Dividing Fractions A Step-by-Step Guide With Examples

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Dividing fractions might seem daunting at first, but with a clear understanding of the underlying principles, it becomes a straightforward process. This comprehensive guide will walk you through the steps involved in dividing fractions, providing detailed explanations and examples to solidify your understanding. We'll cover everything from basic fraction division to handling mixed numbers, ensuring you're well-equipped to tackle any fraction division problem.

Understanding Fraction Division

The core concept behind dividing fractions lies in the idea of reciprocals. The reciprocal of a fraction is simply the fraction flipped upside down. For example, the reciprocal of 2/3 is 3/2. Dividing by a fraction is the same as multiplying by its reciprocal. This fundamental principle is the key to unlocking fraction division.

Think of it this way: division asks how many times one quantity fits into another. When dealing with fractions, this translates to finding out how many times the divisor (the fraction you're dividing by) fits into the dividend (the fraction being divided). By multiplying by the reciprocal, we're essentially converting the division problem into a multiplication problem, which is often easier to handle.

This concept is rooted in the properties of multiplicative inverses. Every number (except zero) has a multiplicative inverse, which is the number that, when multiplied by the original number, results in 1. The reciprocal of a fraction is its multiplicative inverse. When you divide by a fraction, you're essentially multiplying by its inverse, which cancels out the divisor and leaves you with the quotient.

Furthermore, understanding fraction division is crucial for various real-world applications. From measuring ingredients in a recipe to calculating proportions in construction, fractions are an integral part of our daily lives. Mastering fraction division allows you to solve practical problems with confidence and accuracy. It also lays the foundation for more advanced mathematical concepts, such as algebra and calculus, where fractions play a significant role. By grasping the principles of fraction division, you're not just learning a mathematical skill; you're developing a valuable tool for problem-solving in a wide range of contexts.

Steps to Divide Fractions

  1. Identify the fractions: Clearly identify the dividend (the fraction being divided) and the divisor (the fraction you're dividing by).
  2. Find the reciprocal of the divisor: Flip the divisor fraction upside down. This means swapping the numerator and the denominator.
  3. Multiply the dividend by the reciprocal: Multiply the dividend fraction by the reciprocal of the divisor. To multiply fractions, multiply the numerators together and the denominators together.
  4. Simplify the resulting fraction: If necessary, simplify the resulting fraction to its lowest terms. This involves finding the greatest common factor (GCF) of the numerator and denominator and dividing both by the GCF.

Let's illustrate these steps with a simple example: 1/2 ÷ 3/4.

  1. Identify the fractions: Dividend = 1/2, Divisor = 3/4
  2. Find the reciprocal of the divisor: Reciprocal of 3/4 = 4/3
  3. Multiply the dividend by the reciprocal: (1/2) * (4/3) = (1 * 4) / (2 * 3) = 4/6
  4. Simplify the resulting fraction: The GCF of 4 and 6 is 2. Dividing both by 2, we get 2/3.

Therefore, 1/2 ÷ 3/4 = 2/3. This step-by-step approach provides a clear framework for dividing fractions, making the process less intimidating and more manageable. By consistently following these steps, you can confidently solve a wide range of fraction division problems. Remember to always double-check your work and simplify your answer to ensure accuracy.

Examples of Dividing Fractions

Now, let's work through the examples you provided, applying the steps we've discussed.

(a) 2/7 ÷ 8/21

This example focuses on dividing fractions and applying the core principle of multiplying by the reciprocal. Let's break down the solution step by step to ensure clarity and understanding.

