Calculating 1 1/3 Divided By 1 1/9 A Step-by-Step Guide

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Introduction

In the realm of mathematics, mastering the division of mixed fractions is a fundamental skill that unlocks a deeper understanding of numerical relationships. This comprehensive guide delves into the intricacies of dividing mixed fractions, using the specific example of 1 1/3 ÷ 1 1/9 to illustrate the step-by-step process. By the end of this exploration, you will not only be able to solve this particular problem but also possess the knowledge and confidence to tackle any division problem involving mixed fractions.

Understanding Mixed Fractions

Before diving into the division process, it's crucial to grasp the concept of mixed fractions. A mixed fraction is a combination of a whole number and a proper fraction, such as 1 1/3 or 2 1/2. The whole number represents the number of complete units, while the fraction represents a part of a unit. In the mixed fraction 1 1/3, the '1' represents one whole unit, and the '1/3' represents one-third of another unit. Understanding this composition is the first step towards confidently manipulating mixed fractions in mathematical operations.

Converting Mixed Fractions to Improper Fractions

The key to dividing mixed fractions lies in converting them into improper fractions. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number), such as 4/3 or 5/2. To convert a mixed fraction to an improper fraction, follow these steps:

  1. Multiply the whole number by the denominator of the fractional part.
  2. Add the numerator of the fractional part to the result from step 1.
  3. Write the sum from step 2 as the numerator of the improper fraction.
  4. Keep the same denominator as the original fractional part.

Let's apply this to our example, 1 1/3:

  1. Multiply the whole number (1) by the denominator (3): 1 * 3 = 3
  2. Add the numerator (1) to the result: 3 + 1 = 4
  3. Write the sum (4) as the numerator, keeping the denominator (3): 4/3

Therefore, the improper fraction equivalent of 1 1/3 is 4/3. Now, let's convert the other mixed fraction, 1 1/9, to an improper fraction:

  1. Multiply the whole number (1) by the denominator (9): 1 * 9 = 9
  2. Add the numerator (1) to the result: 9 + 1 = 10
  3. Write the sum (10) as the numerator, keeping the denominator (9): 10/9

So, 1 1/9 is equivalent to the improper fraction 10/9. With both mixed fractions converted to improper fractions, we are ready to perform the division.

Dividing Fractions: The Invert and Multiply Rule

The division of fractions is based on a simple yet powerful principle: "invert and multiply." To divide one fraction by another, we invert the second fraction (the divisor) and then multiply the first fraction (the dividend) by the inverted fraction. Inverting a fraction means swapping its numerator and denominator. For example, the inverse of 2/3 is 3/2, and the inverse of 5/4 is 4/5.

Applying this rule to our problem, 4/3 ÷ 10/9, we first invert the second fraction, 10/9, which becomes 9/10. Then, we multiply the first fraction, 4/3, by the inverted fraction, 9/10:

(4/3) * (9/10)

Multiplying Fractions: A Straightforward Process

Multiplying fractions is a straightforward process. We simply multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. In our case:

  • Numerator: 4 * 9 = 36
  • Denominator: 3 * 10 = 30

This gives us the fraction 36/30. While this is a correct answer, it's not in its simplest form. We need to simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator and dividing both by it.

Simplifying Fractions: Finding the Greatest Common Factor

The greatest common factor (GCF) of two numbers is the largest number that divides both of them without leaving a remainder. To find the GCF of 36 and 30, we can list their factors:

  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

The largest number that appears in both lists is 6, so the GCF of 36 and 30 is 6. Now, we divide both the numerator and the denominator of 36/30 by 6:

  • 36 ÷ 6 = 6
  • 30 ÷ 6 = 5

This simplifies the fraction to 6/5. This is an improper fraction, which can be converted back to a mixed fraction for a more intuitive understanding of the result.

