Multiply 6 By 3 5/9 Expressed As A Mixed Number

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In the realm of mathematics, mastering the multiplication of mixed numbers is a fundamental skill. This article delves into a detailed explanation of how to multiply a whole number by a mixed number, providing a step-by-step solution to the specific problem of multiplying 6 by 3 5/9. Whether you're a student seeking to enhance your understanding or an educator looking for a clear teaching resource, this guide will equip you with the knowledge and confidence to tackle such calculations effectively. Mixed numbers, which combine a whole number and a fraction, often appear in everyday situations, from cooking and baking to measuring and construction. Therefore, grasping the concept of multiplying mixed numbers is not only academically valuable but also practically useful in various real-world scenarios. Understanding the underlying principles and techniques involved in this process will empower you to solve more complex mathematical problems and apply these skills in diverse contexts. So, let's embark on this mathematical journey and unravel the intricacies of multiplying mixed numbers.

Understanding Mixed Numbers

Before we dive into the multiplication process, it's crucial to have a solid understanding of what mixed numbers are and how they work. A mixed number is a combination of a whole number and a proper fraction (a fraction where the numerator is less than the denominator). For example, 3 5/9 is a mixed number, where 3 is the whole number part and 5/9 is the fractional part. This mixed number represents 3 whole units plus 5/9 of another unit. To effectively multiply a mixed number, we first need to convert it into an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This conversion is necessary because it allows us to perform multiplication more easily. The process of converting a mixed number to an improper fraction involves multiplying the whole number by the denominator of the fraction, adding the numerator, and then placing the result over the original denominator. This transformation allows us to express the mixed number as a single fraction, which simplifies the multiplication process. By understanding the composition of mixed numbers and mastering the conversion to improper fractions, we lay a strong foundation for tackling multiplication problems involving these numbers. This foundational knowledge is essential for accurately and efficiently solving mathematical problems involving mixed numbers.

Converting Mixed Numbers to Improper Fractions

The first key step in multiplying a mixed number by a whole number is converting the mixed number into an improper fraction. Let's take our example of 3 5/9. To convert this mixed number, we follow these steps:

  1. Multiply the whole number (3) by the denominator of the fraction (9): 3 * 9 = 27
  2. Add the numerator of the fraction (5) to the result: 27 + 5 = 32
  3. Place the result (32) over the original denominator (9): 32/9

Therefore, the mixed number 3 5/9 is equivalent to the improper fraction 32/9. This conversion is crucial because it transforms the mixed number into a single fractional value, which is much easier to work with in multiplication. By converting to an improper fraction, we eliminate the need to deal with separate whole number and fractional parts during the multiplication process. This simplification is a fundamental technique in mixed number arithmetic, making it easier to perform calculations and arrive at accurate solutions. Mastering this conversion step is essential for efficiently multiplying mixed numbers and ensuring that the final answer is correct. The ability to seamlessly convert between mixed numbers and improper fractions is a cornerstone of understanding and working with these numbers effectively.

Multiplying the Whole Number by the Improper Fraction

Now that we've converted the mixed number 3 5/9 into the improper fraction 32/9, we can proceed with the multiplication. We need to multiply 6 (the whole number) by 32/9. To do this, we can express the whole number 6 as a fraction by writing it as 6/1. Multiplying fractions involves multiplying the numerators (the top numbers) and the denominators (the bottom numbers) separately. So, we have:

(6/1) * (32/9)

Multiply the numerators: 6 * 32 = 192 Multiply the denominators: 1 * 9 = 9

This gives us the improper fraction 192/9. This fraction represents the result of the multiplication, but it's an improper fraction, which means the numerator is larger than the denominator. While 192/9 is a correct answer, it's not in its simplest form. To express our answer in the most understandable way, we need to convert this improper fraction back into a mixed number. This conversion will give us a whole number part and a fractional part, making the result easier to interpret and use in practical situations. The next step is to simplify and convert this improper fraction to a mixed number.

