Subtracting Mixed Numbers Step-by-Step Solution For 6 15/12 - 3 3/9

Before diving into the subtraction of mixed numbers, it's crucial to understand what mixed numbers are and how they are structured. Mixed numbers combine a whole number and a proper fraction, such as $6 \frac{15}{12}$ or $3 \frac{3}{9}$. The whole number represents the integer part, while the fraction represents a part of a whole. In the mixed number $6 \frac{15}{12}$, 6 is the whole number, and $\frac{15}{12}$ is the fractional part. Similarly, in $3 \frac{3}{9}$, 3 is the whole number, and $\frac{3}{9}$ is the fraction. A proper understanding of mixed numbers is the foundational step in performing arithmetic operations, especially subtraction. Mixed numbers allow for a more precise representation of quantities that lie between whole numbers, making them essential in various practical applications, from cooking and measurements to finance and engineering. The ability to seamlessly work with mixed numbers is not just a mathematical skill but also a valuable asset in everyday life. Therefore, mastering the basics of mixed numbers is paramount before moving on to more complex operations like subtraction.

Properly identifying the whole and fractional parts is the first step. Next, one must understand that fractions in mixed numbers represent parts of a whole and that these fractions can be manipulated using the rules of fraction arithmetic. It's also essential to recognize that mixed numbers can be converted into improper fractions, which can sometimes simplify calculations. This conversion involves multiplying the whole number by the denominator of the fraction and adding the numerator, all while keeping the same denominator. For example, $6 \frac{15}{12}$ can be converted to an improper fraction by multiplying 6 by 12 (which gives 72), adding 15 (which gives 87), and then placing the result over the original denominator, resulting in $\frac{87}{12}$. Similarly, $3 \frac{3}{9}$ can be converted to $\frac{30}{9}$. Understanding these conversions is critical, as they allow for flexibility in approaching subtraction problems. Whether you choose to work with mixed numbers directly or convert them to improper fractions, the goal is to find the most efficient and accurate method for solving the problem at hand. Ultimately, the goal is to develop a comprehensive understanding that allows for confident manipulation and problem-solving.

When dealing with fractions, especially in mixed numbers, simplifying fractions is a crucial step to make calculations easier. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. This process is also known as reducing the fraction to its lowest terms. Consider the fraction $\frac{15}{12}$. Both 15 and 12 are divisible by 3. Dividing both the numerator and the denominator by 3 gives us $\frac{5}{4}$. This simplified fraction is equivalent to $\frac{15}{12}$ but is easier to work with in calculations. Similarly, in the mixed number $3 \frac{3}{9}$, the fraction $\frac{3}{9}$ can be simplified. Both 3 and 9 are divisible by 3. Dividing both by 3, we get $\frac{1}{3}$. Thus, $3 \frac{3}{9}$ can be simplified to $3 \frac{1}{3}$. Simplifying fractions not only makes the numbers smaller and more manageable but also reduces the chances of making errors in subsequent calculations. It is a fundamental skill in fraction arithmetic and plays a significant role in solving complex problems involving fractions and mixed numbers. Understanding how to identify common factors and reduce fractions to their simplest form is essential for mastering fraction-based mathematics.

The process of simplifying fractions involves finding 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. Once the GCD is found, both the numerator and the denominator are divided by the GCD to obtain the simplified fraction. For example, in the fraction $\frac{15}{12}$, the GCD of 15 and 12 is 3. Dividing both by 3 results in $\frac{5}{4}$. Another method is to successively divide by common factors until no more common factors exist. For instance, in $\frac{3}{9}$, both numbers are divisible by 3, leading to $\frac{1}{3}$. Simplifying fractions is not merely a computational technique; it is a way to express fractions in their most basic and understandable form. This clarity is invaluable when comparing fractions, adding or subtracting them, or using them in real-world applications. Simplified fractions provide a clear representation of the proportion or ratio being described, making mathematical concepts more intuitive and accessible. In the context of mixed numbers, simplifying the fractional part before performing any operations can significantly streamline the problem-solving process and enhance accuracy.

