Mixed Number Subtraction Explained Step-by-Step
In the realm of mathematics, mastering mixed number subtraction is a crucial skill for students and professionals alike. This article aims to provide a comprehensive guide to solving two specific mixed number subtraction problems: 30 7/99 - 25 5/121 and 105 2/17 - 3 1/13. We will break down each problem step-by-step, ensuring a clear understanding of the underlying concepts and techniques involved. By the end of this guide, you will be well-equipped to tackle similar problems with confidence and precision. Whether you are a student preparing for an exam or someone looking to refresh your math skills, this article is designed to help you excel in mixed number subtraction.
Problem 1: 30 7/99 - 25 5/121
To effectively subtract mixed numbers, it's essential to approach the problem methodically. Subtracting mixed numbers involves several key steps, including converting mixed numbers to improper fractions, finding a common denominator, performing the subtraction, and simplifying the result. In this section, we will meticulously walk through each of these steps to solve the problem 30 7/99 - 25 5/121. Our goal is not just to provide the answer, but to ensure you grasp the process thoroughly. By understanding the mechanics behind each step, you'll be better prepared to handle more complex problems in the future. This section will serve as a foundational guide, illustrating the techniques necessary for accurate and efficient mixed number subtraction.
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 allows us to perform subtraction more easily by dealing with single fractions rather than mixed numbers. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fractional part and then add the numerator. This result becomes the new numerator, while the denominator remains the same. Let's apply this to our problem, 30 7/99 - 25 5/121, starting with the first mixed number.
For 30 7/99, we multiply 30 by 99, which equals 2970. Then, we add 7 to get 2977. So, the improper fraction for 30 7/99 is 2977/99. Similarly, for 25 5/121, we multiply 25 by 121, which equals 3025. Adding 5 gives us 3030. Thus, the improper fraction for 25 5/121 is 3030/121. Now, our problem transforms from 30 7/99 - 25 5/121 to 2977/99 - 3030/121. This conversion is a pivotal step because it sets the stage for finding a common denominator, which is necessary for subtraction.
Step 2: Find the Least Common Denominator (LCD)
After converting the mixed numbers to improper fractions, the next essential step is to find the Least Common Denominator (LCD). Finding the LCD is crucial because we cannot subtract fractions unless they have the same denominator. The LCD is the smallest multiple that the denominators of the fractions share. In our problem, we need to find the LCD of 99 and 121. To do this, we can use prime factorization or list multiples of each number until we find a common one. Let's explore finding the LCD for 99 and 121.
The prime factorization of 99 is 3 x 3 x 11 (or 3^2 x 11), and the prime factorization of 121 is 11 x 11 (or 11^2). To find the LCD, we take the highest power of each prime factor that appears in either factorization. This means we take 3^2 and 11^2. Multiplying these together, we get 3^2 x 11^2 = 9 x 121 = 1089. Therefore, the LCD of 99 and 121 is 1089. This LCD will be the common denominator we use to rewrite our fractions so that we can perform the subtraction.
Step 3: Rewrite Fractions with the LCD
With the Least Common Denominator (LCD) determined, the next step is to rewrite each fraction with this new denominator. Rewriting fractions involves finding the appropriate factor to multiply both the numerator and the denominator of each fraction so that the denominator becomes the LCD. This process ensures that the value of the fraction remains unchanged while allowing us to perform subtraction. In our problem, we have the fractions 2977/99 and 3030/121, and the LCD is 1089.
For the first fraction, 2977/99, we need to determine what number to multiply 99 by to get 1089. Since 1089 ÷ 99 = 11, we multiply both the numerator and the denominator of 2977/99 by 11. This gives us (2977 x 11) / (99 x 11) = 32747/1089. For the second fraction, 3030/121, we need to determine what number to multiply 121 by to get 1089. Since 1089 ÷ 121 = 9, we multiply both the numerator and the denominator of 3030/121 by 9. This gives us (3030 x 9) / (121 x 9) = 27270/1089. Now, our subtraction problem is rewritten as 32747/1089 - 27270/1089, which is ready for the next step: performing the subtraction.
Step 4: Subtract the Fractions
Now that both fractions have the same denominator, we can proceed with the subtraction. Subtracting fractions with a common denominator is straightforward: we subtract the numerators and keep the denominator the same. In our problem, we have 32747/1089 - 27270/1089. To subtract these fractions, we subtract the numerators: 32747 - 27270.
Performing the subtraction, 32747 - 27270 = 5477. Therefore, the result of subtracting the fractions is 5477/1089. This fraction represents the difference between our two original fractions. However, it's essential to simplify this result, if possible, and convert it back to a mixed number to make it more understandable. The next step will cover simplifying the fraction and converting it back to a mixed number.
Step 5: Simplify and Convert to Mixed Number
After subtracting the fractions, we obtained the result 5477/1089. The final step is to simplify the fraction and convert it back to a mixed number. Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). Converting an improper fraction to a mixed number involves dividing the numerator by the denominator and expressing the result as a whole number and a fraction.
First, let's simplify 5477/1089. The GCD of 5477 and 1089 is 1. Therefore, the fraction is already in its simplest form. Next, we convert the improper fraction 5477/1089 to a mixed number. To do this, we divide 5477 by 1089. The quotient is 5, and the remainder is 42. Thus, 5477/1089 as a mixed number is 5 42/1089. This is the final simplified answer to the subtraction problem 30 7/99 - 25 5/121. By following these steps, we have successfully subtracted the mixed numbers and expressed the result in its simplest form.