  1. Identify the fractions: Dividend = 2/7, Divisor = 8/21
  2. Find the reciprocal of the divisor: The reciprocal of 8/21 is 21/8. Remember, to find the reciprocal, we simply flip the fraction, swapping the numerator and the denominator. This step is crucial as it transforms the division problem into a multiplication problem, which is often easier to handle.
  3. Multiply the dividend by the reciprocal: Now, we multiply the dividend (2/7) by the reciprocal of the divisor (21/8). This gives us: (2/7) * (21/8) = (2 * 21) / (7 * 8). To perform this multiplication, we multiply the numerators together (2 * 21) and the denominators together (7 * 8).
  4. Calculate the products: 2 * 21 = 42 and 7 * 8 = 56. So, our fraction becomes 42/56. This fraction represents the result of the multiplication, but it's not yet in its simplest form. Simplifying the fraction is the final step to ensure we have the most concise answer.
  5. Simplify the resulting fraction: The fraction 42/56 can be simplified. We need to find the greatest common factor (GCF) of 42 and 56. The GCF is the largest number that divides both 42 and 56 without leaving a remainder. In this case, the GCF is 14. Dividing both the numerator and the denominator by 14, we get: 42 ÷ 14 = 3 and 56 ÷ 14 = 4. Therefore, the simplified fraction is 3/4.

Thus, 2/7 ÷ 8/21 = 3/4. This final result represents the quotient of the original division problem. By following these steps, we have successfully divided the fractions and simplified the answer. This process highlights the importance of understanding reciprocals and simplification in fraction division.

(b) 6/11 ÷ 14/22

This example further illustrates the process of dividing fractions, reinforcing the steps involved in finding the reciprocal and simplifying the result. By working through this example, you'll gain more confidence in your ability to handle fraction division problems.

  1. Identify the fractions: Dividend = 6/11, Divisor = 14/22
  2. Find the reciprocal of the divisor: The reciprocal of 14/22 is 22/14. Remember, flipping the divisor is the key to transforming the division problem into multiplication. This step allows us to apply the rules of fraction multiplication, which are generally more familiar.
  3. Multiply the dividend by the reciprocal: Multiply 6/11 by 22/14: (6/11) * (22/14) = (6 * 22) / (11 * 14). This step involves multiplying the numerators together and the denominators together, as we do in any fraction multiplication problem.
  4. Calculate the products: 6 * 22 = 132 and 11 * 14 = 154. So, the fraction becomes 132/154. While this fraction represents the correct result of the multiplication, it's important to simplify it to its lowest terms for the most concise answer.
  5. Simplify the resulting fraction: The fraction 132/154 needs to be simplified. To do this, we find the greatest common factor (GCF) of 132 and 154. The GCF is the largest number that divides both 132 and 154 without leaving a remainder. In this case, the GCF is 22. Dividing both the numerator and the denominator by 22, we get: 132 ÷ 22 = 6 and 154 ÷ 22 = 7. Therefore, the simplified fraction is 6/7.

Thus, 6/11 ÷ 14/22 = 6/7. This solution demonstrates the complete process of dividing fractions, from identifying the dividend and divisor to simplifying the final result. By practicing these steps, you can master the art of fraction division and solve problems with accuracy and efficiency.

(c) 7 1/9 ÷ 1/3

This example introduces dividing mixed numbers by a fraction. Mixed numbers, which combine a whole number and a fraction, require an extra step before we can apply the standard fraction division procedure. Let's explore how to handle this type of problem.

