Subtracting Mixed Fractions A Step-by-Step Guide With Examples

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This article will guide you through the process of subtracting mixed fractions and simplifying the results. Mixed fractions, which combine whole numbers and fractions, can sometimes seem daunting to subtract. However, by following a few key steps, you can confidently tackle these problems and arrive at simplified answers. We will explore a series of examples, breaking down each step to ensure a clear understanding. Let's dive into the world of fraction subtraction!

1. Understanding Mixed Fractions

Before we delve into subtraction, let's first solidify our understanding of mixed fractions. A mixed fraction is a combination of a whole number and a proper fraction (a fraction where the numerator is less than the denominator). For example, 17 5/6 is a mixed fraction, where 17 is the whole number and 5/6 is the fraction. To effectively subtract mixed fractions, it's essential to be comfortable with converting them into improper fractions and vice versa. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as 11/6. Converting mixed fractions to improper fractions makes the subtraction process much smoother, especially when dealing with borrowing. The process involves multiplying the whole number by the denominator of the fraction and then adding the numerator. This result becomes the new numerator, while the denominator remains the same. For instance, to convert 17 5/6 to an improper fraction, we multiply 17 by 6 (which is 102) and add 5, resulting in 107. Therefore, the improper fraction equivalent of 17 5/6 is 107/6. Understanding this conversion is the foundation for simplifying mixed fraction subtraction problems. Moreover, knowing how to convert back from an improper fraction to a mixed fraction is equally important for presenting the final answer in its simplest form. This involves dividing the numerator by the denominator; the quotient becomes the whole number, the remainder becomes the new numerator, and the denominator stays the same. This reciprocal relationship between mixed and improper fractions is crucial for mastering fraction arithmetic.

2. 17 5/6 - 9 1/3

Let's start with our first example: 17 5/6 - 9 1/3. The initial step is to convert both mixed fractions into improper fractions. As we discussed earlier, this involves multiplying the whole number by the denominator and adding the numerator. For 17 5/6, we have (17 * 6) + 5 = 107, so the improper fraction is 107/6. For 9 1/3, we calculate (9 * 3) + 1 = 28, giving us the improper fraction 28/3. Now, we need to subtract 28/3 from 107/6. However, before we can subtract fractions, they must have a common denominator. The least common multiple (LCM) of 6 and 3 is 6, so we only need to adjust the second fraction. To make the denominator of 28/3 equal to 6, we multiply both the numerator and the denominator by 2, resulting in 56/6. Now we can perform the subtraction: 107/6 - 56/6. Subtracting the numerators, we get 107 - 56 = 51, so the result is 51/6. Finally, we simplify the improper fraction 51/6 by converting it back into a mixed fraction. Dividing 51 by 6, we get a quotient of 8 and a remainder of 3. Therefore, 51/6 is equivalent to 8 3/6. The fraction 3/6 can be further simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us 1/2. So, the final simplified answer is 8 1/2. This comprehensive breakdown illustrates the importance of each step – converting to improper fractions, finding a common denominator, subtracting, and simplifying – in solving mixed fraction subtraction problems.

3. 31 11/12 - 18 3/24

Now, let's tackle the problem 31 11/12 - 18 3/24. Our first task, as with any mixed fraction subtraction, is to convert the mixed fractions into improper fractions. For 31 11/12, we calculate (31 * 12) + 11. 31 multiplied by 12 equals 372, and adding 11 gives us 383. Thus, the improper fraction equivalent of 31 11/12 is 383/12. Next, we convert 18 3/24 into an improper fraction. (18 * 24) + 3 equals 432 plus 3, which is 435. Therefore, 18 3/24 is equivalent to 435/24. Now we have the subtraction problem 383/12 - 435/24. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 24 is 24. We can convert 383/12 to an equivalent fraction with a denominator of 24 by multiplying both the numerator and the denominator by 2. This gives us (383 * 2) / (12 * 2) = 766/24. Now we can perform the subtraction: 766/24 - 435/24. Subtracting the numerators, we get 766 - 435 = 331. So, the result is 331/24. Finally, we simplify the improper fraction 331/24 by converting it back into a mixed fraction. Dividing 331 by 24, we get a quotient of 13 and a remainder of 19. This means 331/24 is equivalent to 13 19/24. In this case, the fraction 19/24 is already in its simplest form, as 19 is a prime number and does not share any common factors with 24 other than 1. Therefore, the final simplified answer for 31 11/12 - 18 3/24 is 13 19/24. This example underscores the importance of finding the LCM efficiently to simplify the subtraction process and highlights the final step of ensuring the fractional part of the mixed number is fully simplified.

4. 20 7/15 - 11 9/20

Let's proceed to our next problem: 20 7/15 - 11 9/20. As with the previous examples, the first step is to convert the mixed numbers into improper fractions. For 20 7/15, we calculate (20 * 15) + 7. 20 multiplied by 15 is 300, and adding 7 gives us 307. Therefore, the improper fraction is 307/15. Next, we convert 11 9/20 into an improper fraction. (11 * 20) + 9 equals 220 plus 9, which is 229. So, 11 9/20 is equivalent to 229/20. Now we have the subtraction problem 307/15 - 229/20. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 20 is 60. To convert 307/15 to an equivalent fraction with a denominator of 60, we multiply both the numerator and the denominator by 4. This gives us (307 * 4) / (15 * 4) = 1228/60. To convert 229/20 to an equivalent fraction with a denominator of 60, we multiply both the numerator and the denominator by 3. This gives us (229 * 3) / (20 * 3) = 687/60. Now we can perform the subtraction: 1228/60 - 687/60. Subtracting the numerators, we get 1228 - 687 = 541. Thus, the result is 541/60. Finally, we simplify the improper fraction 541/60 by converting it back into a mixed fraction. Dividing 541 by 60, we get a quotient of 9 and a remainder of 1. This means 541/60 is equivalent to 9 1/60. In this case, the fraction 1/60 is already in its simplest form. This example reinforces the method of finding the LCM and demonstrates how to handle larger numerators that result from the conversion process. The key takeaway is the systematic approach: convert, find the common denominator, subtract, and simplify.

