Adding Mixed Numbers Step By Step Solution For 8 11/12 + 9 5/18

In this comprehensive guide, we will delve into the process of adding mixed numbers, specifically focusing on the example of $8 \frac{11}{12} + 9 \frac{5}{18}$. Mixed numbers, which combine a whole number and a fraction, often present a challenge when it comes to addition. However, by breaking down the problem into manageable steps, we can easily arrive at the correct solution. This article will not only provide the solution but also offer a detailed explanation of the underlying concepts, ensuring a solid understanding of how to add mixed numbers. Whether you are a student learning the basics or someone looking to brush up on your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems.

Understanding Mixed Numbers

Before we dive into the addition process, it's crucial to understand what mixed numbers are and how they work. A mixed number is a combination of a whole number and a proper fraction. For instance, in the mixed number $8 \frac{11}{12}$, '8' represents the whole number part, and '\frac{11}{12}' represents the fractional part. The fractional part is a proper fraction because the numerator (11) is less than the denominator (12). Understanding this distinction is crucial as it dictates how we approach the addition process. Mixed numbers essentially represent a quantity that is greater than a whole number but not yet another whole number, making them a common and practical way to express measurements and quantities in everyday life.

When we add mixed numbers, we are essentially combining these quantities. To do this effectively, we need to consider both the whole number parts and the fractional parts. The process involves several key steps, including finding a common denominator for the fractions, adding the fractional parts, adding the whole number parts, and then simplifying the result if necessary. Each of these steps plays a critical role in arriving at the correct answer. Without a clear understanding of mixed numbers and their components, the addition process can seem daunting. However, by focusing on the individual parts and following a systematic approach, we can confidently add mixed numbers and solve related problems.

Therefore, the first step in tackling $8 \frac{11}{12} + 9 \frac{5}{18}$ is to recognize each number as a mixed number and understand its composition. This foundational understanding will pave the way for a smooth and accurate addition process, ensuring that we not only get the right answer but also grasp the underlying mathematical principles. With this in mind, let's proceed to the next step, which involves preparing the fractions for addition by finding a common denominator.

Finding a Common Denominator

The crucial step in adding mixed numbers like $8 \frac{11}{12} + 9 \frac{5}{18}$ is finding a common denominator for the fractional parts. The denominators of the fractions, 12 and 18, are different, meaning we cannot directly add the fractions \frac{11}{12} and \frac{5}{18}. A common denominator is a number that both denominators can divide into evenly. This allows us to express the fractions with the same 'size' of pieces, making addition straightforward.

There are several ways to find a common denominator, but the most efficient method is to determine the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators. For 12 and 18, we can list their multiples to find the LCM:

• Multiples of 12: 12, 24, 36, 48, 60, ...
• Multiples of 18: 18, 36, 54, 72, ...

From this list, we can see that the LCM of 12 and 18 is 36. Therefore, 36 will be our common denominator. This means we need to convert both fractions to equivalent fractions with a denominator of 36. To convert \frac{11}{12} to an equivalent fraction with a denominator of 36, we need to determine what number to multiply 12 by to get 36. Since 12 multiplied by 3 equals 36, we multiply both the numerator and the denominator of \frac{11}{12} by 3:

$\frac{11}{12} * \frac{3}{3} = \frac{33}{36}$

Similarly, to convert \frac{5}{18} to an equivalent fraction with a denominator of 36, we need to determine what number to multiply 18 by to get 36. Since 18 multiplied by 2 equals 36, we multiply both the numerator and the denominator of \frac{5}{18} by 2:

$\frac{5}{18} * \frac{2}{2} = \frac{10}{36}$

Now that we have converted both fractions to equivalent fractions with a common denominator, we can rewrite the original problem as:

$8 \frac{33}{36} + 9 \frac{10}{36}$

This transformation is a critical step in the process of adding mixed numbers. By finding a common denominator, we have prepared the fractions for addition, making the next steps significantly easier. With the fractions now in a compatible form, we can proceed to add the fractional parts and the whole number parts separately.

Adding the Fractional and Whole Number Parts

With a common denominator in place, we can proceed to add the fractional parts of the mixed numbers $8 \frac{33}{36} + 9 \frac{10}{36}$. This involves adding the numerators of the fractions while keeping the denominator the same. So, we add 33 and 10, which gives us 43. The fraction becomes \frac{43}{36}.

$\frac{33}{36} + \frac{10}{36} = \frac{33 + 10}{36} = \frac{43}{36}$

Next, we add the whole number parts of the mixed numbers. We have 8 and 9, which when added together, equal 17.

