to PEMDAS, we need to perform the multiplications first. Let's break it down:
- 18 * 5 = 90
- 12 * 5 = 60
Now our expression looks like this: (90 - 60) / 0.3.
In this crucial initial step, accurately performing the multiplications is paramount to arriving at the correct final answer. Multiplication, as a fundamental arithmetic operation, serves as the bedrock for numerous mathematical computations. Understanding and correctly executing multiplication is not just essential for this specific problem but also for a wide range of mathematical challenges. Let's delve deeper into why mastering multiplication is so vital.
Firstly, multiplication is the cornerstone of more advanced mathematical concepts. From algebraic equations to calculus problems, a solid grasp of multiplication is indispensable. When you encounter complex equations, the ability to quickly and accurately multiply numbers allows you to simplify and solve them efficiently. Without this foundational skill, progressing to higher levels of mathematics becomes significantly more challenging.
Secondly, multiplication is intricately linked to real-world applications. Whether you're calculating the total cost of items at a store, determining the area of a room, or estimating travel time, multiplication plays a crucial role. For instance, if you're buying five items that each cost $12, the total cost is easily determined by multiplying 5 by 12. Similarly, if you want to know the area of a rectangular room that measures 10 feet by 15 feet, you would multiply these two dimensions together.
Moreover, multiplication serves as the foundation for understanding other essential mathematical operations such as division and exponentiation. Division, essentially the inverse operation of multiplication, involves splitting a quantity into equal parts. Understanding multiplication helps in grasping the concept of division more intuitively. Exponentiation, on the other hand, is a shorthand way of expressing repeated multiplication. For example, 3 raised to the power of 4 (written as 3^4) means multiplying 3 by itself four times (3 * 3 * 3 * 3).
In addition to its practical and conceptual importance, mastering multiplication also boosts your overall numerical fluency. When you are proficient in multiplication, you can perform mental calculations more quickly and accurately. This numerical agility can be highly beneficial in various situations, from managing personal finances to making quick decisions in professional settings. For example, being able to mentally calculate discounts or estimate expenses can save you time and money.
Furthermore, accurate multiplication is the building block for more advanced mathematical problem-solving. Whether you are working with fractions, decimals, or percentages, multiplication is a key tool in your mathematical arsenal. For instance, to find 25% of 80, you would multiply 80 by 0.25. Similarly, when adding or subtracting fractions, you often need to find a common denominator, which involves multiplying the numerators and denominators of the fractions.
In conclusion, the importance of mastering multiplication cannot be overstated. It is not merely a procedural skill but a fundamental building block for mathematical proficiency. By ensuring that you can perform multiplication accurately and efficiently, you lay a solid foundation for tackling more complex mathematical challenges and applying mathematical concepts in real-world contexts. So, the next time you encounter a multiplication problem, remember that you are not just solving an equation; you are reinforcing a crucial skill that will serve you well in your mathematical journey.
Step 2: Perform the Subtraction
Next, we perform the subtraction inside the parentheses:
- 90 - 60 = 30
Now the expression simplifies to 30 / 0.3.
The importance of accurate subtraction in mathematical calculations cannot be overstressed. In this specific problem, subtracting 60 from 90 is a pivotal step that sets the stage for the final division. However, the significance of subtraction extends far beyond this single example. It is a fundamental arithmetic operation that underpins a vast array of mathematical concepts and real-world applications.
Subtraction, at its core, is the process of finding the difference between two numbers. It tells us how much remains when one quantity is taken away from another. This simple concept is the foundation for understanding more complex mathematical ideas such as negative numbers, algebraic equations, and calculus. A solid grasp of subtraction enables us to solve problems involving reduction, comparison, and allocation, making it an indispensable skill in both academic and practical contexts.
One of the primary reasons subtraction is so crucial is its pervasive presence in everyday life. From managing personal finances to making purchasing decisions, we constantly use subtraction to calculate the remaining amount, compare prices, and determine savings. For instance, when budgeting expenses, we subtract our spending from our income to determine how much money we have left. Similarly, when shopping, we might subtract a discount from the original price to find the sale price.
Subtraction also plays a vital role in various professional fields. In accounting and finance, it is used to calculate profits, losses, and net values. Engineers rely on subtraction to determine the difference between measurements and specifications, ensuring that designs meet the required standards. In project management, subtraction helps in tracking progress, estimating remaining tasks, and managing resources effectively. In scientific research, subtraction is used to analyze data, compare experimental results, and identify significant differences.
