Dividing Fractions How To Solve 7/12 Divided By 2/9

In the realm of mathematics, fractions play a fundamental role, appearing in various calculations and real-world applications. One essential operation involving fractions is division, which might seem daunting at first, but with a clear understanding of the underlying principles, it becomes quite manageable. This comprehensive guide will walk you through the process of dividing fractions, specifically focusing on the example of ${\frac{7}{12}}$ ÷ ${\frac{2}{9}}$, and ensuring that the answer is expressed in its simplest form. So, whether you're a student grappling with homework or an adult brushing up on your math skills, this article will provide you with the knowledge and confidence to tackle fraction division with ease.

Understanding the Basics of Fraction Division

Before diving into the specifics of dividing ${\frac{7}{12}}$ by ${\frac{2}{9}}$, it’s crucial to grasp the core concept of fraction division. At its heart, dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction is simply obtained by swapping its numerator (the top number) and its denominator (the bottom number). For instance, the reciprocal of ${\frac{2}{3}}$ is ${\frac{3}{2}}$. This seemingly simple concept is the key that unlocks the door to effortless fraction division. To truly understand this, let’s consider why this works. Division, in essence, asks the question, "How many times does this number fit into that number?" When we divide by a fraction, we are asking how many times that fractional part fits into the whole. Multiplying by the reciprocal achieves this by effectively reversing the fraction, allowing us to determine how many of the fractional parts are contained within the number we are dividing. This method transforms a potentially complex division problem into a straightforward multiplication problem, making it much easier to solve. This fundamental understanding is the cornerstone of confidently dividing any fraction by another.

Why does this work? Think of dividing by a fraction as asking how many times that fraction fits into the number you're dividing. For example, dividing 1 by ${\frac{1}{2}}$ is asking how many halves are in 1, which is 2. This is the same as multiplying 1 by the reciprocal of ${\frac{1}{2}}$, which is 2. This principle applies to all fraction divisions. Mastering this concept lays a solid foundation for tackling more complex mathematical problems involving fractions, ratios, and proportions. It's not just about memorizing a rule; it's about understanding the underlying logic that makes the rule work. This deeper understanding empowers you to apply the concept in various contexts and solve problems with greater confidence and accuracy. So, take the time to truly internalize this principle, and you'll find that fraction division becomes a much less daunting task.

Step-by-Step Guide to Dividing ${\frac{7}{12}}$ by ${\frac{2}{9}}$

Now, let’s apply this principle to the specific problem at hand: ${\frac{7}{12}}$ ÷ ${\frac{2}{9}}$. We'll break down the process into clear, manageable steps:

Step 1: Find the Reciprocal of the Second Fraction

The first step is to identify the second fraction, which in this case is ${\frac{2}{9}}$, and determine its reciprocal. As we discussed earlier, the reciprocal is found by swapping the numerator and the denominator. So, the reciprocal of ${\frac{2}{9}}$ is ${\frac{9}{2}}$. This seemingly simple step is the cornerstone of our strategy. By flipping the second fraction, we transform the division problem into a multiplication problem, which is generally easier to handle. It's crucial to accurately identify the reciprocal, as any mistake at this stage will propagate through the rest of the calculation, leading to an incorrect answer. Double-check your work to ensure you've correctly swapped the numerator and denominator before proceeding to the next step. This attention to detail will help you maintain accuracy and build confidence in your problem-solving abilities.

Step 2: Change the Division to Multiplication

With the reciprocal in hand, the next step is to transform the division problem into a multiplication problem. This is where the magic happens! We replace the division symbol (÷) with a multiplication symbol (×). So, our original problem, ${\frac{7}{12}}$ ÷ ${\frac{2}{9}}$, now becomes ${\frac{7}{12}}$ × ${\frac{9}{2}}$. This transformation is not just a symbolic change; it reflects the fundamental principle of fraction division we discussed earlier. By multiplying by the reciprocal, we are essentially asking the same question as the division problem but in a way that is mathematically easier to solve. This step highlights the elegance and efficiency of the reciprocal method. It simplifies the process and makes it more intuitive. Make sure you clearly replace the division sign with the multiplication sign to avoid confusion in subsequent steps. This simple change sets the stage for the multiplication process, which is the next key step in solving the problem.

Step 3: Multiply the Fractions

Now, we can proceed with the multiplication of the fractions. To multiply fractions, we multiply the numerators together and the denominators together. So, we have:

• Numerator: 7 × 9 = 63
• Denominator: 12 × 2 = 24

This gives us the fraction ${\frac{63}{24}}$. This step is a straightforward application of the rules of fraction multiplication. It's important to remember that you multiply straight across – numerator times numerator, and denominator times denominator. There's no need to find a common denominator when multiplying fractions, which makes this step relatively simple. However, it's crucial to perform the multiplication accurately to ensure the correct result. Double-check your calculations to avoid any arithmetic errors. Once you've multiplied the numerators and denominators, you'll have a new fraction that represents the result of the multiplication. But, our journey doesn't end here. The next step is to simplify this fraction to its simplest form.

