How To Evaluate The Expression 16 Divided By 12/5
Introduction
In the realm of mathematics, mastering the fundamental operations is crucial for building a strong foundation. Division, particularly when involving fractions, can sometimes seem daunting. However, with a clear understanding of the underlying principles and a step-by-step approach, even complex expressions can be easily evaluated. This article delves into the process of evaluating the expression 16 ÷ (12/5), providing a comprehensive guide that will empower you to confidently tackle similar problems. We will break down the steps involved, explain the reasoning behind each action, and illustrate the concepts with clear examples. By the end of this guide, you will not only be able to solve this specific problem but also gain a deeper understanding of division with fractions, enabling you to apply these skills to a wide range of mathematical scenarios.
Understanding Division with Fractions
Before we dive into the specifics of the problem, let's refresh our understanding of division with fractions. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 2/3 is 3/2. This fundamental concept is the key to solving expressions involving division with fractions. When we encounter an expression like a ÷ (b/c), we can rewrite it as a × (c/b). This transformation simplifies the problem, allowing us to perform multiplication instead of division. This principle is rooted in the inverse relationship between multiplication and division. Understanding this relationship is crucial for manipulating fractions and solving mathematical problems effectively. The ability to convert division into multiplication by using the reciprocal is a cornerstone of fraction arithmetic. This technique not only simplifies calculations but also provides a deeper insight into the nature of fractions and their operations.
Step-by-Step Evaluation of 16 ÷ (12/5)
Now, let's apply this concept to our specific problem: 16 ÷ (12/5). We will break down the evaluation into a series of clear, manageable steps.
Step 1: Rewrite the Division as Multiplication
The first step is to rewrite the division as multiplication by the reciprocal of the fraction. The reciprocal of 12/5 is 5/12. Therefore, we can rewrite the expression as:
16 ÷ (12/5) = 16 × (5/12)
This transformation is the cornerstone of our approach. By converting division into multiplication, we simplify the problem and make it easier to solve. This step leverages the fundamental principle of fraction division, allowing us to proceed with confidence. The ability to recognize and apply this transformation is a key skill in fraction arithmetic. It allows us to manipulate expressions and solve problems more efficiently.
Step 2: Express 16 as a Fraction
To multiply a whole number by a fraction, it's helpful to express the whole number as a fraction. We can write 16 as 16/1. This doesn't change the value of the number but makes the multiplication process clearer.
16 × (5/12) = (16/1) × (5/12)
Expressing whole numbers as fractions is a common technique in fraction arithmetic. It allows us to treat all numbers as fractions, making multiplication and division operations more consistent. This step highlights the importance of representing numbers in different forms to facilitate calculations. The ability to convert between whole numbers and fractions is a fundamental skill in mathematics.
Step 3: Multiply the Fractions
To multiply fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately.
(16/1) × (5/12) = (16 × 5) / (1 × 12)
Performing the multiplication:
(16 × 5) = 80 (1 × 12) = 12
So, the expression becomes:
80/12
Multiplying fractions involves a straightforward process of multiplying the numerators and the denominators. This step emphasizes the mechanical aspect of fraction multiplication. The ability to perform this multiplication accurately is essential for solving fraction-related problems. Understanding the process of multiplying fractions is a cornerstone of fraction arithmetic.
Step 4: Simplify the Fraction
The fraction 80/12 can be simplified. Both 80 and 12 are divisible by 4. Dividing both the numerator and the denominator by 4, we get:
(80 ÷ 4) / (12 ÷ 4) = 20/3
The fraction 20/3 is an improper fraction, meaning the numerator is greater than the denominator. We can convert it to a mixed number.
Simplifying fractions is a crucial step in mathematical problem-solving. It allows us to express the answer in its simplest form, making it easier to understand and interpret. This step highlights the importance of reducing fractions to their lowest terms. The ability to simplify fractions is a fundamental skill in mathematics.
Step 5: Convert to a Mixed Number (Optional)
To convert 20/3 to a mixed number, we divide 20 by 3.
20 ÷ 3 = 6 with a remainder of 2
This means that 20/3 is equal to 6 whole numbers and 2/3. So, the mixed number is:
6 2/3
Converting improper fractions to mixed numbers is often preferred for clarity and ease of interpretation. This step emphasizes the importance of expressing fractions in different forms to suit the context of the problem. The ability to convert between improper fractions and mixed numbers is a valuable skill in mathematics.
Conclusion
In conclusion, evaluating the expression 16 ÷ (12/5) involves several key steps: rewriting the division as multiplication by the reciprocal, expressing whole numbers as fractions, multiplying the fractions, and simplifying the result. By following these steps, we arrive at the solution 20/3, which can also be expressed as the mixed number 6 2/3. This process not only provides the answer to the specific problem but also reinforces the fundamental principles of division with fractions. Mastering these principles will equip you with the skills to confidently tackle a wide range of mathematical challenges. Remember, practice is key to solidifying your understanding. Work through various examples and exercises to further develop your proficiency in fraction arithmetic. The more you practice, the more comfortable and confident you will become in solving these types of problems. This comprehensive guide has provided you with the tools and knowledge necessary to excel in fraction arithmetic. Now, it's time to put your skills to the test and continue your journey of mathematical discovery.
Practice Problems
To further solidify your understanding, try solving these practice problems:
- 24 ÷ (8/3)
- 15 ÷ (5/2)
- 32 ÷ (16/7)
- 10 ÷ (2/5)
- 18 ÷ (9/4)
Further Resources
For additional learning and practice, explore these resources:
- Khan Academy: Fractions
- Mathway: Fraction Calculator
- Purplemath: Dividing Fractions
By utilizing these resources and consistently practicing, you can master the art of dividing fractions and excel in mathematics.