To subtract mixed numbers effectively, it's often helpful to convert mixed numbers to improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Converting mixed numbers to improper fractions makes it easier to perform arithmetic operations like subtraction. For example, let's convert $6 \frac{15}{12}$ to an improper fraction. To do this, we multiply the whole number (6) by the denominator (12) and add the numerator (15). This gives us $(6 imes 12) + 15 = 72 + 15 = 87$. We then place this result over the original denominator, giving us $\frac{87}{12}$. Similarly, to convert $3 \frac{3}{9}$ to an improper fraction, we multiply the whole number (3) by the denominator (9) and add the numerator (3). This gives us $(3 imes 9) + 3 = 27 + 3 = 30$. We then place this result over the original denominator, giving us $\frac{30}{9}$. Converting mixed numbers to improper fractions eliminates the need to handle whole numbers and fractions separately, streamlining the subtraction process. This technique is particularly useful when the fractional parts of the mixed numbers are not easily subtracted directly or when borrowing from the whole number is required.

The conversion process ensures that the entire quantity represented by the mixed number is expressed as a single fraction, making it easier to find a common denominator and perform subtraction. The benefit of using improper fractions is that they allow for direct application of fraction subtraction rules. By converting mixed numbers to this form, the problem becomes a straightforward subtraction of two fractions. This approach simplifies the steps involved and reduces the likelihood of errors, especially when dealing with complex or large numbers. Moreover, improper fractions are invaluable in algebraic manipulations and other advanced mathematical concepts. Their utility extends beyond basic arithmetic operations, making them a cornerstone of mathematical fluency. The ability to fluently convert between mixed numbers and improper fractions is a crucial skill in mathematical proficiency, opening doors to a broader range of problem-solving techniques and applications. Understanding this process not only aids in subtraction but also strengthens overall mathematical competency.

Once the mixed numbers are converted to improper fractions, the next step in subtraction is to find a common denominator. A common denominator is a number that is a multiple of both denominators in the fractions being subtracted. This step is necessary because fractions can only be added or subtracted if they have the same denominator. In our example, we have $\frac{87}{12}$ and $\frac{30}{9}$. The denominators are 12 and 9. To find a common denominator, we need to find the least common multiple (LCM) of 12 and 9. The multiples of 12 are 12, 24, 36, 48, 60, ... and the multiples of 9 are 9, 18, 27, 36, 45, ... The smallest number that appears in both lists is 36, so the least common multiple of 12 and 9 is 36. Therefore, 36 will be our common denominator. Finding a common denominator allows us to express both fractions with the same denominator, making it possible to subtract them directly. This step is a critical foundation for accurate fraction arithmetic.

To express each fraction with the common denominator, we multiply both the numerator and the denominator of each fraction by the number that makes the denominator equal to the common denominator. For $\frac{87}{12}$, we need to multiply the denominator 12 by 3 to get 36. So, we multiply both the numerator and the denominator by 3: $\frac{87 imes 3}{12 imes 3} = \frac{261}{36}$. For $\frac{30}{9}$, we need to multiply the denominator 9 by 4 to get 36. So, we multiply both the numerator and the denominator by 4: $\frac{30 imes 4}{9 imes 4} = \frac{120}{36}$. Now we have both fractions with the same denominator: $\frac{261}{36}$ and $\frac{120}{36}$. With a common denominator in place, we can proceed with the subtraction. The process of finding a common denominator is not just a procedural step; it fundamentally alters the fractions so they represent parts of the same whole, enabling accurate comparison and subtraction. It ensures that we are subtracting like quantities, which is essential for mathematical validity and real-world applications. The ability to quickly and accurately find common denominators is a hallmark of strong fraction arithmetic skills and is crucial for success in more advanced mathematical topics.

With a common denominator, subtracting the fractions becomes a straightforward process. We simply subtract the numerators while keeping the denominator the same. In our example, we have $\frac{261}{36}$ and $\frac{120}{36}$. To subtract these fractions, we subtract the numerators: $261 - 120 = 141$. The denominator remains 36. So, the result of the subtraction is $\frac{141}{36}$. This fraction represents the difference between the two original fractions, now expressed as a single fraction. The step of subtracting the numerators builds upon the previous steps of converting to improper fractions and finding a common denominator, illustrating the sequential nature of fraction arithmetic. This process highlights the importance of understanding each step and its contribution to the overall solution.

The resulting fraction $\frac{141}{36}$ is an improper fraction, which means the numerator is larger than the denominator. While this is a valid representation of the difference, it is often more useful to convert it back to a mixed number or simplify it further. The flexibility to work with fractions in both improper and mixed forms is a key skill in mathematical fluency. The subtraction of fractions is not just an isolated operation; it is a foundational concept that underlies many mathematical calculations and problem-solving scenarios. Whether calculating quantities in recipes, determining proportions in construction, or analyzing financial data, the ability to subtract fractions accurately is essential. By mastering this skill, individuals gain a powerful tool for navigating a wide range of practical applications and deepening their understanding of mathematical relationships.

