Step-by-Step Guide To Evaluating $1 \frac{2}{3} \div 2 \frac{1}{3}$

Evaluate the expression $1 \frac{2}{3} \div 2 \frac{1}{3}$.

Introduction to Dividing Mixed Numbers

In the realm of mathematics, particularly in arithmetic, dividing mixed numbers is a fundamental skill. When you're faced with the problem of dividing one mixed number by another, such as the expression $1 \frac{2}{3} \div 2 \frac{1}{3}$, it may seem daunting at first. However, by following a systematic approach and understanding the underlying principles, this task becomes quite manageable. This article aims to provide a comprehensive guide on how to evaluate such divisions, ensuring clarity and precision in each step. To effectively divide mixed numbers, we must first transform them into improper fractions. This conversion is crucial because it allows us to apply the standard division rule for fractions, which involves multiplying by the reciprocal. The process entails multiplying the whole number part of the mixed number by the denominator of the fractional part, adding the numerator, and then placing this result over the original denominator. Once both mixed numbers are converted into improper fractions, we can proceed with the division operation. The division of fractions is performed by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by simply swapping its numerator and denominator. This seemingly simple step is the cornerstone of fraction division. The final step involves simplifying the resulting fraction, if possible. Simplification ensures that the answer is presented in its most concise and understandable form. This might involve reducing the fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD). In some cases, the final result might be an improper fraction, which can be converted back into a mixed number to provide a more intuitive representation of the quantity. By mastering these techniques, you'll be able to confidently tackle division problems involving mixed numbers.

Step-by-Step Conversion to Improper Fractions

To initiate the division of mixed numbers, the initial crucial step involves transforming each mixed number into its equivalent improper fraction form. This transformation is a cornerstone in simplifying the division process, making it more amenable to standard fraction arithmetic. Let's delve into this process with a detailed, step-by-step approach, using the mixed numbers from our original expression, $1 \frac{2}{3}$ and $2 \frac{1}{3}$, as illustrative examples.

Converting $1 \frac{2}{3}$ to an Improper Fraction

To convert the mixed number $1 \frac{2}{3}$ into an improper fraction, we follow a specific procedure. First, we multiply the whole number part, which is 1, by the denominator of the fractional part, which is 3. This yields the product 1 * 3 = 3. Next, we add the numerator of the fractional part, which is 2, to this product. So, we have 3 + 2 = 5. This sum becomes the new numerator of our improper fraction. The denominator remains the same as in the original mixed number, which is 3. Therefore, the improper fraction equivalent of $1 \frac{2}{3}$ is $\frac{5}{3}$. This conversion is critical because it allows us to express the quantity as a single fraction, simplifying subsequent calculations.

Converting $2 \frac{1}{3}$ to an Improper Fraction

Similarly, we apply the same method to convert the mixed number $2 \frac{1}{3}$ into an improper fraction. We begin by multiplying the whole number part, 2, by the denominator, 3, resulting in 2 * 3 = 6. Then, we add the numerator, 1, to this product, giving us 6 + 1 = 7. This sum becomes the new numerator of the improper fraction. The denominator remains 3, as it was in the original mixed number. Consequently, the improper fraction equivalent of $2 \frac{1}{3}$ is $\frac{7}{3}$. Now that both mixed numbers have been successfully converted into improper fractions, $\frac{5}{3}$ and $\frac{7}{3}$, we are well-prepared to proceed with the division operation. This conversion is a fundamental step in the process of dividing mixed numbers, as it transforms the problem into a more manageable form for calculation.

Performing the Division of Improper Fractions

With the mixed numbers $1 \frac{2}{3}$ and $2 \frac{1}{3}$ now transformed into their improper fraction equivalents, $\frac{5}{3}$ and $\frac{7}{3}$ respectively, we are poised to perform the division operation. The division of fractions follows a specific rule: to divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. This seemingly simple rule is the key to effectively dividing fractions.

Understanding the Reciprocal

The reciprocal of a fraction is obtained by simply swapping its numerator and denominator. For instance, the reciprocal of $\frac{7}{3}$ is $\frac{3}{7}$. This concept of reciprocals is crucial in the division process because it transforms a division problem into a multiplication problem, which is often easier to handle. By multiplying by the reciprocal, we are essentially finding out how many times the second fraction fits into the first fraction, which is the fundamental idea behind division.