Step 4: Simplify the Fraction

The final step is to simplify the fraction ${\frac{63}{24}}$ to its simplest form. This means finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. The GCD of 63 and 24 is 3. So, we divide both the numerator and the denominator by 3:

• 63 ÷ 3 = 21
• 24 ÷ 3 = 8

This gives us the simplified fraction ${\frac{21}{8}}$. Simplifying fractions is a crucial step in expressing the answer in its most concise and understandable form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. Finding the GCD can sometimes be a bit challenging, but there are various methods you can use, such as listing the factors of each number or using the prime factorization method. Once you've found the GCD, dividing both the numerator and denominator by it will reduce the fraction to its simplest form. This step not only makes the answer more elegant but also makes it easier to work with in future calculations. In this case, ${\frac{21}{8}}$ is an improper fraction (numerator is greater than the denominator), so we can also express it as a mixed number for better understanding.

Step 5: Convert to a Mixed Number (Optional)

Since ${\frac{21}{8}}$ is an improper fraction (the numerator is greater than the denominator), we can convert it to a mixed number. To do this, we divide 21 by 8. The quotient is 2, and the remainder is 5. So, the mixed number is 2 ${\frac{5}{8}}$. Converting an improper fraction to a mixed number can provide a more intuitive understanding of the quantity. A mixed number combines a whole number and a proper fraction, making it easier to visualize the value. To convert, you divide the numerator by the denominator. The quotient becomes the whole number part of the mixed number, the remainder becomes the numerator of the fractional part, and the denominator remains the same. This optional step adds clarity to the answer and is often preferred, especially in real-world applications where mixed numbers can be more easily interpreted. In our example, 2 ${\frac{5}{8}}$ clearly shows that the result is two whole units and five-eighths of another unit.

The Final Answer

Therefore, ${\frac{7}{12}}$ ÷ ${\frac{2}{9}}$ = ${\frac{21}{8}}$ or 2 ${\frac{5}{8}}$ in simplest form. Congratulations! You've successfully navigated the process of dividing fractions and expressing the answer in its simplest form. This final answer represents the solution to our original problem. Whether you choose to express it as an improper fraction or a mixed number depends on the context and the preference of your instructor or the requirements of the problem. Both ${\frac{21}{8}}$ and 2 ${\frac{5}{8}}$ are mathematically equivalent and represent the same quantity. The key takeaway is that you've mastered the steps involved in dividing fractions and simplifying the result. This skill will serve you well in various mathematical contexts and real-world applications. So, take pride in your accomplishment and continue to practice and refine your understanding of fractions and other mathematical concepts.

Practice Problems to Sharpen Your Skills

To solidify your understanding of dividing fractions, try these practice problems:

1. ${\frac{3}{4}}$ ÷ ${\frac{1}{2}}$
2. ${\frac{5}{8}}$ ÷ ${\frac{3}{4}}$
3. ${\frac{9}{10}}$ ÷ ${\frac{2}{5}}$

Working through these practice problems will reinforce your understanding of the steps involved in dividing fractions and simplifying the results. Each problem presents a slightly different scenario, allowing you to apply the concepts in various contexts. Make sure to follow the step-by-step guide we've discussed: find the reciprocal of the second fraction, change the division to multiplication, multiply the fractions, and simplify the result. Don't be afraid to make mistakes; they are valuable learning opportunities. If you get stuck, revisit the explanations and examples in this article. With consistent practice, you'll develop confidence and fluency in dividing fractions. Consider these problems as a stepping stone to mastering more complex mathematical concepts that rely on a solid understanding of fractions.

Conclusion

Dividing fractions might seem tricky at first, but by understanding the concept of reciprocals and following the steps outlined above, it becomes a straightforward process. Remember to always simplify your answer to its simplest form. With practice, you'll master this essential math skill! Mastering fraction division is a significant achievement in your mathematical journey. It's a skill that not only helps you in academic settings but also in everyday life, from cooking and baking to measuring and problem-solving. The key is to remember the fundamental principle of multiplying by the reciprocal and to consistently apply the steps we've discussed. Don't be discouraged by challenges; view them as opportunities to learn and grow. The more you practice, the more confident and proficient you'll become. Embrace the power of fractions and their applications, and you'll find that they are an invaluable tool in your mathematical arsenal. So, keep practicing, keep exploring, and keep expanding your mathematical horizons!