  1. Convert the mixed number to an improper fraction: First, we need to convert the mixed number 7 1/9 into an improper fraction. An improper fraction is one where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number (7) by the denominator (9) and add the numerator (1). This result becomes the new numerator, and we keep the same denominator. So, 7 * 9 + 1 = 63 + 1 = 64. Therefore, 7 1/9 is equivalent to 64/9.
  2. Identify the fractions: Now we have the problem as 64/9 ÷ 1/3. Dividend = 64/9, Divisor = 1/3. With the mixed number converted to an improper fraction, we can proceed with the standard fraction division steps.
  3. Find the reciprocal of the divisor: The reciprocal of 1/3 is 3/1. Remember, we flip the fraction to find its reciprocal, swapping the numerator and denominator. This step is essential for converting the division problem into a multiplication problem.
  4. Multiply the dividend by the reciprocal: Multiply 64/9 by 3/1: (64/9) * (3/1) = (64 * 3) / (9 * 1). This involves multiplying the numerators together and the denominators together, as we do in any fraction multiplication problem.
  5. Calculate the products: 64 * 3 = 192 and 9 * 1 = 9. So, the fraction becomes 192/9. This fraction represents the result of the multiplication, but it's important to simplify it to its lowest terms and, if possible, convert it back to a mixed number.
  6. Simplify the resulting fraction: The fraction 192/9 can be simplified. To do this, we find the greatest common factor (GCF) of 192 and 9. The GCF is the largest number that divides both 192 and 9 without leaving a remainder. In this case, the GCF is 3. Dividing both the numerator and the denominator by 3, we get: 192 ÷ 3 = 64 and 9 ÷ 3 = 3. Therefore, the simplified fraction is 64/3.
  7. Convert the improper fraction to a mixed number (optional): Since the original problem involved a mixed number, it's often helpful to convert the improper fraction back to a mixed number. To do this, we divide the numerator (64) by the denominator (3). 64 ÷ 3 = 21 with a remainder of 1. The quotient (21) becomes the whole number part of the mixed number, the remainder (1) becomes the numerator, and we keep the same denominator (3). Therefore, 64/3 is equivalent to 21 1/3.

Thus, 7 1/9 ÷ 1/3 = 64/3 or 21 1/3. This example demonstrates the complete process of dividing a mixed number by a fraction, including the crucial step of converting the mixed number to an improper fraction before proceeding with the division. By mastering this technique, you can confidently solve more complex fraction division problems.

(d) 26/25 ÷ 8/15

This example provides another opportunity to practice dividing fractions and simplifying the resulting fraction. Let's break down the steps to ensure a clear understanding of the process.

  1. Identify the fractions: Dividend = 26/25, Divisor = 8/15
  2. Find the reciprocal of the divisor: The reciprocal of 8/15 is 15/8. Remember, flipping the divisor is the key to converting the division problem into a multiplication problem.
  3. Multiply the dividend by the reciprocal: Multiply 26/25 by 15/8: (26/25) * (15/8) = (26 * 15) / (25 * 8). This step involves multiplying the numerators together and the denominators together, as we do in any fraction multiplication problem.
  4. Calculate the products: 26 * 15 = 390 and 25 * 8 = 200. So, the fraction becomes 390/200. This fraction represents the result of the multiplication, but it's important to simplify it to its lowest terms for the most concise answer.
  5. Simplify the resulting fraction: The fraction 390/200 needs to be simplified. To do this, we find the greatest common factor (GCF) of 390 and 200. The GCF is the largest number that divides both 390 and 200 without leaving a remainder. In this case, the GCF is 10. Dividing both the numerator and the denominator by 10, we get: 390 ÷ 10 = 39 and 200 ÷ 10 = 20. Therefore, the simplified fraction is 39/20.
  6. Convert the improper fraction to a mixed number (optional): Since the resulting fraction is an improper fraction (numerator is greater than the denominator), it can be converted to a mixed number. To do this, we divide the numerator (39) by the denominator (20). 39 ÷ 20 = 1 with a remainder of 19. The quotient (1) becomes the whole number part of the mixed number, the remainder (19) becomes the numerator, and we keep the same denominator (20). Therefore, 39/20 is equivalent to 1 19/20.

Thus, 26/25 ÷ 8/15 = 39/20 or 1 19/20. This solution demonstrates the complete process of dividing fractions, including simplification and conversion to a mixed number when appropriate. By practicing these steps, you can improve your ability to solve fraction division problems accurately and efficiently.

(e) 3 1/9 ÷ 20/18

This example involves dividing a mixed number by a fraction, similar to example (c). The key is to first convert the mixed number into an improper fraction before proceeding with the division steps. Let's walk through the solution.

