Solving (144/72 + 10/12) ÷ (1 1/3 - (2/3 × 3/4)) A Step-by-Step Guide

Jul 21, 2025 by ADMIN 70 views

Understanding and Solving the Mathematical Expression (144/72 + 10/12) ÷ (1 1/3 - (2/3 × 3/4))

This article delves into the step-by-step solution of the mathematical expression (144/72 + 10/12) ÷ (1 1/3 - (2/3 × 3/4)). We will break down each component of the expression, explaining the arithmetic operations involved and the order in which they must be performed to arrive at the correct answer. This comprehensive guide aims to enhance your understanding of fraction manipulation, mixed numbers, and the fundamental principles of arithmetic operations. By the end of this article, you'll be well-equipped to tackle similar mathematical problems with confidence and accuracy. Mastering these concepts is crucial for success in various areas of mathematics and real-world applications.

Breaking Down the Expression

To begin, let’s dissect the given expression: (144/72 + 10/12) ÷ (1 1/3 - (2/3 × 3/4)). This expression involves fractions, mixed numbers, division, multiplication, addition, and subtraction. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is paramount in solving this correctly. We must first address the operations within the parentheses, then handle multiplication and division, and finally, addition and subtraction.

Step 1: Simplify within the First Parenthesis (144/72 + 10/12)

The first parenthesis contains the expression 144/72 + 10/12. The initial step here is to simplify each fraction individually. 144/72 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 72. Thus, 144/72 simplifies to 2. Next, we simplify 10/12. The GCD of 10 and 12 is 2. Dividing both the numerator and the denominator by 2 gives us 5/6. Now, our expression within the first parenthesis becomes 2 + 5/6.

To add these two terms, we need a common denominator. Since 2 can be written as 2/1, we need to find a common denominator for 1 and 6, which is 6. Converting 2 to a fraction with a denominator of 6 gives us 12/6. So, the expression now looks like 12/6 + 5/6. Adding the numerators, we get 17/6. Therefore, the simplified form of the first parenthesis is 17/6. Understanding how to simplify fractions and find common denominators is a key skill in mathematical operations. This step is fundamental in ensuring that the subsequent calculations are accurate and streamlined. The simplification process not only makes the numbers easier to work with but also reduces the chances of errors in the later stages of the calculation.

Step 2: Simplify within the Second Parenthesis (1 1/3 - (2/3 × 3/4))

The second parenthesis contains the expression 1 1/3 - (2/3 × 3/4). Following the order of operations, we must first address the multiplication within the parenthesis. The expression 2/3 × 3/4 involves multiplying two fractions. To multiply fractions, we multiply the numerators together and the denominators together. So, 2/3 × 3/4 = (2 × 3) / (3 × 4) = 6/12. This fraction can be simplified by dividing both the numerator and the denominator by their GCD, which is 6. This gives us 1/2.

Now, our expression within the second parenthesis is 1 1/3 - 1/2. We need to convert the mixed number 1 1/3 into an improper fraction. To do this, we multiply the whole number (1) by the denominator (3) and add the numerator (1), then place the result over the original denominator. So, 1 1/3 = (1 × 3 + 1) / 3 = 4/3. Now the expression becomes 4/3 - 1/2. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 2 is 6. Converting 4/3 to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 2, giving us 8/6. Converting 1/2 to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 3, giving us 3/6.

The expression is now 8/6 - 3/6. Subtracting the numerators, we get 5/6. Thus, the simplified form of the second parenthesis is 5/6. The ability to convert mixed numbers into improper fractions and to find the least common multiple is essential for handling operations involving fractions and mixed numbers. This step-by-step breakdown ensures that even complex expressions can be managed effectively. Mastering these skills allows for more accurate and efficient problem-solving in mathematics.

Step 3: Perform the Division Operation (17/6) ÷ (5/6)

Now that we have simplified both parentheses, our expression is reduced to (17/6) ÷ (5/6). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 5/6 is 6/5. So, the expression becomes 17/6 × 6/5. To multiply these fractions, we multiply the numerators together and the denominators together: (17 × 6) / (6 × 5) = 102/30.

This fraction can be simplified by dividing both the numerator and the denominator by their GCD, which is 6. Dividing 102 by 6 gives us 17, and dividing 30 by 6 gives us 5. So, the simplified fraction is 17/5. This can be converted into a mixed number by dividing 17 by 5. The quotient is 3, and the remainder is 2. Therefore, 17/5 is equal to 3 2/5. Understanding the concept of reciprocals and how they apply to division of fractions is a fundamental aspect of arithmetic. The process of simplifying fractions after multiplication ensures that the answer is in its simplest form, making it easier to interpret and use in further calculations. This step highlights the importance of converting between improper fractions and mixed numbers to fully understand the magnitude of the result.

