Subtracting Mixed Numbers A Step By Step Guide For 9 1/6 - 5 1/4

Introduction

In the realm of mathematics, mastering the subtraction of mixed numbers is a crucial skill, especially when working with fractions. This article delves into the step-by-step process of subtracting mixed numbers, focusing on the specific example of $9 \frac{1}{6} - 5 \frac{1}{4}$. We will explore the underlying principles, techniques, and nuances involved in simplifying the result to its simplest form. Whether you are a student grappling with fractions or an educator seeking a clear explanation, this guide will provide you with a comprehensive understanding of mixed number subtraction.

Understanding Mixed Numbers

Before we dive into the subtraction process, it's essential to grasp the concept of mixed numbers. A mixed number is a combination of a whole number and a proper fraction (where the numerator is less than the denominator). For instance, $9 \frac{1}{6}$ is a mixed number comprising the whole number 9 and the proper fraction $\frac{1}{6}$. Similarly, $5 \frac{1}{4}$ consists of the whole number 5 and the fraction $\frac{1}{4}$. These numbers are used to represent quantities greater than one whole unit, offering a more intuitive way to express values between whole numbers. Understanding the composition of mixed numbers is the foundation for performing arithmetic operations such as subtraction.

To effectively subtract mixed numbers, it’s often necessary to convert them into improper fractions. An improper fraction is one where the numerator is greater than or equal to the denominator. This conversion simplifies the subtraction process, especially when the fractional parts of the mixed numbers have different denominators. 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 this result over the original denominator. This process ensures that we are working with a single fractional representation of the quantity, making subtraction more straightforward. For instance, to convert $9 \frac{1}{6}$ to an improper fraction, we multiply 9 by 6 (which equals 54), add 1 (resulting in 55), and place this over the denominator 6, giving us $\frac{55}{6}$. This conversion technique is a cornerstone of mixed number arithmetic.

Step-by-Step Subtraction of $9 \frac{1}{6} - 5 \frac{1}{4}$

Step 1: Convert Mixed Numbers to Improper Fractions

The first crucial step in subtracting mixed numbers is to convert them into improper fractions. This conversion simplifies the process and allows for easier manipulation of the fractional parts. To convert $9 \frac{1}{6}$ into an improper fraction, we multiply the whole number 9 by the denominator 6, which gives us 54. Then, we add the numerator 1 to this result, obtaining 55. We place this sum over the original denominator 6, resulting in the improper fraction $\frac{55}{6}$. Similarly, for $5 \frac{1}{4}$, we multiply 5 by 4, which equals 20, and add the numerator 1, giving us 21. Placing this over the denominator 4 yields the improper fraction $\frac{21}{4}$. Now, our subtraction problem is transformed into $\frac{55}{6} - \frac{21}{4}$, which is easier to manage arithmetically. This initial conversion is a pivotal step in accurately solving the subtraction problem.

Step 2: Find the Least Common Denominator (LCD)

After converting the mixed numbers to improper fractions, the next critical step is to find the Least Common Denominator (LCD). The LCD is the smallest common multiple of the denominators of the fractions involved. In our case, we have the fractions $\frac{55}{6}$ and $\frac{21}{4}$, so we need to find the LCD of 6 and 4. The multiples of 6 are 6, 12, 18, 24, and so on, while the multiples of 4 are 4, 8, 12, 16, and so on. The smallest number that appears in both lists is 12, making it the LCD. Finding the LCD is crucial because it allows us to rewrite the fractions with a common denominator, enabling straightforward subtraction. This ensures that we are subtracting equivalent parts, leading to an accurate result. Once we have the LCD, we can proceed to rewrite each fraction with this new denominator.

Step 3: Rewrite Fractions with the LCD

Once we have identified the Least Common Denominator (LCD), the next step is to rewrite each fraction using this common denominator. This ensures that we are subtracting like quantities, which is essential for accurate calculations. In our example, the LCD of 6 and 4 is 12. To rewrite $\frac{55}{6}$ with a denominator of 12, we need to determine what number to multiply the original denominator 6 by to get 12. Since 6 multiplied by 2 equals 12, we also multiply the numerator 55 by 2, resulting in 110. Therefore, $\frac{55}{6}$ becomes $\frac{110}{12}$. For the fraction $\frac{21}{4}$, we need to find what number to multiply 4 by to get 12. Since 4 multiplied by 3 equals 12, we multiply the numerator 21 by 3, resulting in 63. Thus, $\frac{21}{4}$ becomes $\frac{63}{12}$. Now, our subtraction problem is rewritten as $\frac{110}{12} - \frac{63}{12}$, which has a common denominator and is ready for subtraction. This step is crucial for ensuring the accuracy of the final result.

Step 4: Subtract the Fractions

With the fractions now sharing a common denominator, the next step is to subtract the numerators. This is a straightforward process once the fractions are expressed in terms of the same denominator. In our example, we have $\frac{110}{12} - \frac{63}{12}$. To subtract these fractions, we subtract the numerators: 110 - 63 = 47. The denominator remains the same, so we have $\frac{47}{12}$. This improper fraction represents the result of our subtraction. The subtraction of the numerators is a direct application of the principle that fractions with common denominators can be subtracted by simply subtracting their numerators. This step brings us closer to the final answer, but it's important to remember that we may need to simplify the result further to express it in its simplest form.

Step 5: Simplify the Result

After performing the subtraction, the resulting fraction is $\frac{47}{12}$, which is an improper fraction. To express this in the simplest form, we need to convert it back into a mixed number. This involves dividing the numerator (47) by the denominator (12). When we divide 47 by 12, we get a quotient of 3 and a remainder of 11. This means that 12 goes into 47 three times with 11 left over. Therefore, we write the mixed number as 3 (the quotient) and $\frac{11}{12}$ (the remainder over the original denominator). So, $\frac{47}{12}$ is equivalent to $3 \frac{11}{12}$. Next, we need to check if the fractional part, $\frac{11}{12}$, can be simplified further. The factors of 11 are 1 and 11, and the factors of 12 are 1, 2, 3, 4, 6, and 12. Since the only common factor between 11 and 12 is 1, the fraction $\frac{11}{12}$ is already in its simplest form. Thus, the final answer, expressed as a mixed number in its simplest form, is $3 \frac{11}{12}$.

Conclusion

In conclusion, subtracting mixed numbers requires a series of steps to ensure accuracy. Starting with the conversion of mixed numbers to improper fractions, finding the least common denominator, rewriting fractions, subtracting, and finally, simplifying the result, each step plays a vital role in arriving at the correct answer. In the specific case of $9 \frac{1}{6} - 5 \frac{1}{4}$, we have methodically demonstrated how to arrive at the simplified mixed number result of $3 \frac{11}{12}$. By mastering these steps, you can confidently tackle subtraction problems involving mixed numbers. This comprehensive guide serves as a valuable resource for students and educators alike, fostering a deeper understanding of fraction arithmetic.

Keywords

Mixed numbers, improper fractions, subtraction, least common denominator, simplify, $9 \frac{1}{6}$, $5 \frac{1}{4}$, $3 \frac{11}{12}$, fraction arithmetic, conversion, numerator, denominator