Solving (1 1/2) + (6 1/3 - 4.75) × 2 2/5 A Step-by-Step Guide

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In this comprehensive guide, we will delve into the intricacies of solving a complex mixed number arithmetic problem. Our focus will be on the expression: (1 1/2) + (6 1/3 - 4.75) × 2 2/5. This problem encompasses various mathematical operations, including addition, subtraction, multiplication, and the manipulation of mixed numbers and decimals. By systematically breaking down each step, we will ensure a clear understanding of the underlying principles and techniques involved. This exploration will not only enhance your arithmetic skills but also provide a solid foundation for tackling more advanced mathematical challenges. This problem highlights the necessity of understanding the order of operations, also known as PEMDAS/BODMAS, which dictates the sequence in which mathematical operations must be performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). The ability to convert between mixed numbers, fractions, and decimals is also crucial for solving this problem efficiently and accurately. This article aims to provide a step-by-step solution, reinforcing these fundamental mathematical concepts and empowering you to confidently handle similar problems in the future. Whether you are a student looking to improve your grades, a professional seeking to brush up on your arithmetic skills, or simply someone who enjoys solving mathematical puzzles, this guide will offer valuable insights and practical techniques.

1. Converting Mixed Numbers to Improper Fractions

To effectively tackle the given arithmetic problem, converting mixed numbers to improper fractions is a crucial initial step. This conversion allows us to perform arithmetic operations more seamlessly, particularly multiplication and division. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). To convert a mixed number to an improper fraction, we follow a simple procedure: multiply the whole number by the denominator of the fractional part, add the numerator, and then place the result over the original denominator. This process transforms the mixed number into a single fraction, where the numerator can be greater than the denominator. For instance, let’s consider the mixed number 1 1/2. To convert it to an improper fraction, we multiply the whole number (1) by the denominator (2), which gives us 2. We then add the numerator (1), resulting in 3. Finally, we place this sum over the original denominator (2), yielding the improper fraction 3/2. This conversion simplifies the arithmetic operations, especially when dealing with multiplication or division. Another example is the mixed number 2 2/5. Multiplying the whole number (2) by the denominator (5) gives us 10. Adding the numerator (2) results in 12. Placing this over the denominator (5) gives us the improper fraction 12/5. This methodical conversion ensures that we are working with fractions that are easier to manipulate in subsequent calculations. Similarly, the mixed number 6 1/3 can be converted into an improper fraction. Multiply 6 by 3 to get 18, then add 1 to get 19. Place this over the original denominator 3, resulting in the improper fraction 19/3. Converting mixed numbers to improper fractions not only simplifies calculations but also reduces the risk of errors. By working with a single fraction instead of a combination of a whole number and a fraction, we create a more uniform and manageable mathematical expression. This step is fundamental to solving complex arithmetic problems involving mixed numbers, setting the stage for accurate and efficient computation.

2. Converting Decimals to Fractions

Converting decimals to fractions is another essential skill for solving our problem. Decimals and fractions are two different ways of representing non-whole numbers, and being able to convert between them allows for greater flexibility in calculations. A decimal is a number written in base-10 notation, where the digits after the decimal point represent fractions with denominators that are powers of 10 (e.g., tenths, hundredths, thousandths). To convert a decimal to a fraction, we identify the place value of the last digit, write the decimal as a fraction with the appropriate power of 10 as the denominator, and then simplify the fraction if possible. For example, the decimal 4.75 can be converted to a fraction by recognizing that the last digit (5) is in the hundredths place. We can write 4.75 as 475/100. This fraction can then be simplified by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 475 and 100 is 25. Dividing both the numerator and the denominator by 25, we get 19/4. This fraction is in its simplest form. Another example to illustrate this process is the decimal 0.25. Since the last digit (5) is in the hundredths place, we can write 0.25 as 25/100. The GCD of 25 and 100 is 25. Dividing both the numerator and the denominator by 25 simplifies the fraction to 1/4. In the context of our original problem, converting the decimal 4.75 to the fraction 19/4 allows us to work with fractions consistently, which can simplify the arithmetic operations. Similarly, if we had a decimal like 0.125, we would write it as 125/1000. The GCD of 125 and 1000 is 125. Dividing both by 125 gives us the simplified fraction 1/8. The ability to convert decimals to fractions is not only crucial for this specific problem but also for a wide range of mathematical applications. It enhances our understanding of number representation and provides us with the tools to manipulate numbers in different forms, making complex calculations more manageable. By mastering this conversion, we can approach mathematical problems with greater confidence and precision, ensuring accurate solutions.

