Evaluating $\frac{7}{9}+\frac{1}{6} \div \frac{3}{7}$ In Simplest Form

Evaluating mathematical expressions often requires a solid understanding of the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this comprehensive guide, we will methodically break down the evaluation of the expression $\frac{7}{9} + \frac{1}{6} \div \frac{3}{7}$, ensuring we arrive at the answer in its simplest form. This involves not only applying the correct order of operations but also understanding fraction arithmetic, including division and addition. By the end of this discussion, you will have a clear understanding of each step involved and how to apply these principles to similar mathematical problems. We will delve deep into why each step is necessary and how it contributes to the overall solution. Fraction arithmetic, including division and addition, can sometimes seem daunting, but with a methodical approach and a clear understanding of the rules, it becomes much more manageable. So, let’s embark on this mathematical journey to simplify and evaluate the given expression.

Understanding the Order of Operations

Before we dive into the specifics of the problem, it’s crucial to understand the order of operations. The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed. The acronym PEMDAS serves as a handy reminder: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order ensures that mathematical expressions are evaluated consistently, leading to a unique and correct answer. Ignoring the order of operations can result in incorrect evaluations, highlighting the importance of adhering to this fundamental principle. In our expression, $\frac{7}{9} + \frac{1}{6} \div \frac{3}{7}$, we have addition and division. According to PEMDAS, division must be performed before addition. This means we first tackle the division of fractions, followed by the addition of the resulting fraction to another. Understanding this order is paramount to achieving the correct simplified answer. The order of operations isn't just a mathematical convention; it's a necessary framework that ensures clarity and consistency in mathematical calculations, allowing mathematicians and students alike to arrive at the same correct solutions. Each operation has its place in the hierarchy, and respecting this order is a cornerstone of accurate mathematical problem-solving. Now, with this principle firmly in mind, we can move on to the actual evaluation of our expression.

Step 1: Dividing Fractions

The first crucial step in evaluating $\frac{7}{9} + \frac{1}{6} \div \frac{3}{7}$ is to address the division of fractions. According to the order of operations (PEMDAS), division takes precedence over addition. Therefore, we must first evaluate $\frac{1}{6} \div \frac{3}{7}$. Dividing fractions involves a simple yet crucial rule: to divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. In this case, the reciprocal of $\frac{3}{7}$ is $\frac{7}{3}$. Thus, the division $\frac{1}{6} \div \frac{3}{7}$ becomes $\frac{1}{6} \times \frac{7}{3}$. Now we multiply the numerators together and the denominators together: $(1 \times 7) / (6 \times 3) = \frac{7}{18}$. This means that $\frac{1}{6}$ divided by $\frac{3}{7}$ equals $\frac{7}{18}$. Understanding this rule of reciprocals is vital in fraction arithmetic. It transforms a division problem into a multiplication problem, which is often easier to handle. This step is not only essential for this particular problem but is also a fundamental concept in mathematics. Mastering the division of fractions opens the door to solving more complex equations and problems involving rational numbers. By correctly executing this step, we pave the way for the next operation: adding the resulting fraction to the remaining term in the expression.

Step 2: Adding Fractions

Now that we have simplified the division part of the expression, we move on to the addition of fractions. Our expression now looks like this: $\frac{7}{9} + \frac{7}{18}$. To add fractions, they must have a common denominator. This means we need to find the least common multiple (LCM) of the denominators 9 and 18. The multiples of 9 are 9, 18, 27, and so on, while the multiples of 18 are 18, 36, and so on. The least common multiple of 9 and 18 is 18. So, we need to convert $\frac{7}{9}$ to an equivalent fraction with a denominator of 18. To do this, we multiply both the numerator and the denominator of $\frac{7}{9}$ by 2, which gives us $\frac{7 \times 2}{9 \times 2} = \frac{14}{18}$. Now our expression is $\frac{14}{18} + \frac{7}{18}$. Adding fractions with the same denominator involves adding the numerators and keeping the denominator the same. So, $\frac{14}{18} + \frac{7}{18} = \frac{14 + 7}{18} = \frac{21}{18}$. This result gives us a fraction that can potentially be simplified further. Finding the common denominator is a critical skill in fraction arithmetic. It allows us to combine fractions in a meaningful way, paving the path to a simplified and accurate final answer. Without a common denominator, the numerators cannot be directly added, emphasizing the importance of this step. Our next move is to simplify the resulting fraction to its lowest terms.

