Dividing Mixed Numbers: A Simple Guide

Let's tackle the problem of dividing mixed numbers: $4 \frac{2}{3} \div 3 \frac{1}{3}$. We'll simplify the answer and express it as a mixed number. This comprehensive guide will walk you through each step, ensuring you grasp the concept thoroughly. So, buckle up, math enthusiasts; let's dive in!

Converting Mixed Numbers to Improper Fractions

Before we can divide these mixed numbers, we need to convert them into improper fractions. Converting mixed numbers to improper fractions is a crucial initial step. It allows us to perform division more easily. So, let's break down the process step by step. For the first mixed number, $4 \frac{2}{3}$, we multiply the whole number (4) by the denominator (3), which gives us 12. Then, we add the numerator (2) to the result, giving us 14. We keep the same denominator (3). Thus, $4 \frac{2}{3}$ becomes $\frac{14}{3}$. See how we transformed a mixed number into a fraction? Now, let's apply the same technique to the second mixed number, $3 \frac{1}{3}$. Multiply the whole number (3) by the denominator (3), resulting in 9. Next, add the numerator (1) to get 10. Again, we retain the original denominator (3). Therefore, $3 \frac{1}{3}$ transforms into $\frac{10}{3}$. By converting both mixed numbers to improper fractions, we now have a division problem that looks like this: $\frac{14}{3} \div \frac{10}{3}$. This conversion is essential because it sets the stage for straightforward fraction division. Understanding this process is key to mastering operations with mixed numbers and fractions. Practice converting more mixed numbers to improper fractions to solidify your understanding. With a bit of practice, you'll be converting like a pro in no time! Remember, converting to improper fractions simplifies the division process and makes it much easier to handle. So, don't skip this important step! Now that we've successfully converted both mixed numbers into improper fractions, we're ready to proceed with the division. The next step involves understanding how to divide fractions, which we'll cover in the following section. Keep practicing, and you'll become a master of mixed number conversions!

Dividing Improper Fractions

Dividing fractions might sound intimidating, but it's actually quite simple! When dividing fractions, we use a trick: we multiply by the reciprocal. To divide $\frac{14}{3}$ by $\frac{10}{3}$, we multiply $\frac{14}{3}$ by the reciprocal of $\frac{10}{3}$. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of $\frac{10}{3}$ is $\frac{3}{10}$. Therefore, our division problem transforms into a multiplication problem: $\frac{14}{3} \times \frac{3}{10}$. This transformation is the key to dividing fractions. Remember, instead of dividing, you multiply by the reciprocal. This makes the calculation much easier and more straightforward. Now that we have a multiplication problem, we can proceed with multiplying the fractions. To multiply fractions, we multiply the numerators together and the denominators together. So, $\frac{14}{3} \times \frac{3}{10}$ becomes $\frac{14 \times 3}{3 \times 10}$, which simplifies to $\frac{42}{30}$. This fraction represents the result of our division, but it's not yet in its simplest form. We need to simplify it to get the final answer. Simplifying fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by the GCD. In the next section, we'll focus on simplifying this fraction to its simplest form. Keep in mind that dividing fractions is all about multiplying by the reciprocal. Once you understand this concept, you can confidently tackle any fraction division problem. So, practice multiplying by reciprocals, and you'll become a pro at dividing fractions in no time!

Simplifying the Fraction

Now that we have $\frac{42}{30}$, we need to simplify it. Simplifying fractions involves reducing them to their lowest terms. To simplify the fraction, we look for the greatest common divisor (GCD) of the numerator (42) and the denominator (30). The GCD is the largest number that divides both 42 and 30 without leaving a remainder. Let's list the factors of 42: 1, 2, 3, 6, 7, 14, 21, and 42. Now, let's list the factors of 30: 1, 2, 3, 5, 6, 10, 15, and 30. Comparing the two lists, we find that the greatest common divisor of 42 and 30 is 6. Now that we know the GCD is 6, we divide both the numerator and the denominator by 6. So, $\frac{42}{30}$ becomes $\frac{42 \div 6}{30 \div 6}$, which simplifies to $\frac{7}{5}$. This is the simplest form of the fraction. However, the original question asked us to write the answer as a mixed number. So, we need to convert the improper fraction $\frac{7}{5}$ into a mixed number. To convert an improper fraction to a mixed number, we divide the numerator (7) by the denominator (5). 7 divided by 5 is 1 with a remainder of 2. The quotient (1) becomes the whole number part of the mixed number, the remainder (2) becomes the numerator, and we keep the same denominator (5). Therefore, $\frac{7}{5}$ converts to $1 \frac{2}{5}$. This is our final answer in the form of a mixed number. Simplifying fractions is an essential skill in mathematics. It allows us to express fractions in their most concise form, making them easier to understand and work with. Remember to always look for the greatest common divisor and divide both the numerator and the denominator by it. With practice, you'll become proficient at simplifying fractions and converting between improper fractions and mixed numbers.

Writing as a Mixed Number

We've simplified our fraction to $\frac{7}{5}$. The final step is to express this as a mixed number. Converting an improper fraction to a mixed number involves dividing the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays the same. In our case, we divide 7 by 5. 7 divided by 5 is 1 with a remainder of 2. So, the whole number part is 1, the new numerator is 2, and the denominator remains 5. Therefore, $\frac{7}{5}$ is equal to $1 \frac{2}{5}$. This mixed number represents the simplified answer to our original division problem. Writing the answer as a mixed number is often preferred, especially when dealing with real-world quantities. It provides a clearer understanding of the amount we have. For example, $1 \frac{2}{5}$ is easier to visualize than $\frac{7}{5}$. Practice converting improper fractions to mixed numbers to improve your understanding and skills. Remember, the key is to divide the numerator by the denominator and use the quotient and remainder to form the mixed number. With a little practice, you'll be converting improper fractions to mixed numbers like a pro! Now that we've successfully converted the improper fraction to a mixed number, we have the final answer to our original division problem. This completes our step-by-step solution. Understanding how to convert between improper fractions and mixed numbers is a fundamental skill in mathematics and is essential for solving a wide range of problems.

Final Answer

Therefore, $4 \frac{2}{3} \div 3 \frac{1}{3} = 1 \frac{2}{5}$. This completes our step-by-step solution. We started by converting mixed numbers to improper fractions, then divided the fractions by multiplying by the reciprocal, simplified the resulting fraction, and finally, converted the simplified improper fraction back to a mixed number. Each step is crucial for solving the problem accurately. Remember to practice these steps to improve your skills in dividing mixed numbers. With consistent practice, you'll become more confident and proficient in solving similar problems. Understanding the underlying concepts and applying them systematically will help you succeed in mathematics. So, keep practicing, and you'll master the art of dividing mixed numbers in no time! This comprehensive guide has provided you with a solid foundation for tackling division problems involving mixed numbers. Now, go forth and conquer more mathematical challenges!