3. Applying the Order of Operations (PEMDAS/BODMAS)

Applying the order of operations, often remembered by the acronyms PEMDAS or BODMAS, is a fundamental principle in mathematics. This principle dictates the sequence in which mathematical operations should be performed to ensure accurate and consistent results. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS, a common variant, stands for Brackets, Orders (exponents), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Both acronyms convey the same order of operations. Parentheses or brackets are always addressed first, as they group operations that should be performed before others. This ensures that complex expressions within the parentheses are simplified before they are used in further calculations. Next, exponents or orders (such as squares and cubes) are evaluated. After exponents, multiplication and division are performed from left to right. This means that if a multiplication operation appears before a division operation, it is performed first, and vice versa. This left-to-right rule is crucial for accuracy. Finally, addition and subtraction are carried out, also from left to right. This ensures that these operations are performed in the correct sequence. In the context of our problem, (1 1/2) + (6 1/3 - 4.75) × 2 2/5, we must first address the operations within the parentheses. This means subtracting 4.75 from 6 1/3 before proceeding with other operations. Once the parentheses are resolved, we then perform multiplication before addition. This involves multiplying the result of the subtraction by 2 2/5. Finally, we add the result of the multiplication to 1 1/2. If we were to disregard the order of operations, we would likely arrive at an incorrect answer. For example, if we added 1 1/2 to 6 1/3 first and then subtracted 4.75, we would be violating the established rules of arithmetic. PEMDAS/BODMAS provides a systematic approach to solving mathematical problems, ensuring that we follow the correct sequence of operations. This not only helps us arrive at the correct answer but also promotes consistency and clarity in our calculations. Mastering the order of operations is essential for success in mathematics, as it underpins accurate problem-solving in various mathematical contexts. By adhering to this principle, we can confidently tackle complex arithmetic expressions and ensure reliable results.

4. Solving the Expression Step-by-Step

To effectively solve the expression (1 1/2) + (6 1/3 - 4.75) × 2 2/5 step-by-step, we will apply the principles and techniques discussed in the previous sections. This involves converting mixed numbers and decimals to fractions, adhering to the order of operations (PEMDAS/BODMAS), and performing each operation methodically. Let's begin by converting the mixed numbers to improper fractions and the decimal to a fraction:

  • 1 1/2 = (1 × 2 + 1) / 2 = 3/2
  • 6 1/3 = (6 × 3 + 1) / 3 = 19/3
  • 2 2/5 = (2 × 5 + 2) / 5 = 12/5
  • 4.75 = 475/100 = 19/4 (simplified)

Now we can rewrite the expression using these fractions:

(3/2) + (19/3 - 19/4) × (12/5)

Next, we follow the order of operations and address the parentheses first. This involves subtracting 19/4 from 19/3. To do this, we need a common denominator, which is the least common multiple (LCM) of 3 and 4, which is 12:

19/3 - 19/4 = (19 × 4) / (3 × 4) - (19 × 3) / (4 × 3) = 76/12 - 57/12 = 19/12

Now our expression looks like this:

(3/2) + (19/12) × (12/5)

According to PEMDAS/BODMAS, we perform multiplication before addition. So, we multiply 19/12 by 12/5:

(19/12) × (12/5) = (19 × 12) / (12 × 5) = 228/60

We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 12:

228/60 = (228 ÷ 12) / (60 ÷ 12) = 19/5

Now our expression is:

(3/2) + (19/5)

To add these fractions, we need a common denominator, which is the LCM of 2 and 5, which is 10:

(3/2) + (19/5) = (3 × 5) / (2 × 5) + (19 × 2) / (5 × 2) = 15/10 + 38/10 = 53/10

Finally, we can convert the improper fraction 53/10 back to a mixed number:

53/10 = 5 3/10

Therefore, the solution to the expression (1 1/2) + (6 1/3 - 4.75) × 2 2/5 is 5 3/10. This step-by-step approach ensures accuracy and clarity, breaking down a complex problem into manageable parts.