After subtracting the fractions, the result is often an improper fraction. Converting back to a mixed number provides a clearer understanding of the quantity, as it separates the whole number part from the fractional part. In our example, we have the improper fraction $\frac{141}{36}$. To convert this to a mixed number, we divide the numerator (141) by the denominator (36). The quotient will be the whole number part, and the remainder will be the numerator of the fractional part. Dividing 141 by 36 gives us a quotient of 3 and a remainder of 33. This means that $\frac{141}{36}$ is equal to 3 whole units and $\frac{33}{36}$. Therefore, the mixed number is $3 \frac{33}{36}$. Converting back to a mixed number allows for a more intuitive understanding of the quantity and is often preferred in real-world applications where the whole number and fractional parts have distinct meanings.

The process of converting an improper fraction back to a mixed number not only provides a more human-readable form of the result but also reinforces the connection between fractions and mixed numbers. It demonstrates that these are simply two different ways of representing the same quantity. The ability to move seamlessly between improper fractions and mixed numbers is a hallmark of mathematical fluency and enables individuals to choose the representation that is most convenient for the task at hand. The mixed number $3 \frac{33}{36}$ conveys the magnitude of the quantity more directly than the improper fraction $\frac{141}{36}$, making it easier to visualize and compare. This visual aspect is crucial in practical contexts, such as measuring ingredients or interpreting financial data. The conversion process highlights the importance of both forms of representation and underscores the interconnectedness of mathematical concepts. In mathematical problem-solving, the ability to flexibly switch between different representations is a powerful tool that enhances comprehension and accuracy.

The final step in subtracting mixed numbers is often simplifying the final fraction. This involves reducing the fractional part of the mixed number to its lowest terms. In our example, we have the mixed number $3 \frac{33}{36}$. The fraction $\frac{33}{36}$ can be simplified because both 33 and 36 have a common factor, which is 3. Dividing both the numerator and the denominator by 3, we get $\frac{11}{12}$. Therefore, the simplified mixed number is $3 \frac{11}{12}$. Simplifying the fraction ensures that the answer is expressed in its simplest form, which is mathematically elegant and easier to understand. This step is a final check to ensure the answer is in the most concise and clear representation.

The simplified form not only makes the result easier to interpret but also reduces the chance of errors in future calculations. A simplified fraction represents the quantity in its most basic form, making it easier to compare with other fractions and use in subsequent operations. The process of simplifying the fraction reinforces the concept of equivalent fractions and the importance of reducing fractions to their lowest terms. In the context of real-world problems, simplified fractions often provide a clearer understanding of the proportions involved. For example, in a recipe, $3 \frac{11}{12}$ cups of flour is a more readily understandable quantity than $3 \frac{33}{36}$ cups. The simplification process highlights the mathematical principle that quantities can be expressed in multiple equivalent forms, and choosing the simplest form enhances both accuracy and understanding. The ability to simplify fractions is a valuable skill in mathematical literacy, empowering individuals to work with fractions confidently and effectively in various contexts.

In conclusion, subtracting mixed numbers involves several key steps: understanding mixed numbers, simplifying fractions, converting mixed numbers to improper fractions, finding a common denominator, subtracting the fractions, converting back to a mixed number, and simplifying the final fraction. By following these steps carefully, one can accurately subtract mixed numbers and arrive at the correct answer. In our example, $6 \frac{15}{12} - 3 \frac{3}{9} = 3 \frac{11}{12}$. This process not only provides the solution but also reinforces the foundational concepts of fraction arithmetic. Mastering these skills is essential for success in mathematics and in various real-world applications where fractions and mixed numbers are commonly encountered.

The ability to confidently and accurately subtract mixed numbers is a testament to a strong mathematical foundation. It requires a comprehensive understanding of fractions, mixed numbers, and the principles of arithmetic operations. The sequential nature of the steps involved emphasizes the importance of attention to detail and methodical problem-solving. The skills developed in this process extend beyond mere calculation; they foster critical thinking, analytical reasoning, and the ability to break down complex problems into manageable steps. Moreover, the practical applications of mixed number subtraction are vast, ranging from everyday tasks like cooking and measuring to more advanced fields such as engineering and finance. The journey from understanding the basics to mastering the subtraction of mixed numbers is a journey of mathematical empowerment, equipping individuals with the tools and confidence to tackle a wide range of challenges.