  1. Convert the mixed number to an improper fraction: We begin by converting the mixed number 3 1/9 into an improper fraction. Multiply the whole number (3) by the denominator (9) and add the numerator (1): 3 * 9 + 1 = 27 + 1 = 28. Keep the same denominator (9). Thus, 3 1/9 is equivalent to 28/9.
  2. Identify the fractions: Now we have the division problem as 28/9 ÷ 20/18. Dividend = 28/9, Divisor = 20/18. With the mixed number converted, we can proceed with fraction division.
  3. Find the reciprocal of the divisor: The reciprocal of 20/18 is 18/20. Remember, to find the reciprocal, we simply flip the fraction, swapping the numerator and denominator.
  4. Multiply the dividend by the reciprocal: Multiply 28/9 by 18/20: (28/9) * (18/20) = (28 * 18) / (9 * 20). This involves multiplying the numerators together and the denominators together.
  5. Calculate the products: 28 * 18 = 504 and 9 * 20 = 180. So, the fraction becomes 504/180. This fraction is the result of the multiplication, but simplification is necessary for the most concise answer.
  6. Simplify the resulting fraction: The fraction 504/180 can be simplified. We need to find the greatest common factor (GCF) of 504 and 180. The GCF is the largest number that divides both 504 and 180 without leaving a remainder. In this case, the GCF is 36. Dividing both the numerator and the denominator by 36, we get: 504 ÷ 36 = 14 and 180 ÷ 36 = 5. Therefore, the simplified fraction is 14/5.
  7. Convert the improper fraction to a mixed number (optional): Since the result is an improper fraction, we can convert it to a mixed number. Divide the numerator (14) by the denominator (5): 14 ÷ 5 = 2 with a remainder of 4. The quotient (2) is the whole number part, the remainder (4) is the numerator, and the denominator (5) stays the same. Therefore, 14/5 is equivalent to 2 4/5.

Thus, 3 1/9 ÷ 20/18 = 14/5 or 2 4/5. This example reinforces the process of dividing a mixed number by a fraction, emphasizing the importance of converting to improper fractions and simplifying the result. By practicing these steps, you can confidently tackle similar problems.

(f) 5 7/4 ÷ 3/16

This example presents a unique challenge because the mixed number 5 7/4 has a fraction part (7/4) that is itself an improper fraction. Before converting the mixed number to an improper fraction, we need to simplify it. This highlights the importance of simplifying mixed numbers before performing any operations. Let's break down the solution step-by-step.

  1. Simplify the mixed number: The fraction part of the mixed number, 7/4, is an improper fraction. We can convert it to a mixed number: 7 ÷ 4 = 1 with a remainder of 3. So, 7/4 is equivalent to 1 3/4. Now, we can rewrite the original mixed number as 5 + 1 3/4, which equals 6 3/4. Simplifying the mixed number first makes the subsequent steps easier.
  2. Convert the mixed number to an improper fraction: Now that we have the simplified mixed number 6 3/4, we can convert it to an improper fraction. Multiply the whole number (6) by the denominator (4) and add the numerator (3): 6 * 4 + 3 = 24 + 3 = 27. Keep the same denominator (4). Thus, 6 3/4 is equivalent to 27/4.
  3. Identify the fractions: The division problem is now 27/4 ÷ 3/16. Dividend = 27/4, Divisor = 3/16. With the mixed number converted, we can proceed with fraction division.
  4. Find the reciprocal of the divisor: The reciprocal of 3/16 is 16/3. Remember, flipping the divisor transforms the division problem into a multiplication problem.
  5. Multiply the dividend by the reciprocal: Multiply 27/4 by 16/3: (27/4) * (16/3) = (27 * 16) / (4 * 3). This involves multiplying the numerators together and the denominators together.
  6. Calculate the products: 27 * 16 = 432 and 4 * 3 = 12. So, the fraction becomes 432/12. This fraction represents the result of the multiplication, but simplification is crucial for the most concise answer.
  7. Simplify the resulting fraction: The fraction 432/12 can be simplified. We need to find the greatest common factor (GCF) of 432 and 12. In this case, 12 divides evenly into 432, so the GCF is 12. Dividing both the numerator and the denominator by 12, we get: 432 ÷ 12 = 36 and 12 ÷ 12 = 1. Therefore, the simplified fraction is 36/1, which is equal to 36.

