Solving 7 2/3 + 6 3/4 - 3 4/15 - 2 17/25 + 11/20 A Step-by-Step Guide

Jul 11, 2025 by ADMIN 70 views

Decoding the Mixed Fractions Puzzle: A Step-by-Step Solution to 7 2/3 + 6 3/4 - 3 4/15 - 2 17/25 + 11/20

Navigating the world of mixed fractions can sometimes feel like solving a complex puzzle. In this comprehensive guide, we'll break down the arithmetic expression 7 2/3 + 6 3/4 - 3 4/15 - 2 17/25 + 11/20 step by step. We will transform mixed fractions into improper fractions, find a common denominator, perform the necessary operations, and simplify the result. By the end of this explanation, you'll have a clear understanding of how to solve similar problems and a deeper appreciation for the elegance of fraction arithmetic.

1. Understanding the Challenge: Mixed Fractions and the Order of Operations

Before we dive into the solution, let's first understand the challenge. We're dealing with mixed fractions, which combine whole numbers and proper fractions (where the numerator is less than the denominator). To perform arithmetic operations like addition and subtraction, it's generally easier to work with improper fractions. We also need to adhere to the order of operations, which dictates the sequence in which operations are performed (in this case, we'll be working from left to right since we have only addition and subtraction).

When working with mixed fractions, it’s essential to grasp the fundamental principles that govern them. A mixed fraction, by definition, combines a whole number and a proper fraction, representing a quantity greater than one. The expression 7 2/3 is a mixed fraction, with 7 being the whole number and 2/3 being the proper fraction. To effectively manipulate mixed fractions in arithmetic operations, we often convert them into improper fractions. This conversion simplifies the process of finding common denominators and performing calculations. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us in what sequence to perform the operations. In our expression, we primarily deal with addition and subtraction, which are performed from left to right. This ensures that we tackle the problem systematically, reducing the risk of errors. Understanding these basics sets a strong foundation for solving the problem and similar arithmetic expressions involving fractions.

The first key step in simplifying this expression is to convert each mixed fraction into an improper fraction. An improper fraction is one where the numerator is greater than or equal to the denominator. This conversion makes it easier to find a common denominator and perform the necessary arithmetic operations. To convert a mixed fraction to an improper fraction, we multiply the whole number by the denominator of the fraction and add the numerator. This result becomes the new numerator, and the denominator remains the same. For instance, the mixed fraction 7 2/3 is converted to an improper fraction as follows: Multiply the whole number 7 by the denominator 3, which gives 21. Then, add the numerator 2 to get 23. So, the improper fraction is 23/3. Similarly, 6 3/4 becomes (6 * 4 + 3)/4 = 27/4. Applying this conversion to all mixed fractions in the expression is a critical step towards simplification. It eliminates the complexity of handling mixed numbers directly in addition and subtraction, paving the way for a more straightforward calculation process. This initial conversion is not just a mathematical technique; it’s a strategy for making the problem more accessible and less daunting.

2. Converting Mixed Fractions to Improper Fractions

The first step is to convert all mixed fractions into improper fractions:

7 2/3 = (7 * 3 + 2) / 3 = 23/3
6 3/4 = (6 * 4 + 3) / 4 = 27/4
3 4/15 = (3 * 15 + 4) / 15 = 49/15
2 17/25 = (2 * 25 + 17) / 25 = 67/25

Now our expression looks like this: 23/3 + 27/4 - 49/15 - 67/25 + 11/20

The conversion of mixed fractions to improper fractions is a critical step in simplifying complex arithmetic expressions. This process involves transforming each mixed fraction into a single fraction where the numerator is larger than the denominator. By converting to improper fractions, we eliminate the whole number component, making it easier to find a common denominator and perform addition and subtraction. Let's delve deeper into the mechanics and reasoning behind this conversion. Take the mixed fraction 7 2/3 as an example. To convert this, we multiply the whole number (7) by the denominator of the fraction (3), resulting in 21. We then add this product to the numerator (2), giving us 23. This new value becomes the numerator of the improper fraction, while the denominator remains the same. Thus, 7 2/3 converts to 23/3. This process effectively represents the mixed fraction as a single fraction, streamlining the subsequent calculations. Similarly, applying this method to 6 3/4 yields (6 * 4 + 3) / 4 = 27/4. For 3 4/15, we get (3 * 15 + 4) / 15 = 49/15, and for 2 17/25, it's (2 * 25 + 17) / 25 = 67/25. These conversions are not just about changing the form of the numbers; they fundamentally simplify the arithmetic process. By working with improper fractions, we create a unified format that allows us to apply standard fraction arithmetic rules more easily. This step is indispensable in tackling the complexities of mixed fraction arithmetic.