Applying the Division Rule

Now, let's apply this rule to our problem. We have the division $\frac{5}{3} \div \frac{7}{3}$. To perform this division, we multiply $\frac{5}{3}$ by the reciprocal of $\frac{7}{3}$, which is $\frac{3}{7}$. This gives us the expression $\frac{5}{3} \times \frac{3}{7}$. To multiply fractions, we multiply the numerators together and the denominators together. So, we have (5 * 3) / (3 * 7), which equals $\frac{15}{21}$. This fraction represents the result of our division, but it is not yet in its simplest form. The next step is to simplify this fraction to its lowest terms.

Simplifying the Resulting Fraction

Simplifying a fraction involves reducing it to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). In the case of $\frac{15}{21}$, the GCD of 15 and 21 is 3. Dividing both the numerator and the denominator by 3, we get $\frac{15 \div 3}{21 \div 3}$, which simplifies to $\frac{5}{7}$. This is the simplified form of the fraction, and it represents the final result of the division. Therefore, $1 \frac{2}{3} \div 2 \frac{1}{3}$ equals $\frac{5}{7}$. This comprehensive approach ensures that the division is performed accurately and the result is presented in its most understandable form.

Simplification and Final Result

After performing the division of improper fractions, the subsequent and equally vital step involves simplifying the resulting fraction to its lowest terms. This simplification is crucial for presenting the answer in its most concise and understandable form. In our example, we arrived at the fraction $\frac{15}{21}$ after multiplying $\frac{5}{3}$ by the reciprocal of $\frac{7}{3}$. Now, we need to simplify this fraction.

Identifying the Greatest Common Divisor (GCD)

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. In the case of $\frac{15}{21}$, we need to find the GCD of 15 and 21. The factors of 15 are 1, 3, 5, and 15, while the factors of 21 are 1, 3, 7, and 21. The common factors of 15 and 21 are 1 and 3, and the greatest among them is 3. Therefore, the GCD of 15 and 21 is 3. This GCD will be used to simplify the fraction.

Dividing by the GCD

Once we have identified the GCD, we divide both the numerator and the denominator of the fraction by this value. In our case, we divide both 15 and 21 by 3. This gives us $\frac{15 \div 3}{21 \div 3}$, which simplifies to $\frac{5}{7}$. This fraction, $\frac{5}{7}$, is in its simplest form because 5 and 7 have no common factors other than 1. This means that the fraction cannot be reduced any further. The simplification process ensures that the fraction is represented in its most basic form, making it easier to interpret and use in further calculations.

The Final Result

Therefore, the final result of the division $1 \frac{2}{3} \div 2 \frac{1}{3}$ is $\frac{5}{7}$. This is a proper fraction, meaning that the numerator is less than the denominator. In some cases, the simplified fraction might be an improper fraction, where the numerator is greater than or equal to the denominator. In such cases, it is often preferable to convert the improper fraction back into a mixed number. However, in our example, the result is a proper fraction, so no further conversion is necessary. The final answer, $\frac{5}{7}$, represents the quotient of the original division problem, expressed in its simplest and most understandable form. This comprehensive simplification process is a crucial step in solving mathematical problems involving fractions.

Conclusion

In summary, evaluating the division of mixed numbers, such as $1 \frac{2}{3} \div 2 \frac{1}{3}$, involves a series of carefully executed steps. The process begins with converting the mixed numbers into improper fractions, which transforms the problem into a more manageable form. This conversion is a fundamental step, as it allows us to apply the standard rules of fraction division. Once the mixed numbers are converted, the division is performed by multiplying the first fraction by the reciprocal of the second fraction. This step is crucial, as it effectively changes the division problem into a multiplication problem, which is often easier to solve. The resulting fraction is then simplified to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). Simplification ensures that the final answer is presented in its most concise and understandable form. In our example, we found that $1 \frac{2}{3} \div 2 \frac{1}{3}$ equals $\frac{5}{7}$. This result is a proper fraction, meaning it is already in its simplest form and no further conversion is needed. Mastering the division of mixed numbers is a valuable skill in mathematics, applicable in various contexts. Whether you are working on arithmetic problems, algebraic equations, or real-world applications, the ability to confidently divide mixed numbers is essential. By understanding the underlying principles and practicing the steps involved, you can successfully tackle these types of problems with ease and accuracy. The key is to approach each step systematically, ensuring that the conversions, multiplications, and simplifications are performed correctly. With practice, dividing mixed numbers becomes a straightforward and routine task, enhancing your overall mathematical proficiency.