Converting Improper Fractions to Mixed Fractions

To convert the improper fraction 6/5 back to a mixed fraction, we divide the numerator (6) by the denominator (5). The quotient (the result of the division) becomes the whole number part of the mixed fraction, and the remainder becomes the numerator of the fractional part, keeping the same denominator.

  • 6 ÷ 5 = 1 with a remainder of 1

Therefore, the mixed fraction equivalent of 6/5 is 1 1/5. This means that 1 1/3 divided by 1 1/9 equals 1 1/5.

A Step-by-Step Recap

Let's summarize the steps involved in dividing mixed fractions using our example, 1 1/3 ÷ 1 1/9:

  1. Convert mixed fractions to improper fractions:
    • 1 1/3 = (1 * 3 + 1) / 3 = 4/3
    • 1 1/9 = (1 * 9 + 1) / 9 = 10/9
  2. Invert the second fraction (the divisor):
    • 10/9 becomes 9/10
  3. Multiply the first fraction by the inverted fraction:
    • (4/3) * (9/10) = 36/30
  4. Simplify the resulting fraction:
    • GCF of 36 and 30 is 6
    • 36/30 = (36 ÷ 6) / (30 ÷ 6) = 6/5
  5. Convert the improper fraction back to a mixed fraction (if desired):
    • 6/5 = 1 1/5

Practical Applications and Real-World Examples

The ability to divide mixed fractions is not just a theoretical exercise; it has practical applications in various real-world scenarios. Here are a few examples:

  1. Cooking and Baking: Recipes often call for fractional amounts of ingredients. If you need to halve a recipe that uses mixed fractions, you'll need to divide mixed fractions. For instance, if a recipe calls for 2 1/2 cups of flour and you want to make half the recipe, you would divide 2 1/2 by 2.
  2. Construction and Measurement: In construction, measurements often involve mixed fractions. If you need to divide a length of wood that is 5 3/4 feet long into three equal pieces, you would divide 5 3/4 by 3.
  3. Time Management: If you have 3 1/4 hours to complete four tasks, you can divide 3 1/4 by 4 to determine how much time you can spend on each task.
  4. Sharing and Distribution: If you have 10 1/2 pizzas to share among 7 people, you would divide 10 1/2 by 7 to find out how much pizza each person gets.

These examples demonstrate that dividing mixed fractions is a valuable skill that can be applied in everyday situations, making it an essential part of mathematical literacy.

Tips and Tricks for Mastering Mixed Fraction Division

To further solidify your understanding and improve your proficiency in dividing mixed fractions, consider the following tips and tricks:

  1. Practice Regularly: The key to mastering any mathematical skill is consistent practice. Work through a variety of problems involving different mixed fractions to build your confidence and speed.
  2. Use Visual Aids: Visual aids, such as diagrams or number lines, can help you visualize the process of dividing mixed fractions, especially when you are first learning the concept.
  3. Break Down Problems: If you encounter a complex problem, break it down into smaller, more manageable steps. This can make the problem less daunting and easier to solve.
  4. Check Your Work: Always check your work to ensure that you have performed the calculations correctly. This can help you identify and correct any errors.
  5. Seek Help When Needed: If you are struggling with dividing mixed fractions, don't hesitate to seek help from a teacher, tutor, or online resources. There are many resources available to support your learning.
  6. Understand the "Why": Focus on understanding the underlying principles of fraction division rather than just memorizing the steps. This will allow you to apply the concept to a wider range of problems.

Conclusion

Dividing mixed fractions might seem challenging at first, but with a clear understanding of the steps involved and consistent practice, it becomes a manageable and even enjoyable mathematical skill. By converting mixed fractions to improper fractions, applying the "invert and multiply" rule, simplifying fractions, and converting back to mixed fractions when necessary, you can confidently solve any division problem involving mixed fractions. Remember to practice regularly, use visual aids when needed, and seek help when you encounter difficulties. Mastering this skill will not only enhance your mathematical abilities but also equip you with a valuable tool for solving real-world problems. So, embrace the challenge, delve into the world of fractions, and unlock the power of mathematical understanding.