5. Verification and Alternative Methods

After solving a complex arithmetic problem, it is crucial to verify the solution and explore alternative methods to ensure accuracy and deepen understanding. Verification involves checking the final answer using a different approach or re-evaluating the steps taken. Alternative methods can provide a fresh perspective and confirm the correctness of the solution. One way to verify our solution of 5 3/10 is to use a calculator. By inputting the original expression (1 1/2) + (6 1/3 - 4.75) × 2 2/5 into a calculator that supports mixed numbers and decimals, we can directly compare the result with our calculated answer. If the calculator output matches 5 3/10 (or its decimal equivalent, 5.3), this confirms our solution. Another method of verification involves working backwards through the steps. We start with the final answer, 5 3/10, and reverse the operations to see if we arrive back at the original expression. This process can highlight any errors in our calculations. For instance, we can subtract 1 1/2 from 5 3/10, then divide the result by 2 2/5, and finally check if we obtain the value of (6 1/3 - 4.75). Exploring alternative methods can also provide a deeper understanding of the problem. Instead of converting all numbers to fractions, we could choose to convert all numbers to decimals and perform the operations using decimal arithmetic. This approach would involve converting 1 1/2 to 1.5, 6 1/3 to approximately 6.333, and 2 2/5 to 2.4. The expression would then become 1.5 + (6.333 - 4.75) × 2.4. Performing the calculations in this format and comparing the result with our previous answer serves as an excellent verification method. Furthermore, we can break down the problem into smaller parts and solve each part independently before combining the results. For example, we could calculate (6 1/3 - 4.75) separately and then multiply it by 2 2/5 before adding 1 1/2. This approach can simplify the problem and make it easier to identify potential errors. Verification and the use of alternative methods are integral parts of problem-solving in mathematics. They not only ensure the accuracy of the solution but also enhance our mathematical reasoning and problem-solving skills. By employing these techniques, we can approach complex problems with greater confidence and precision.

Conclusion

In conclusion, the process of solving the complex arithmetic expression (1 1/2) + (6 1/3 - 4.75) × 2 2/5 has provided a comprehensive exploration of fundamental mathematical principles and techniques. This journey involved converting mixed numbers to improper fractions, converting decimals to fractions, adhering to the order of operations (PEMDAS/BODMAS), performing step-by-step calculations, and verifying the solution through alternative methods. Each step highlighted the importance of precision and methodical problem-solving. The conversion of mixed numbers and decimals to fractions allowed us to work with a consistent mathematical representation, simplifying the arithmetic operations. Applying the order of operations ensured that we performed the calculations in the correct sequence, avoiding potential errors and arriving at the accurate solution. The step-by-step approach broke down the complex expression into manageable parts, making the problem more accessible and less daunting. Verification, through the use of calculators and alternative methods, provided a crucial check on our solution, confirming its accuracy and deepening our understanding of the problem. The final answer, 5 3/10, represents the culmination of these efforts and demonstrates the power of applying mathematical principles systematically. This exercise underscores the importance of mastering basic arithmetic skills, as they form the foundation for more advanced mathematical concepts. The ability to confidently manipulate fractions, decimals, and mixed numbers, along with a thorough understanding of the order of operations, is essential for success in mathematics and related fields. Furthermore, the problem-solving strategies employed in this context, such as breaking down complex problems into smaller parts and verifying solutions, are valuable skills that can be applied in various areas of life. By engaging with problems like this, we not only enhance our mathematical abilities but also develop critical thinking and analytical skills that are essential for lifelong learning and problem-solving. This exploration serves as a testament to the beauty and practicality of mathematics, showcasing its ability to provide clear and concise solutions to complex problems.