3. Finding the Least Common Denominator (LCD)

To add or subtract fractions, they must have a common denominator. To make the calculations easier, we find the least common denominator (LCD), which is the smallest number that is a multiple of all the denominators. Our denominators are 3, 4, 15, 25, and 20. To find the LCD, we can use prime factorization:

3 = 3
4 = 2 * 2
15 = 3 * 5
25 = 5 * 5
20 = 2 * 2 * 5

The LCD is the product of the highest powers of all prime factors present: 2^2 * 3 * 5^2 = 4 * 3 * 25 = 300

So, our LCD is 300.

The cornerstone of adding or subtracting fractions lies in the concept of a common denominator. Fractions can only be directly added or subtracted if they share the same denominator. This is because the denominator represents the number of equal parts into which a whole is divided, and to combine fractions, we need to ensure we are dealing with parts of the same size. The least common denominator (LCD) is the smallest number that is a multiple of all the denominators in the set of fractions we are working with. Using the LCD simplifies calculations and reduces the need for further simplification at the end. In our expression, the denominators are 3, 4, 15, 25, and 20. Finding the LCD is crucial for efficiently adding and subtracting these fractions. The prime factorization method is one of the most reliable ways to determine the LCD. We break down each denominator into its prime factors: 3 remains as 3, 4 is 2 * 2, 15 is 3 * 5, 25 is 5 * 5, and 20 is 2 * 2 * 5. The LCD is then constructed by taking the highest power of each prime factor present in the factorizations. In this case, we have 2^2 (from 4 and 20), 3 (from 3 and 15), and 5^2 (from 25). Multiplying these together, we get 2^2 * 3 * 5^2 = 4 * 3 * 25 = 300. Thus, 300 is the LCD for our fractions. This number serves as the common ground upon which we can rewrite each fraction, allowing us to perform the necessary arithmetic operations. The LCD is not just a number; it’s a key that unlocks the door to solving fraction addition and subtraction problems.

4. Rewriting Fractions with the LCD

Now we rewrite each fraction with the LCD of 300:

23/3 = (23 * 100) / (3 * 100) = 2300/300
27/4 = (27 * 75) / (4 * 75) = 2025/300
49/15 = (49 * 20) / (15 * 20) = 980/300
67/25 = (67 * 12) / (25 * 12) = 804/300
11/20 = (11 * 15) / (20 * 15) = 165/300

Our expression now looks like this: 2300/300 + 2025/300 - 980/300 - 804/300 + 165/300

The process of rewriting fractions with the least common denominator (LCD) is a crucial step in preparing them for addition and subtraction. This involves adjusting each fraction so that it has the LCD as its denominator, while maintaining its value. The fundamental principle here is that we multiply both the numerator and the denominator of each fraction by the same number, which is equivalent to multiplying the fraction by 1, thus preserving its value. This process transforms the fractions into equivalent forms that can be easily combined. In our problem, the LCD is 300. To rewrite each fraction, we determine what number we need to multiply the original denominator by to get 300, and then multiply both the numerator and denominator by that same number. For 23/3, we multiply both numerator and denominator by 100 (since 3 * 100 = 300), resulting in 2300/300. For 27/4, we multiply by 75 (since 4 * 75 = 300), yielding 2025/300. Similarly, 49/15 is multiplied by 20, giving us 980/300; 67/25 is multiplied by 12, resulting in 804/300; and 11/20 is multiplied by 15, which gives us 165/300. By rewriting each fraction in this manner, we ensure that they all have the same denominator, allowing us to perform addition and subtraction by simply adding or subtracting the numerators. This step is a testament to the power of equivalent fractions and their role in simplifying arithmetic operations.