Thus, 5 7/4 ÷ 3/16 = 36. This example highlights the importance of simplifying mixed numbers, especially when the fraction part is an improper fraction, before proceeding with division. By mastering this technique, you can avoid unnecessary complications and solve problems more efficiently.

(g) 4 5/9 ÷ 2 4/27

This example involves dividing two mixed numbers. To solve this type of problem, we need to convert both mixed numbers into improper fractions before performing the division. This is a crucial step in simplifying the process and ensuring an accurate result. Let's break down the solution step-by-step.

  1. Convert the first mixed number to an improper fraction: We begin by converting 4 5/9 into an improper fraction. Multiply the whole number (4) by the denominator (9) and add the numerator (5): 4 * 9 + 5 = 36 + 5 = 41. Keep the same denominator (9). Thus, 4 5/9 is equivalent to 41/9.
  2. Convert the second mixed number to an improper fraction: Next, we convert 2 4/27 into an improper fraction. Multiply the whole number (2) by the denominator (27) and add the numerator (4): 2 * 27 + 4 = 54 + 4 = 58. Keep the same denominator (27). Thus, 2 4/27 is equivalent to 58/27.
  3. Identify the fractions: The division problem is now 41/9 ÷ 58/27. Dividend = 41/9, Divisor = 58/27. With both mixed numbers converted to improper fractions, we can proceed with fraction division.
  4. Find the reciprocal of the divisor: The reciprocal of 58/27 is 27/58. Remember, flipping the divisor transforms the division problem into a multiplication problem.
  5. Multiply the dividend by the reciprocal: Multiply 41/9 by 27/58: (41/9) * (27/58) = (41 * 27) / (9 * 58). This involves multiplying the numerators together and the denominators together.
  6. Calculate the products: 41 * 27 = 1107 and 9 * 58 = 522. So, the fraction becomes 1107/522. This fraction represents the result of the multiplication, but simplification is essential for the most concise answer.
  7. Simplify the resulting fraction: The fraction 1107/522 can be simplified. To do this, we find the greatest common factor (GCF) of 1107 and 522. In this case, the GCF is 9. Dividing both the numerator and the denominator by 9, we get: 1107 ÷ 9 = 123 and 522 ÷ 9 = 58. Therefore, the simplified fraction is 123/58.
  8. Convert the improper fraction to a mixed number (optional): Since the result is an improper fraction, we can convert it to a mixed number. Divide the numerator (123) by the denominator (58): 123 ÷ 58 = 2 with a remainder of 7. The quotient (2) is the whole number part, the remainder (7) is the numerator, and the denominator (58) stays the same. Therefore, 123/58 is equivalent to 2 7/58.

Thus, 4 5/9 ÷ 2 4/27 = 123/58 or 2 7/58. This example demonstrates the complete process of dividing two mixed numbers, emphasizing the importance of converting to improper fractions and simplifying the result. By practicing these steps, you can confidently tackle similar problems.

(h) 7 1/3 ÷ 4/9

This example focuses on dividing a mixed number by a fraction, a scenario we've encountered before. The key is to first convert the mixed number to an improper fraction before applying the division rules. This ensures we're working with fractions in their proper form for multiplication and division. Let's walk through the solution step by step.