5. 23 11/15 - 12 3/10

Let's move on to the next subtraction problem: 23 11/15 - 12 3/10. As with the previous examples, the first step involves converting the mixed numbers into improper fractions. For 23 11/15, we calculate (23 * 15) + 11. 23 multiplied by 15 equals 345, and adding 11 results in 356. So, the improper fraction is 356/15. For 12 3/10, we calculate (12 * 10) + 3, which equals 120 plus 3, giving us 123. Thus, 12 3/10 is equivalent to 123/10. Now we have the subtraction problem 356/15 - 123/10. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 10 is 30. To convert 356/15 to an equivalent fraction with a denominator of 30, we multiply both the numerator and the denominator by 2. This gives us (356 * 2) / (15 * 2) = 712/30. To convert 123/10 to an equivalent fraction with a denominator of 30, we multiply both the numerator and the denominator by 3. This gives us (123 * 3) / (10 * 3) = 369/30. Now we can perform the subtraction: 712/30 - 369/30. Subtracting the numerators, we get 712 - 369 = 343. Thus, the result is 343/30. Finally, we simplify the improper fraction 343/30 by converting it back into a mixed fraction. Dividing 343 by 30, we get a quotient of 11 and a remainder of 13. This means 343/30 is equivalent to 11 13/30. In this case, the fraction 13/30 is already in its simplest form. This example further illustrates the process of converting mixed numbers to improper fractions, finding the LCM, and simplifying the result back into a mixed number. Each step is crucial in arriving at the correct, simplified answer.

6. 65 10/15 - 50 2/3

Let's continue with our practice and solve 65 10/15 - 50 2/3. As before, the first step is to convert the mixed numbers into improper fractions. For 65 10/15, we calculate (65 * 15) + 10. 65 multiplied by 15 is 975, and adding 10 gives us 985. Thus, the improper fraction is 985/15. Next, we convert 50 2/3 into an improper fraction. (50 * 3) + 2 equals 150 plus 2, which is 152. So, 50 2/3 is equivalent to 152/3. Now we have the subtraction problem 985/15 - 152/3. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 3 is 15. This simplifies our work since we only need to convert the second fraction. To convert 152/3 to an equivalent fraction with a denominator of 15, we multiply both the numerator and the denominator by 5. This gives us (152 * 5) / (3 * 5) = 760/15. Now we can perform the subtraction: 985/15 - 760/15. Subtracting the numerators, we get 985 - 760 = 225. Thus, the result is 225/15. Finally, we simplify the improper fraction 225/15. We can see that 225 is divisible by 15. Dividing 225 by 15, we get 15 with no remainder. This means 225/15 simplifies to 15, which is a whole number. This example demonstrates that sometimes, after subtracting and simplifying, the result can be a whole number, further emphasizing the importance of always simplifying the final answer.

7. 78 7/9 - 45 1/4

Let's tackle our final example: 78 7/9 - 45 1/4. As with all previous problems, the initial step is to convert the mixed numbers into improper fractions. For 78 7/9, we calculate (78 * 9) + 7. 78 multiplied by 9 equals 702, and adding 7 gives us 709. Thus, the improper fraction is 709/9. Next, we convert 45 1/4 into an improper fraction. (45 * 4) + 1 equals 180 plus 1, which is 181. So, 45 1/4 is equivalent to 181/4. Now we have the subtraction problem 709/9 - 181/4. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 9 and 4 is 36. To convert 709/9 to an equivalent fraction with a denominator of 36, we multiply both the numerator and the denominator by 4. This gives us (709 * 4) / (9 * 4) = 2836/36. To convert 181/4 to an equivalent fraction with a denominator of 36, we multiply both the numerator and the denominator by 9. This gives us (181 * 9) / (4 * 9) = 1629/36. Now we can perform the subtraction: 2836/36 - 1629/36. Subtracting the numerators, we get 2836 - 1629 = 1207. Thus, the result is 1207/36. Finally, we simplify the improper fraction 1207/36 by converting it back into a mixed fraction. Dividing 1207 by 36, we get a quotient of 33 and a remainder of 19. This means 1207/36 is equivalent to 33 19/36. In this case, the fraction 19/36 is already in its simplest form. This final example reinforces the comprehensive process of mixed fraction subtraction, showcasing the importance of accurately converting to improper fractions, finding the LCM, performing the subtraction, and simplifying the result back into a mixed number.

Conclusion

In conclusion, subtracting mixed fractions involves a series of steps that, when followed methodically, lead to accurate and simplified answers. The key steps include converting mixed fractions to improper fractions, finding the least common multiple of the denominators, converting the fractions to have a common denominator, subtracting the numerators, and finally, simplifying the resulting fraction back into a mixed number or a whole number if possible. Each example discussed in this article highlights the importance of these steps and provides a clear pathway for solving various subtraction problems involving mixed fractions. By practicing these steps, you can build confidence and proficiency in handling mixed fraction subtraction, a fundamental skill in mathematics.