$8 + 9 = 17$

Now, we combine the results from adding the fractional parts and the whole number parts. We have 17 and \frac{43}{36}. This gives us a preliminary sum of $17 \frac{43}{36}$.

However, it's important to notice that the fraction \frac{43}{36} is an improper fraction because the numerator (43) is greater than the denominator (36). This means that \frac{43}{36} represents a value greater than 1. To simplify this, we need to convert the improper fraction into a mixed number. To do this, we divide 43 by 36. 36 goes into 43 once, with a remainder of 7. This means that \frac{43}{36} is equivalent to $1 \frac{7}{36}$.

Now we can rewrite $17 \frac{43}{36}$ as $17 + 1 \frac{7}{36}$. We then add the whole numbers together: 17 + 1 = 18. This leaves us with the simplified sum of $18 \frac{7}{36}$. This step of converting the improper fraction and adding it to the whole number is crucial for expressing the answer in its simplest form. It ensures that the fractional part of the mixed number is a proper fraction, making the answer more easily understood and interpreted.

Therefore, by adding the fractional and whole number parts separately and then simplifying the result, we have made significant progress in solving the problem. The next and final step involves checking if the fraction can be further simplified and presenting the final answer.

Simplifying the Result

After adding the fractional and whole number parts, we arrived at $18 \frac{7}{36}$. The final step in solving $8 \frac{11}{12} + 9 \frac{5}{18}$ is to simplify the result. This involves checking if the fractional part of the mixed number, in this case, \frac{7}{36}, can be further reduced. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. In other words, the fraction cannot be divided down any further.

To determine if \frac{7}{36} can be simplified, we need to find the factors of both the numerator (7) and the denominator (36). Factors are numbers that divide evenly into a given number.

• Factors of 7: 1, 7
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

By comparing the factors, we can see that the only common factor between 7 and 36 is 1. This means that \frac{7}{36} is already in its simplest form. There are no other numbers that can divide evenly into both 7 and 36, so we cannot reduce the fraction any further.

Since the fractional part is already simplified, our final answer is $18 \frac{7}{36}$. This is the simplest form of the sum of $8 \frac{11}{12}$ and $9 \frac{5}{18}$. This final check is essential to ensure that the answer is presented in the most concise and understandable way. Simplification not only makes the answer mathematically correct but also makes it easier to interpret and use in practical applications.

In summary, we have successfully added the mixed numbers by finding a common denominator, adding the fractional and whole number parts separately, converting improper fractions if necessary, and finally, simplifying the result. This systematic approach ensures accuracy and clarity in solving addition problems involving mixed numbers.

Therefore, the sum of $8 \frac{11}{12}$ and $9 \frac{5}{18}$ is $18 \frac{7}{36}$.

Conclusion

In conclusion, adding mixed numbers like $8 \frac{11}{12} + 9 \frac{5}{18}$ involves a series of steps that, when followed carefully, lead to the correct solution. We began by understanding the concept of mixed numbers, which are a combination of a whole number and a proper fraction. This foundational knowledge is crucial for grasping the process of addition. The next critical step was finding a common denominator for the fractional parts, which allows us to add the fractions accurately. We determined the Least Common Multiple (LCM) of the denominators and converted the fractions to equivalent fractions with the common denominator.

After preparing the fractions, we proceeded to add the fractional parts and the whole number parts separately. This involves adding the numerators of the fractions while keeping the denominator the same, and then adding the whole numbers. We then combined the results to get a preliminary sum. However, we encountered an improper fraction in our sum, which required us to convert it into a mixed number. This conversion is a crucial step in simplifying the result and expressing it in its most understandable form.

Finally, we simplified the resulting mixed number by checking if the fractional part could be further reduced. We identified the factors of both the numerator and the denominator and determined that they had no common factors other than 1, meaning the fraction was already in its simplest form. This final check ensures that the answer is presented in its most concise and mathematically correct form.

By following this systematic approach, we successfully added $8 \frac{11}{12}$ and $9 \frac{5}{18}$ and arrived at the final answer of $18 \frac{7}{36}$. This process not only provides the solution but also reinforces the underlying mathematical principles involved in adding mixed numbers. Whether you are a student learning this concept or someone looking to review your skills, this guide offers a comprehensive understanding of how to tackle similar problems with confidence. The key takeaways are the importance of finding a common denominator, adding fractional and whole number parts separately, simplifying improper fractions, and ensuring the final answer is in its simplest form. With practice, these steps become second nature, making adding mixed numbers a manageable and even enjoyable mathematical task.