Problem 2: 105 2/17 - 3 1/13
Now, let's tackle the second problem: 105 2/17 - 3 1/13. Similar to the previous problem, we will break this down into manageable steps, ensuring a clear and thorough understanding of the process. This problem offers another opportunity to reinforce the techniques of converting mixed numbers to improper fractions, finding the Least Common Denominator (LCD), rewriting fractions with the LCD, subtracting the fractions, and simplifying the result. By working through this example, you will further solidify your skills in mixed number subtraction and gain confidence in your ability to solve such problems independently. Each step will be explained in detail to ensure clarity and comprehension.
Step 1: Convert Mixed Numbers to Improper Fractions
The initial step in subtracting the mixed numbers 105 2/17 and 3 1/13 is to convert each into an improper fraction. Converting mixed numbers to improper fractions allows us to perform subtraction more easily by dealing with a single fractional value for each number. To convert a mixed number, we multiply the whole number part by the denominator and then add the numerator. The result becomes the new numerator, and the denominator remains the same.
For 105 2/17, we multiply 105 by 17, which equals 1785. Adding 2 to this gives us 1787. Therefore, the improper fraction for 105 2/17 is 1787/17. Next, we convert 3 1/13 to an improper fraction. We multiply 3 by 13, which equals 39. Adding 1 to this gives us 40. So, the improper fraction for 3 1/13 is 40/13. Now, our problem is transformed from 105 2/17 - 3 1/13 to 1787/17 - 40/13. This conversion is crucial as it sets the stage for finding a common denominator, which is required for subtracting fractions.
Step 2: Find the Least Common Denominator (LCD)
After converting the mixed numbers to improper fractions, the next step is to find the Least Common Denominator (LCD) of the fractions. Finding the Least Common Denominator is essential because we cannot subtract fractions unless they have the same denominator. The LCD is the smallest multiple that both denominators share. In this case, we need to find the LCD of 17 and 13. Since 17 and 13 are both prime numbers, their only common multiple is their product.
To find the LCD of 17 and 13, we simply multiply them together: 17 x 13 = 221. Therefore, the LCD is 221. This will be the common denominator we use to rewrite our fractions, allowing us to perform the subtraction. Understanding how to find the LCD is a fundamental skill in fraction arithmetic, and it’s crucial for accurate calculations.
Step 3: Rewrite Fractions with the LCD
Now that we have found the Least Common Denominator (LCD), which is 221, the next step is to rewrite both fractions with this new denominator. Rewriting fractions with a common denominator is essential for performing subtraction. We need to determine what factor to multiply both the numerator and denominator of each fraction by to get the LCD as the new denominator. In our problem, we have the fractions 1787/17 and 40/13, and the LCD is 221.
For the first fraction, 1787/17, we need to find the factor that, when multiplied by 17, gives us 221. We know that 17 x 13 = 221, so we multiply both the numerator and the denominator of 1787/17 by 13. This gives us (1787 x 13) / (17 x 13) = 23231/221. For the second fraction, 40/13, we need to find the factor that, when multiplied by 13, gives us 221. We know that 13 x 17 = 221, so we multiply both the numerator and the denominator of 40/13 by 17. This gives us (40 x 17) / (13 x 17) = 680/221. Now, our subtraction problem is rewritten as 23231/221 - 680/221, which sets us up for the next step: subtracting the fractions.
Step 4: Subtract the Fractions
With both fractions now having the same denominator of 221, we can proceed with the subtraction. Subtracting fractions with a common denominator involves subtracting the numerators while keeping the denominator the same. In our problem, we have 23231/221 - 680/221. To subtract these fractions, we subtract the numerators: 23231 - 680.
Performing the subtraction, 23231 - 680 = 22551. Therefore, the result of subtracting the fractions is 22551/221. This fraction represents the difference between the two original fractions, but it is an improper fraction. To make it more understandable, we need to convert it back to a mixed number and, if possible, simplify it. The next step will cover these processes.
Step 5: Simplify and Convert to Mixed Number
After subtracting the fractions, we obtained the result 22551/221. The final step is to simplify the fraction and convert it back to a mixed number. Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). Converting an improper fraction to a mixed number involves dividing the numerator by the denominator and expressing the result as a whole number and a fraction.
First, let's simplify 22551/221. The GCD of 22551 and 221 is 1. Therefore, the fraction is already in its simplest form. Next, we convert the improper fraction 22551/221 to a mixed number. To do this, we divide 22551 by 221. The quotient is 102, and the remainder is 29. Thus, 22551/221 as a mixed number is 102 29/221. This is the final simplified answer to the subtraction problem 105 2/17 - 3 1/13. By following each step carefully, we have successfully subtracted the mixed numbers and expressed the result in its simplest form.
Conclusion
In conclusion, mastering mixed number subtraction involves a series of steps: converting mixed numbers to improper fractions, finding the Least Common Denominator (LCD), rewriting fractions with the LCD, subtracting the fractions, and simplifying the result. This article has provided a comprehensive guide to solving two specific problems, 30 7/99 - 25 5/121 and 105 2/17 - 3 1/13, illustrating each step in detail. By understanding and practicing these techniques, you can confidently tackle similar problems and enhance your mathematical proficiency. Effective mixed number subtraction is a valuable skill in various mathematical contexts, and with the knowledge gained from this guide, you are well-prepared to excel in this area.