Step 3: Simplifying the Fraction

The final step in evaluating $\frac{7}{9} + \frac{1}{6} \div \frac{3}{7}$ is to simplify the fraction. We have arrived at the fraction $\frac{21}{18}$. To simplify a fraction, we look for the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. The factors of 21 are 1, 3, 7, and 21, while the factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common divisor of 21 and 18 is 3. Thus, we divide both the numerator and the denominator by 3: $\frac{21 \div 3}{18 \div 3} = \frac{7}{6}$. The fraction $\frac{7}{6}$ is now in its simplest form because 7 and 6 have no common factors other than 1. This simplified fraction represents our final answer to the given expression. Simplifying fractions is crucial in mathematics. It presents the answer in its most concise and understandable form. A fraction that is not simplified, while mathematically correct, can be seen as incomplete. Reducing to lowest terms is a standard practice and demonstrates a complete understanding of fraction manipulation. In this case, $\frac{7}{6}$ is an improper fraction, meaning the numerator is greater than the denominator. While it's mathematically accurate, it can also be expressed as a mixed number to further clarify its value.

Expressing the Answer as a Mixed Number (Optional)

Although $\frac{7}{6}$ is the simplest form of the fraction, we can also express it as a mixed number for a clearer understanding of its magnitude. To convert an improper fraction to a mixed number, we divide the numerator by the denominator. In this case, we divide 7 by 6. 7 divided by 6 gives us a quotient of 1 and a remainder of 1. The quotient becomes the whole number part of the mixed number, the remainder becomes the new numerator, and the denominator stays the same. So, $\frac{7}{6}$ as a mixed number is $1 \frac{1}{6}$. This means that $\frac{7}{6}$ is equal to one whole and one-sixth. Expressing a fraction as a mixed number can provide a more intuitive sense of its value, especially for those less familiar with fractions. It bridges the gap between fractions and whole numbers, making it easier to visualize and compare different quantities. While not always necessary, converting to a mixed number is a helpful step in some contexts, particularly when communicating the result to a broader audience. In our case, both $\frac{7}{6}$ and $1 \frac{1}{6}$ are correct and simplified answers, but the mixed number form may offer a clearer sense of the quantity.

Conclusion

In conclusion, we have successfully evaluated the expression $\frac{7}{9} + \frac{1}{6} \div \frac{3}{7}$ and arrived at the simplified answer of $\frac{7}{6}$, which can also be expressed as the mixed number $1 \frac{1}{6}$. This process involved several key steps, each crucial to achieving the correct result. We began by understanding the order of operations (PEMDAS), which dictated that we perform the division before the addition. We then tackled the division of fractions by multiplying by the reciprocal, followed by the addition of fractions, which required finding a common denominator. Finally, we simplified the resulting fraction to its lowest terms and optionally converted it to a mixed number. This exercise highlights the importance of a methodical approach to mathematical problems. By breaking down the problem into smaller, manageable steps and adhering to the rules of arithmetic, we can confidently solve complex expressions. Fraction arithmetic is a fundamental skill in mathematics, and mastering these operations is essential for further studies in algebra, calculus, and other advanced topics. The ability to accurately manipulate fractions is not only valuable in academic settings but also in everyday life, from cooking and baking to financial calculations. By understanding and applying these principles, you can tackle a wide range of mathematical challenges with greater confidence and precision.