Final Answer

Therefore, the final answer to the mathematical expression (144/72 + 10/12) ÷ (1 1/3 - (2/3 × 3/4)) is 3 2/5 or 17/5 in improper fraction form. This comprehensive breakdown illustrates the importance of following the order of operations and the individual steps required to manipulate fractions and mixed numbers effectively. The meticulous process ensures accuracy and enhances the understanding of fundamental mathematical principles. The ability to solve complex expressions like this is a testament to a strong foundation in arithmetic and algebra.

Importance of Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is a fundamental principle in mathematics. It dictates the sequence in which arithmetic operations must be performed to ensure a correct solution. Neglecting this order can lead to drastically different and incorrect results. In the given expression, we strictly adhered to PEMDAS, first simplifying the expressions within the parentheses, then performing the multiplication and division, and finally, carrying out the addition and subtraction. This systematic approach is crucial for maintaining accuracy and avoiding errors.

PEMDAS Explained

Parentheses: Operations within parentheses (or brackets) are always performed first. This includes any arithmetic operations contained within the parentheses, such as addition, subtraction, multiplication, or division.
Exponents: After handling parentheses, any exponents (or powers) in the expression should be evaluated.
Multiplication and Division: These operations are performed from left to right. It’s important to note that multiplication and division have equal precedence, so the order in which they appear in the expression determines the sequence of their execution.
Addition and Subtraction: Similar to multiplication and division, addition and subtraction are performed from left to right. They also have equal precedence, so their order in the expression is crucial.

Consequences of Ignoring PEMDAS

Ignoring the order of operations can lead to significant errors in mathematical calculations. For instance, if we were to perform addition before multiplication in an expression, we would likely arrive at an incorrect answer. The order of operations ensures consistency and accuracy in mathematical problem-solving, making it an indispensable rule in arithmetic and algebra.

Real-World Applications

Understanding and being able to solve mathematical expressions involving fractions, mixed numbers, and the order of operations is not just an academic exercise; it has numerous real-world applications. These skills are essential in various fields, from everyday tasks to professional settings. Mastering these mathematical concepts provides a solid foundation for problem-solving and critical thinking in a wide range of situations.

Practical Examples

Cooking and Baking: Recipes often involve fractions and ratios. Knowing how to adjust ingredient quantities based on fractions is crucial for successful cooking and baking. For example, if a recipe calls for 2/3 cup of flour and you want to double the recipe, you need to be able to calculate 2/3 × 2.
Construction and Carpentry: Measurements in construction and carpentry frequently involve fractions and mixed numbers. Accurately calculating lengths, areas, and volumes requires a solid understanding of fraction manipulation and arithmetic operations. For instance, determining the amount of lumber needed for a project often involves adding and subtracting fractional measurements.
Financial Planning: Managing personal finances involves working with percentages, which are essentially fractions. Calculating interest rates, budgeting expenses, and understanding investments all require a strong grasp of mathematical concepts, including fractions and mixed numbers.
Engineering and Science: Many scientific and engineering calculations involve complex mathematical expressions with fractions, mixed numbers, and various operations. Accurate calculations are essential for designing structures, conducting experiments, and analyzing data.

Professional Settings

In professional settings, the ability to solve mathematical problems accurately and efficiently is highly valued. Whether it’s calculating costs in business, analyzing data in research, or designing systems in engineering, a strong foundation in mathematics is essential for success. Developing these skills enhances problem-solving capabilities and contributes to professional competence.

Practice Problems

To reinforce your understanding of solving mathematical expressions involving fractions and mixed numbers, it’s essential to practice. Solving various problems helps solidify your skills and builds confidence in your ability to tackle complex calculations. Here are a few practice problems to get you started:

Solve: (3/4 + 1/2) ÷ (2 1/3 - 1/6)
Evaluate: (5/8 × 4/5) + (1 1/2 ÷ 3/4)
Simplify: (2 3/4 - 1 1/8) × (2/3 + 1/4)

Tips for Practice

Break down complex problems: Divide the problem into smaller, manageable steps. This makes the overall calculation less daunting and reduces the chances of errors.
Show your work: Write down each step of your calculation. This helps you track your progress and identify any mistakes more easily.
Check your answers: After solving a problem, double-check your answer. Use estimation or alternative methods to verify the accuracy of your solution.
Seek feedback: If you’re unsure about your solution or the process you used, ask for help from a teacher, tutor, or online resources. Feedback is valuable for improving your understanding and skills.

Additional Resources

There are numerous resources available online and in libraries that can help you practice and improve your mathematical skills. Websites like Khan Academy, Coursera, and edX offer courses and tutorials on various mathematical topics. Additionally, textbooks, workbooks, and practice tests can provide further opportunities to hone your skills. Utilizing these resources can significantly enhance your learning and mastery of mathematical concepts.

By consistently practicing and applying the principles discussed in this article, you can develop a strong foundation in mathematics and confidently solve complex expressions involving fractions, mixed numbers, and the order of operations. This skill set is not only valuable in academic settings but also essential for success in various real-world applications and professional fields. The key is to approach each problem systematically, follow the order of operations, and continuously practice to reinforce your understanding. Remember, consistent effort and practice are the cornerstones of mathematical proficiency.