5. Performing the Addition and Subtraction

Now that all fractions have the same denominator, we can perform the addition and subtraction from left to right:

(2300/300 + 2025/300) - 980/300 - 804/300 + 165/300

= 4325/300 - 980/300 - 804/300 + 165/300

= 3345/300 - 804/300 + 165/300

= 2541/300 + 165/300

= 2706/300

With all the fractions sharing a common denominator, the arithmetic operations of addition and subtraction become significantly simplified. The fundamental principle here is that when fractions have the same denominator, we can add or subtract them by simply adding or subtracting their numerators, while keeping the denominator constant. This process is straightforward and allows us to combine the fractions efficiently. In our expression, we now have 2300/300 + 2025/300 - 980/300 - 804/300 + 165/300. We perform the operations from left to right, starting with addition: 2300/300 + 2025/300 equals 4325/300. Next, we subtract 980/300 from 4325/300, which results in 3345/300. Then, we subtract 804/300 from 3345/300, giving us 2541/300. Finally, we add 165/300 to 2541/300, which equals 2706/300. This step-by-step approach ensures accuracy and clarity in the calculations. By performing the addition and subtraction in this methodical manner, we avoid potential errors and arrive at the simplified fraction 2706/300. This methodical approach highlights how crucial the common denominator is in simplifying the process of combining fractions.

6. Simplifying the Result

The final step is to simplify the fraction 2706/300. We can simplify this fraction by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. Both 2706 and 300 are divisible by 6:

2706 / 6 = 451

300 / 6 = 50

So, 2706/300 simplifies to 451/50

We can also express this as a mixed fraction: 451/50 = 9 1/50

The final step in solving any fraction arithmetic problem is to simplify the result. Simplification involves reducing the fraction to its lowest terms, making it easier to understand and work with. This is achieved by finding the greatest common divisor (GCD) of the numerator and denominator and then dividing both by this GCD. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In our case, we have the fraction 2706/300. To find the GCD of 2706 and 300, we can use methods such as listing factors or prime factorization. By inspection, we can see that both numbers are divisible by 6. Dividing 2706 by 6 gives us 451, and dividing 300 by 6 gives us 50. Thus, the simplified fraction is 451/50. This fraction is now in its lowest terms because 451 and 50 have no common factors other than 1. In addition to simplifying the fraction, it is often helpful to convert an improper fraction (where the numerator is greater than the denominator) back into a mixed fraction. To do this, we divide the numerator by the denominator. The quotient becomes the whole number part of the mixed fraction, and the remainder becomes the numerator of the fractional part, with the denominator remaining the same. Dividing 451 by 50 gives us a quotient of 9 and a remainder of 1. Therefore, 451/50 can be expressed as the mixed fraction 9 1/50. This final conversion provides a more intuitive understanding of the quantity, as it clearly shows the whole number part and the fractional part. Simplification and conversion to mixed fractions are crucial for presenting the answer in its most concise and understandable form.

7. Conclusion: The Answer and the Process

Therefore, 7 2/3 + 6 3/4 - 3 4/15 - 2 17/25 + 11/20 = 9 1/50

We've successfully navigated the mixed fractions puzzle! By converting mixed fractions to improper fractions, finding the LCD, rewriting fractions, performing the operations, and simplifying the result, we arrived at the answer: 9 1/50. This process demonstrates the importance of understanding the fundamental principles of fraction arithmetic and applying them systematically. Solving complex mathematical problems like this one not only sharpens our calculation skills but also enhances our problem-solving abilities in general.

In conclusion, the journey through this mathematical expression has not only provided us with the final answer but has also reinforced the importance of a structured approach in mathematics. We started with the mixed fraction expression 7 2/3 + 6 3/4 - 3 4/15 - 2 17/25 + 11/20, which initially seemed daunting. However, by breaking it down into manageable steps, we were able to tackle each component methodically. The first critical step was converting mixed fractions into improper fractions, which set the stage for easier manipulation. Following this, we identified the least common denominator (LCD) as 300, a pivotal step that allowed us to rewrite each fraction with a common base. Rewriting the fractions with the LCD enabled us to perform the addition and subtraction operations seamlessly, following the order of operations from left to right. Finally, we simplified the resulting fraction, 2706/300, to its lowest terms, 451/50, and then converted it back into a mixed fraction, 9 1/50. This final form provides a clear and concise representation of the solution. The process highlights the interconnectedness of mathematical concepts and the significance of each step in achieving the final answer. Mastering these steps not only builds confidence in fraction arithmetic but also nurtures a deeper understanding of mathematical principles. The elegance of this problem lies not just in the answer but in the systematic approach required to reach it. By understanding and applying these techniques, we are better equipped to tackle a wide range of mathematical challenges, fostering both accuracy and efficiency in our problem-solving endeavors.