Solving 32 - [15/20 + 12/20] A Step-by-Step Mathematical Guide
Introduction
In this comprehensive guide, we will delve into the world of mathematical expressions, specifically focusing on how to solve the equation 32 - [15/20 + 12/20]. This seemingly simple equation encompasses fundamental arithmetic operations, including fractions and brackets, which are essential building blocks for more advanced mathematical concepts. Our aim is to break down each step of the calculation, providing clear explanations and insights to ensure a thorough understanding. Whether you are a student looking to strengthen your math skills or simply someone interested in refreshing your knowledge, this guide will provide you with the tools and confidence to tackle similar problems with ease. We will explore the order of operations, the rules for adding fractions, and the importance of brackets in determining the correct answer. So, let's embark on this mathematical journey together and unravel the solution to this equation step by step.
Breaking Down the Equation
To effectively solve the equation 32 - [15/20 + 12/20], we need to dissect it into manageable parts and understand the role of each component. The equation consists of three main elements: a whole number (32), a subtraction operation (-), and an expression enclosed in brackets ([15/20 + 12/20]). The expression within the brackets involves the addition of two fractions, 15/20 and 12/20. The brackets play a crucial role in dictating the order of operations, indicating that the addition of the fractions must be performed before the subtraction from 32. This is in line with the universally accepted mathematical convention known as the order of operations, often remembered by the acronym PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding this hierarchy is paramount to arriving at the correct solution. By carefully examining each part of the equation, we can develop a strategic approach to solving it, ensuring that we follow the correct order of operations and minimize the chances of error. In the subsequent sections, we will delve deeper into the individual steps involved in solving this equation, starting with the addition of the fractions within the brackets.
Step 1: Adding Fractions Inside the Brackets
The first critical step in solving the equation 32 - [15/20 + 12/20] is to address the operation within the brackets. Specifically, we need to add the two fractions, 15/20 and 12/20. When adding fractions, a fundamental principle is that they must have a common denominator. Fortunately, in this case, both fractions already share the same denominator, which is 20. This simplifies the addition process significantly. To add fractions with a common denominator, we simply add their numerators (the numbers above the fraction line) and keep the denominator the same. Therefore, we have 15/20 + 12/20. Adding the numerators, 15 and 12, gives us 27. Thus, the sum of the two fractions is 27/20. This fraction represents the result of the operation within the brackets. It is an improper fraction, meaning that the numerator is greater than the denominator. While 27/20 is a valid answer, it can also be expressed as a mixed number, which consists of a whole number and a proper fraction (where the numerator is less than the denominator). Converting an improper fraction to a mixed number can sometimes make it easier to visualize and work with. In the next step, we will use this result, 27/20, to continue solving the equation, subtracting it from 32.
Step 2: Subtracting the Result from 32
Having successfully added the fractions within the brackets, we now move on to the next operation in the equation 32 - [15/20 + 12/20]. We have determined that 15/20 + 12/20 equals 27/20. Now, we need to subtract this result from 32. This step involves subtracting a fraction from a whole number. To perform this subtraction effectively, we need to express the whole number, 32, as a fraction with the same denominator as the fraction we are subtracting (27/20). This allows us to combine the two quantities easily. To convert 32 into a fraction with a denominator of 20, we multiply it by 20/20, which is equivalent to 1. This gives us 32 * (20/20) = 640/20. Now, we can rewrite the subtraction as 640/20 - 27/20. With both numbers expressed as fractions with the same denominator, we can subtract the numerators while keeping the denominator constant. This means we subtract 27 from 640, which gives us 613. Therefore, the result of the subtraction is 613/20. This fraction is an improper fraction, as the numerator (613) is larger than the denominator (20). To better understand the value of this fraction, it is often helpful to convert it into a mixed number. This will give us a whole number part and a fractional part, making it easier to interpret the magnitude of the result.
Step 3: Converting the Improper Fraction to a Mixed Number
In the previous step, we arrived at the improper fraction 613/20 as the result of subtracting 27/20 from 32. While 613/20 is a mathematically correct answer, it is often more intuitive and practical to express it as a mixed number. A mixed number consists of a whole number and a proper fraction, making it easier to visualize the quantity. To convert the improper fraction 613/20 into a mixed number, we need to determine how many whole times 20 goes into 613 and what remainder is left over. This is essentially a division problem. We divide 613 by 20. 20 goes into 613 a total of 30 times (30 * 20 = 600). This means that the whole number part of our mixed number is 30. Now, we need to find the remainder. We subtract the product of 30 and 20 (which is 600) from 613. The remainder is 613 - 600 = 13. This remainder becomes the numerator of the fractional part of our mixed number, and the denominator remains the same (20). Therefore, the mixed number equivalent of 613/20 is 30 and 13/20. This mixed number provides a clearer understanding of the value: it is 30 whole units plus 13/20 of another unit. This conversion is the final step in solving the equation, giving us a clear and easily interpretable answer.
Final Answer
After meticulously working through each step of the equation 32 - [15/20 + 12/20], we have arrived at the final answer. We began by simplifying the expression within the brackets, adding the fractions 15/20 and 12/20 to get 27/20. Next, we subtracted this result from 32, which required us to express 32 as a fraction with a denominator of 20, resulting in 640/20. Subtracting 27/20 from 640/20 gave us the improper fraction 613/20. Finally, to provide a more intuitive answer, we converted this improper fraction into a mixed number. By dividing 613 by 20, we found that 20 goes into 613 thirty times with a remainder of 13. This led us to the mixed number 30 and 13/20. Therefore, the final answer to the equation 32 - [15/20 + 12/20] is 30 and 13/20. This comprehensive solution demonstrates the importance of following the order of operations and the techniques for adding and subtracting fractions. By breaking down the problem into smaller, manageable steps, we have successfully solved the equation and gained a deeper understanding of the underlying mathematical principles.
Conclusion
In conclusion, solving the equation 32 - [15/20 + 12/20] has been a valuable exercise in reinforcing fundamental mathematical concepts. We have navigated the intricacies of the order of operations, fraction addition, and subtraction involving whole numbers and fractions. By methodically breaking down the equation into smaller, digestible steps, we were able to arrive at the solution of 30 and 13/20 with confidence. This process highlights the importance of a structured approach to problem-solving in mathematics. Each step, from adding the fractions within the brackets to converting the improper fraction to a mixed number, required a clear understanding of the relevant rules and principles. Furthermore, this exercise underscores the interconnectedness of various mathematical concepts. The ability to work with fractions, whole numbers, and mixed numbers seamlessly is crucial for success in more advanced mathematical pursuits. The skills honed in solving this equation, such as attention to detail, adherence to the order of operations, and the ability to manipulate fractions, are transferable to a wide range of mathematical problems. Ultimately, mastering these foundational skills not only enhances mathematical proficiency but also cultivates critical thinking and problem-solving abilities that are valuable in all aspects of life.