Furthermore, subtraction is intrinsically linked to other essential mathematical operations such as addition, multiplication, and division. It is the inverse operation of addition, meaning that subtraction can be used to undo addition and vice versa. This relationship is crucial in solving equations and simplifying expressions. For example, if we have the equation x + 5 = 10, we can subtract 5 from both sides to isolate x and find its value.
Subtraction is also essential for understanding negative numbers and the number line. Negative numbers represent values less than zero, and subtraction helps us to understand how they interact with positive numbers. By subtracting a larger number from a smaller number, we can move to the negative side of the number line. This concept is fundamental for understanding advanced topics such as algebra and calculus, where negative numbers are frequently used.
The ability to subtract accurately also enhances our problem-solving skills. When faced with a complex problem, we often need to break it down into smaller, more manageable parts. Subtraction allows us to quantify the differences between these parts, helping us to identify patterns, make comparisons, and draw conclusions. For example, in a statistical analysis, we might subtract the mean from individual data points to calculate deviations and assess the variability of the data.
In summary, accurate subtraction is a cornerstone of mathematical competence. Its importance extends beyond basic arithmetic to encompass a wide range of mathematical concepts and real-world applications. By mastering subtraction, we equip ourselves with a powerful tool for solving problems, making informed decisions, and navigating the complexities of the world around us. Whether it's balancing a checkbook, calculating a discount, or analyzing scientific data, the ability to subtract accurately is an invaluable asset. Therefore, dedicating time to practicing and perfecting subtraction skills is a worthwhile investment in our mathematical proficiency and overall problem-solving abilities.
Step 3: Perform the Division
Finally, we perform the division:
- 30 / 0.3 = 100
Therefore, the final answer is 100.
The final step in solving the expression (18 * 5 - 12 * 5) / 0.3 is performing the division. This operation not only provides the ultimate solution to the problem but also underscores the critical role of division in mathematics. Division, as a foundational arithmetic operation, allows us to understand how quantities are distributed, shared, or broken down into equal parts. Its mastery is essential for both academic success and practical application in daily life.
At its core, division is the inverse operation of multiplication. While multiplication combines quantities, division separates them. This fundamental relationship highlights the complementary nature of these operations and their mutual dependence in solving mathematical problems. For instance, if we know that 5 multiplied by 2 equals 10, then we also know that 10 divided by 5 equals 2.
The significance of division extends far beyond simple arithmetic. It is a critical component in various mathematical disciplines, including algebra, geometry, and calculus. In algebra, division is used to solve equations and simplify expressions. For example, if we have the equation 3x = 15, we divide both sides by 3 to find the value of x. In geometry, division helps calculate ratios, proportions, and scale factors. Determining the scale of a map or the proportions of a building requires division to ensure accurate representation. In calculus, division is used to find derivatives and integrals, which are essential concepts in understanding rates of change and accumulation.
Moreover, division is omnipresent in real-world scenarios. From managing personal finances to planning events, division enables us to allocate resources, calculate costs, and make informed decisions. For example, when splitting a restaurant bill among friends, we divide the total amount by the number of people. Similarly, when calculating the miles per gallon of a car, we divide the number of miles driven by the amount of fuel consumed.
In the business world, division is a crucial tool for financial analysis and operational management. Companies use division to calculate profit margins, return on investment, and market share. Retailers divide total sales by the number of transactions to determine the average transaction size. Manufacturers divide total production costs by the number of units produced to calculate the cost per unit. These calculations are essential for making strategic decisions and optimizing business performance.
In the realm of science and engineering, division is indispensable for data analysis and problem-solving. Scientists use division to calculate rates, ratios, and proportions in experiments. Engineers use division to determine stresses, strains, and efficiencies in structural designs. For example, when calculating the speed of an object, we divide the distance traveled by the time taken.
The mastery of division also fosters critical thinking and problem-solving skills. When faced with a complex problem, understanding division allows us to break it down into smaller, more manageable parts. This analytical approach is crucial for identifying patterns, making comparisons, and drawing conclusions. For example, in statistics, division is used to calculate averages, percentages, and probabilities, which are fundamental tools for analyzing data and making predictions.