Converting the Improper Fraction to a Mixed Number

We now have the improper fraction 192/9, and our goal is to convert it back into a mixed number. To do this, we divide the numerator (192) by the denominator (9). The quotient (the result of the division) will be the whole number part of the mixed number, and the remainder will be the numerator of the fractional part. The denominator of the fractional part will remain the same as the original denominator (9).

  1. Divide 192 by 9: 192 ÷ 9 = 21 with a remainder of 3
  2. The quotient, 21, becomes the whole number part of the mixed number.
  3. The remainder, 3, becomes the numerator of the fractional part.
  4. The denominator of the fractional part remains 9.

Therefore, the improper fraction 192/9 is equivalent to the mixed number 21 3/9. This mixed number represents the final result of our multiplication, but we can simplify it further. The fraction 3/9 can be reduced to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This simplification will give us the most concise and easily understandable representation of our answer. The ability to convert between improper fractions and mixed numbers is a vital skill in mathematics, allowing us to express results in the most appropriate and simplified form.

Simplifying the Fraction

Our result so far is the mixed number 21 3/9. However, the fraction 3/9 can be simplified. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by that GCD. The GCD of 3 and 9 is 3. Dividing both the numerator and the denominator by 3, we get:

3 ÷ 3 = 1 9 ÷ 3 = 3

So, the simplified fraction is 1/3. Now, we can rewrite our mixed number with the simplified fraction:

21 3/9 becomes 21 1/3

This is the final answer in its simplest form. Simplifying fractions is a crucial step in mathematics because it presents the result in the most concise and easily understandable way. A simplified fraction is easier to compare with other fractions and use in further calculations. The process of finding the greatest common divisor and dividing both the numerator and denominator by it ensures that the fraction is reduced to its lowest terms. In our example, simplifying 3/9 to 1/3 made the mixed number 21 1/3 the most elegant and practical representation of the solution. This step demonstrates the importance of not only arriving at a correct answer but also presenting it in its most simplified form.

Final Answer

Therefore, 6 multiplied by 3 5/9 is equal to 21 1/3. This final answer represents the product of the whole number and the mixed number, expressed in its simplest mixed number form. We arrived at this solution by following a series of steps: converting the mixed number to an improper fraction, multiplying the whole number by the improper fraction, converting the resulting improper fraction back to a mixed number, and finally, simplifying the fraction part of the mixed number. Each of these steps is essential to ensure the accuracy and clarity of the final result. Understanding and mastering these steps will enable you to confidently tackle similar problems involving the multiplication of mixed numbers. The ability to work with mixed numbers is a valuable skill in various mathematical contexts and real-world applications. This comprehensive solution demonstrates the importance of breaking down complex problems into manageable steps and applying the fundamental principles of arithmetic to arrive at the correct answer. The mixed number 21 1/3 is the most concise and understandable way to express the result of this multiplication.

Conclusion

In conclusion, multiplying a whole number by a mixed number involves a series of steps that, when followed carefully, lead to the correct solution. We began by understanding the concept of mixed numbers and the necessity of converting them into improper fractions for multiplication. Then, we performed the multiplication by treating the whole number as a fraction and multiplying the numerators and denominators. The resulting improper fraction was then converted back into a mixed number, and finally, the fraction part was simplified to its lowest terms. This step-by-step process not only provides the correct answer but also enhances understanding of the underlying mathematical principles. The example of multiplying 6 by 3 5/9, which resulted in 21 1/3, illustrates the importance of each step in the process. Mastering these steps will equip you with the skills to confidently solve similar problems involving mixed numbers and fractions. The ability to work with mixed numbers is a fundamental skill in mathematics, with applications in various fields, from basic arithmetic to more advanced mathematical concepts. By understanding and practicing these techniques, you can improve your mathematical proficiency and problem-solving abilities.

  • Converting mixed numbers to improper fractions is a fundamental skill.
  • Multiplying fractions involves multiplying the numerators and denominators.
  • Simplifying fractions is crucial for presenting the answer in its simplest form.