Dividing Fractions Simplifying 8/7 Divided By 10/9

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In the realm of mathematics, understanding fractions and their operations is fundamental. Fractions represent parts of a whole, and mastering their manipulation is crucial for various mathematical concepts. Among these operations, division of fractions often poses a challenge for learners. This article aims to provide a comprehensive guide on how to perform the operation and simplify the answer fully for the expression 8/7 ÷ 10/9. We will delve into the underlying principles, step-by-step methods, and practical examples to ensure a clear understanding of this essential skill.

Before we dive into the division of fractions, let's recap the basics. A fraction consists of two parts: the numerator and the denominator. The numerator represents the number of parts we have, while the denominator represents the total number of parts that make up the whole. For instance, in the fraction 8/7, 8 is the numerator, and 7 is the denominator. Understanding the roles of these components is vital for grasping fraction operations.

Dividing fractions might seem daunting at first, but it's actually quite straightforward once you understand the underlying concept. When dividing fractions, we are essentially asking how many times one fraction fits into another. To perform this operation, we use a simple yet powerful trick: we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, the reciprocal of 10/9 is 9/10. This method allows us to convert a division problem into a multiplication problem, which is generally easier to handle.

Let's apply this concept to our specific problem: 8/7 ÷ 10/9. We'll break down the process into manageable steps to ensure clarity and accuracy.

Step 1: Identify the Fractions

The first step is to clearly identify the two fractions involved in the division. In this case, we have 8/7 as the dividend (the fraction being divided) and 10/9 as the divisor (the fraction we are dividing by).

Step 2: Find the Reciprocal of the Divisor

The next step is to find the reciprocal of the divisor, which is 10/9. To do this, we simply swap the numerator and the denominator. Thus, the reciprocal of 10/9 is 9/10. This reciprocal will be crucial in the next step, as it transforms the division problem into a multiplication problem.

Step 3: Multiply the Dividend by the Reciprocal of the Divisor

Now that we have the reciprocal, we can rewrite the division problem as a multiplication problem. We multiply the dividend (8/7) by the reciprocal of the divisor (9/10). This gives us:

(8/7) * (9/10)

To multiply fractions, we multiply the numerators together and the denominators together:

(8 * 9) / (7 * 10) = 72/70

So, the result of multiplying the fractions is 72/70.

Step 4: Simplify the Resulting Fraction

The final step is to simplify the resulting fraction, 72/70. Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by the GCD. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the Greatest Common Divisor (GCD)

There are several methods to find the GCD, but one common method is listing the factors of both numbers and identifying the largest factor they have in common. Let's list the factors of 72 and 70:

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70

By comparing the factors, we can see that the greatest common factor of 72 and 70 is 2.

Dividing by the GCD

Now that we have the GCD, we divide both the numerator and the denominator by 2:

72 ÷ 2 = 36

70 ÷ 2 = 35

So, the simplified fraction is 36/35.

Step 5: Convert to a Mixed Number (if necessary)

In some cases, it may be necessary to convert an improper fraction (where the numerator is greater than the denominator) to a mixed number. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). Since 36/35 is an improper fraction, we can convert it to a mixed number.

To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator remains the same.

36 ÷ 35 = 1 with a remainder of 1

So, the mixed number is 1 1/35. This is the simplest form of the fraction.

To recap, here are the steps we followed to divide 8/7 by 10/9 and simplify the answer:

  1. Identify the fractions: 8/7 and 10/9.
  2. Find the reciprocal of the divisor (10/9): 9/10.
  3. Multiply the dividend (8/7) by the reciprocal (9/10): (8/7) * (9/10) = 72/70.
  4. Simplify the resulting fraction (72/70) by dividing both numerator and denominator by their GCD (2): 36/35.
  5. Convert the improper fraction (36/35) to a mixed number: 1 1/35.

To further solidify your understanding, let's look at a few more examples of dividing fractions.

Example 1: 3/4 ÷ 2/5

  1. Identify the fractions: 3/4 and 2/5.
  2. Find the reciprocal of the divisor (2/5): 5/2.
  3. Multiply the dividend (3/4) by the reciprocal (5/2): (3/4) * (5/2) = 15/8.
  4. Simplify the resulting fraction (15/8): The GCD of 15 and 8 is 1, so the fraction is already in its simplest form.
  5. Convert the improper fraction (15/8) to a mixed number: 15 ÷ 8 = 1 with a remainder of 7, so the mixed number is 1 7/8.

Example 2: 5/6 ÷ 1/3

  1. Identify the fractions: 5/6 and 1/3.
  2. Find the reciprocal of the divisor (1/3): 3/1.
  3. Multiply the dividend (5/6) by the reciprocal (3/1): (5/6) * (3/1) = 15/6.
  4. Simplify the resulting fraction (15/6) by dividing both numerator and denominator by their GCD (3): 5/2.
  5. Convert the improper fraction (5/2) to a mixed number: 5 ÷ 2 = 2 with a remainder of 1, so the mixed number is 2 1/2.

When dividing fractions, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate calculations.

Mistake 1: Forgetting to Find the Reciprocal

The most common mistake is forgetting to find the reciprocal of the divisor before multiplying. Remember, dividing by a fraction is the same as multiplying by its reciprocal. Always swap the numerator and denominator of the second fraction before multiplying.

Mistake 2: Multiplying Numerators and Denominators Incorrectly

When multiplying fractions, multiply the numerators together and the denominators together. Avoid the mistake of adding or subtracting the numerators or denominators.

Mistake 3: Not Simplifying the Resulting Fraction

Always simplify the resulting fraction to its lowest terms. This means dividing both the numerator and the denominator by their greatest common divisor. If the result is an improper fraction, convert it to a mixed number.

Mistake 4: Confusing Dividend and Divisor

Ensure you correctly identify the dividend (the fraction being divided) and the divisor (the fraction you are dividing by). The order matters, so be careful to swap the correct fraction when finding the reciprocal.

Dividing fractions is a fundamental skill in mathematics that can be mastered with practice and understanding. By following the step-by-step guide outlined in this article, you can confidently perform the operation and simplify the answer fully. Remember to find the reciprocal of the divisor, multiply the fractions, simplify the result, and convert to a mixed number if necessary. With consistent practice and attention to detail, you will excel in dividing fractions and build a strong foundation for more advanced mathematical concepts. The expression 8/7 ÷ 10/9 serves as an excellent example to illustrate this process, leading to the simplified result of 1 1/35. Keep practicing, and you'll find that dividing fractions becomes second nature. Understanding fractions operations are crucial, and by mastering them, you enhance your mathematical prowess.