  1. Convert the mixed number to an improper fraction: We begin by converting the mixed number 7 1/3 into an improper fraction. To do this, we multiply the whole number (7) by the denominator (3) and add the numerator (1): 7 * 3 + 1 = 21 + 1 = 22. We keep the same denominator (3). Therefore, 7 1/3 is equivalent to 22/3.
  2. Identify the fractions: Now we have the division problem as 22/3 ÷ 4/9. Dividend = 22/3, Divisor = 4/9. With the mixed number successfully converted to an improper fraction, we can proceed with the standard fraction division steps.
  3. Find the reciprocal of the divisor: The reciprocal of 4/9 is 9/4. Remember, to find the reciprocal, we simply flip the fraction, swapping the numerator and the denominator. This step is crucial for transforming the division problem into a multiplication problem, which is often easier to solve.
  4. Multiply the dividend by the reciprocal: Multiply 22/3 by 9/4: (22/3) * (9/4) = (22 * 9) / (3 * 4). This involves multiplying the numerators together and the denominators together, as we do in any fraction multiplication problem.
  5. Calculate the products: 22 * 9 = 198 and 3 * 4 = 12. So, the fraction becomes 198/12. This fraction represents the result of the multiplication, but it's important to simplify it to its lowest terms for the most concise and understandable answer.
  6. Simplify the resulting fraction: The fraction 198/12 needs to be simplified. To do this, we find the greatest common factor (GCF) of 198 and 12. The GCF is the largest number that divides both 198 and 12 without leaving a remainder. In this case, the GCF is 6. Dividing both the numerator and the denominator by 6, we get: 198 ÷ 6 = 33 and 12 ÷ 6 = 2. Therefore, the simplified fraction is 33/2.
  7. Convert the improper fraction to a mixed number (optional): Since the original problem included a mixed number, it's often helpful to convert the improper fraction back to a mixed number. To do this, we divide the numerator (33) by the denominator (2). 33 ÷ 2 = 16 with a remainder of 1. The quotient (16) becomes the whole number part of the mixed number, the remainder (1) becomes the numerator, and we keep the same denominator (2). Therefore, 33/2 is equivalent to 16 1/2.

Thus, 7 1/3 ÷ 4/9 = 33/2 or 16 1/2. This example demonstrates the complete process of dividing a mixed number by a fraction, including the essential step of converting the mixed number to an improper fraction before proceeding with the division. By mastering these steps, you can confidently solve a wide range of fraction division problems involving mixed numbers.

(i) 48/35 ÷ 16/25

This example provides another opportunity to practice the fundamental steps of dividing fractions and simplifying the result. By working through this problem, you can further solidify your understanding of the process and improve your calculation skills. Let's break down the solution step by step.

  1. Identify the fractions: Dividend = 48/35, Divisor = 16/25. The first step in any fraction division problem is to clearly identify the dividend (the fraction being divided) and the divisor (the fraction you're dividing by). This helps to avoid confusion and ensures you're applying the correct operations.
  2. Find the reciprocal of the divisor: The reciprocal of 16/25 is 25/16. Remember, to find the reciprocal of a fraction, you simply flip it, swapping the numerator and the denominator. This step is crucial because dividing by a fraction is the same as multiplying by its reciprocal. This transformation allows us to use the familiar rules of fraction multiplication.
  3. Multiply the dividend by the reciprocal: Multiply 48/35 by 25/16: (48/35) * (25/16) = (48 * 25) / (35 * 16). This step involves multiplying the numerators together and the denominators together, as we do in any fraction multiplication problem. It's important to perform this multiplication accurately to arrive at the correct result.
  4. Calculate the products: 48 * 25 = 1200 and 35 * 16 = 560. So, the fraction becomes 1200/560. This fraction represents the result of the multiplication, but it's not yet in its simplest form. Simplifying the fraction is the final step to ensure we have the most concise and understandable answer.
  5. Simplify the resulting fraction: The fraction 1200/560 can be simplified. To do this, we need to find the greatest common factor (GCF) of 1200 and 560. The GCF is the largest number that divides both 1200 and 560 without leaving a remainder. In this case, the GCF is 80. Dividing both the numerator and the denominator by 80, we get: 1200 ÷ 80 = 15 and 560 ÷ 80 = 7. Therefore, the simplified fraction is 15/7.
  6. Convert the improper fraction to a mixed number (optional): Since the resulting fraction, 15/7, is an improper fraction (the numerator is greater than the denominator), it can be converted to a mixed number. This is often preferred for clarity, especially in real-world applications. To convert 15/7 to a mixed number, we divide the numerator (15) by the denominator (7): 15 ÷ 7 = 2 with a remainder of 1. The quotient (2) becomes the whole number part of the mixed number, the remainder (1) becomes the numerator, and we keep the same denominator (7). Therefore, 15/7 is equivalent to 2 1/7.