Furthermore, division is closely linked to fractions and decimals, two essential concepts in mathematics. A fraction is essentially a division problem, where the numerator is divided by the denominator. Understanding division helps in comprehending the relationship between fractions and whole numbers. Similarly, decimals are a way of representing fractions, and division is used to convert fractions to decimals.
In conclusion, division is a cornerstone of mathematical proficiency, and its significance extends far beyond the classroom. It is an essential tool for solving problems, making informed decisions, and navigating the complexities of the world around us. By mastering division, we equip ourselves with a fundamental skill that enhances our mathematical abilities and empowers us to tackle a wide range of challenges. Therefore, investing time and effort in practicing and perfecting division skills is a valuable endeavor that will yield significant benefits in both our academic and professional lives.
Conclusion
By following the order of operations (PEMDAS) and breaking down the expression into smaller steps, we have successfully calculated the value of (18 * 5 - 12 * 5) / 0.3, which is 100. This exercise demonstrates the importance of understanding and applying the correct order of operations in mathematical calculations. The meticulous approach to solving this problem not only ensures an accurate answer but also reinforces the critical role of each arithmetic operation in mathematical problem-solving. Understanding the order of operations, encapsulated by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is not merely a procedural guideline; it is the foundational structure upon which mathematical accuracy is built.
This principle ensures that complex expressions are evaluated consistently and correctly, regardless of who is performing the calculation. The correct application of PEMDAS is vital in various fields, including engineering, finance, computer science, and even everyday calculations, as it provides a systematic approach to problem-solving. In the context of the given expression, (18 * 5 - 12 * 5) / 0.3, the adherence to PEMDAS was instrumental in achieving the accurate final result of 100. Let’s delve deeper into why each component of PEMDAS is crucial and how they collectively contribute to mathematical precision.
Firstly, the "P" in PEMDAS stands for Parentheses, which denote the highest priority in the order of operations. Parentheses group terms together, indicating that the operations within them must be performed before any other operations in the expression. This grouping is essential because it allows us to override the default order and specify which calculations should be done first. In the given expression, the terms (18 * 5 - 12 * 5) are enclosed in parentheses, signaling that the multiplication and subtraction within should be addressed before the division by 0.3. By solving the operations within the parentheses first, we maintain the logical flow of the calculation and ensure an accurate intermediate result.
Next, the "E" stands for Exponents, which are evaluated after parentheses. Exponents represent repeated multiplication and are a critical part of many mathematical and scientific calculations. Although exponents were not present in the specific expression we solved, understanding their place in the order of operations is vital for handling more complex problems. When an expression includes exponents, they must be evaluated before multiplication, division, addition, or subtraction, thus adhering to the structured sequence of PEMDAS.
The "MD" in PEMDAS represents Multiplication and Division. These operations are of equal priority and are performed from left to right. The decision to treat multiplication and division with the same precedence is based on their inverse relationship; division can be thought of as multiplication by the reciprocal. In the expression (18 * 5 - 12 * 5) / 0.3, the multiplication operations (18 * 5) and (12 * 5) were performed first within the parentheses. After calculating these products, the expression was simplified to (90 - 60) / 0.3. The final division step, 30 / 0.3, was then performed to arrive at the ultimate answer. The left-to-right rule for multiplication and division ensures that the calculations are performed in the correct sequence, further enhancing accuracy.
Lastly, the "AS" in PEMDAS stands for Addition and Subtraction, which are also of equal priority and are performed from left to right. Similar to multiplication and division, addition and subtraction are inverse operations, making their equal precedence logical. In our expression, after the multiplication operations were completed, the subtraction (90 - 60) was performed within the parentheses, resulting in 30. This step was crucial in simplifying the expression before the final division. By consistently applying the left-to-right rule for addition and subtraction, we avoid ambiguity and maintain the integrity of the calculation.
In conclusion, the successful calculation of (18 * 5 - 12 * 5) / 0.3 = 100 underscores the indispensable role of PEMDAS in mathematical problem-solving. By adhering to this structured approach, we ensure that complex expressions are evaluated accurately and consistently. Each component of PEMDAS—Parentheses, Exponents, Multiplication and Division, Addition and Subtraction—serves a specific purpose in the order of operations, and their collective application is essential for mathematical precision. Whether in academic pursuits, professional fields, or everyday calculations, the mastery of PEMDAS is a fundamental skill that empowers us to tackle a wide range of mathematical challenges with confidence and accuracy.