Thus, 48/35 ÷ 16/25 = 15/7 or 2 1/7. This solution demonstrates the complete process of dividing fractions, from identifying the fractions and finding the reciprocal to multiplying and simplifying the result. By consistently following these steps, you can confidently solve a wide range of fraction division problems. Remember to always check your work and simplify your answer to ensure accuracy.

(j) 72/39 ÷ 18/13

This final example reinforces the process of dividing fractions and provides an opportunity to practice simplifying the result. By working through this problem, you can further strengthen your understanding of the concepts and techniques involved. Let's break down the solution step by step.

  1. Identify the fractions: Dividend = 72/39, Divisor = 18/13. The first step in dividing fractions is to clearly identify the dividend and the divisor. This helps to ensure that you're applying the correct procedure and avoids any potential confusion.
  2. Find the reciprocal of the divisor: The reciprocal of 18/13 is 13/18. Remember, the reciprocal of a fraction is obtained by flipping the fraction, swapping the numerator and the denominator. This step is essential because dividing by a fraction is the same as multiplying by its reciprocal, which simplifies the calculation.
  3. Multiply the dividend by the reciprocal: Multiply 72/39 by 13/18: (72/39) * (13/18) = (72 * 13) / (39 * 18). This step involves multiplying the numerators together and the denominators together, as we do in any fraction multiplication problem. Accurate multiplication is crucial for arriving at the correct answer.
  4. Calculate the products: 72 * 13 = 936 and 39 * 18 = 702. So, the fraction becomes 936/702. This fraction represents the result of the multiplication, but it's important to simplify it to its lowest terms for the most concise and understandable answer. Simplifying fractions makes them easier to work with and compare.
  5. Simplify the resulting fraction: The fraction 936/702 needs to be simplified. To do this, we find the greatest common factor (GCF) of 936 and 702. The GCF is the largest number that divides both 936 and 702 without leaving a remainder. In this case, the GCF is 234. Dividing both the numerator and the denominator by 234, we get: 936 ÷ 234 = 4 and 702 ÷ 234 = 3. Therefore, the simplified fraction is 4/3.
  6. Convert the improper fraction to a mixed number (optional): Since the resulting fraction, 4/3, is an improper fraction (the numerator is greater than the denominator), it can be converted to a mixed number. This is often preferred for clarity, especially in practical applications. To convert 4/3 to a mixed number, we divide the numerator (4) by the denominator (3): 4 ÷ 3 = 1 with a remainder of 1. The quotient (1) becomes the whole number part of the mixed number, the remainder (1) becomes the numerator, and we keep the same denominator (3). Therefore, 4/3 is equivalent to 1 1/3.

Thus, 72/39 ÷ 18/13 = 4/3 or 1 1/3. This final example demonstrates the complete process of dividing fractions, from identifying the fractions and finding the reciprocal to multiplying and simplifying the result. By mastering these steps and practicing consistently, you can develop a strong understanding of fraction division and confidently solve a variety of problems.

Conclusion

Dividing fractions is a fundamental skill in mathematics with wide-ranging applications. By understanding the concept of reciprocals and following the steps outlined in this guide, you can confidently tackle any fraction division problem. Remember to always simplify your answers to their lowest terms and, when dealing with mixed numbers, convert them to improper fractions before dividing. With practice, you'll master this essential skill and unlock new possibilities in your mathematical journey. This guide has provided a comprehensive overview of dividing fractions, equipping you with the knowledge and tools necessary to succeed. Whether you're a student learning the basics or someone looking to refresh your skills, this information will serve as a valuable resource. Keep practicing, and you'll become proficient in dividing fractions in no time!